Research ArticleDynamic Scalable Stochastic Petri NetA Novel Model for Designing and Analysis ofResource Scheduling in Cloud Computing

Hua He1 Shanchen Pang2 and Zenghua Zhao1

1School of Computer Science and Technology Tianjin University Tianjin 300072 China2College of Computer and Communication Engineering China University of Petroleum Qingdao 266580 China

Correspondence should be addressed to Shanchen Pang shanchenpangsohucom

Received 19 January 2016 Revised 21 June 2016 Accepted 4 July 2016

Academic Editor Fabrizio Messina

Copyright copy 2016 Hua He et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Performance evaluation of cloud computing systems studies the relationships among system configuration system load and per-formance indicators However such evaluation is not feasible by dint of measurement methods or simulation methods due tothe properties of cloud computing such as large scale diversity and dynamics To overcome those challenges we present a novelDynamic Scalable Stochastic Petri Net (DSSPN) tomodel and analyze the performance of cloud computing systems DSSPN can notonly clearly depict system dynamic behaviors in an intuitive and efficient way but also easily discover performance deficiencies andbottlenecks of systems In this study we further elaborate some properties of DSSPN In addition we improve fair scheduling takinginto consideration job diversity and resource heterogeneity To validate the improved algorithm and the applicability of DSSPNwe conduct extensive experiments through Stochastic Petri Net Package (SPNP) The performance results show that the improvedalgorithm is better than fair scheduling in some key performance indicators such as average throughput response time and averagecompletion time

1 Introduction

Cloud computing provides shared configurable resourcesto users as services with pay-as-you-go scheme [1] Theseservices that consisted of set of componentsmay be offered bydifferent providers [2] Tomeet the needs of customers cloudservice providers have to ensure that their profit and returnon investment are not rapidly decreased due to increasedcosts while maintaining a desirable level of the quality ofservice (QoS) of consumers such as execution time delaytime and budget restrictions [2ndash4] To address this problemmost researches of cloud computing have focused on per-formance improvement and satisfaction of the QoS require-ments and developed some efficient solutions But for nowlittle work has been done about finding a convenient methodof modeling analyzing and evaluating the performanceof scheduling algorithms or systems in cloud environment

without spending toomuch time on comparison and analysis[5 6]

Performance evaluation is important to cloud computingdevelopment It is primarily aimed at selecting schemes thatmeet the requirements of consumers finding out perfor-mance defects predicting performance of the designed sys-tems in future and discovering better ways to achieve optimalresource allocation In other words performance evaluationis important in selecting improving and designing systemsor scheduling algorithms in cloud computing environment[7] The methods of performance evaluation are approxi-mately divided into three types measurement method sim-ulation method and model method However measurementand simulation methods are only applicable to existing andrunning systems and might be time consuming In additionthe two methods are incapable of finding out performancebottlenecks and analyzing large-scale and complicated cloud

Hindawi Publishing CorporationScientific ProgrammingVolume 2016 Article ID 9259248 13 pageshttpdxdoiorg10115520169259248

2 Scientific Programming

computing systems In this study we only focus on the modelmethod

Model method is a kind of analysis method of perfor-mance evaluation by studying and describing the relation-ships among performance system and load based onmathe-matical theories To facilitate mathematical descriptions andcalculation it usually requires simplification of the systemmodel and making some rational assumptions about thestatus of the system Compared to the other two methodsmodel method is based on a mature theoretical foundationand can clearly describe the relationship among all factorswith a lower cost

Stochastic Petri Net (SPN) is a powerful tool of modelmethod and can be applied to graphic modeling and mathe-matical analysis of many systems and areas such as computerscience communication network andmultiprocessor system[8ndash13] It can not only easily describe the properties ofsystems that have concurrency and synchronization charac-teristics but also clearly depict dynamic behaviors of systemsin an intuitive and efficient way In this way it is easy todiscover performance deficiencies and bottlenecks by usingSPN in analysis

However SPN is still not entirely suitable for modelingand performance evaluation of cloud computing systems (i)cloud computing offers scalable infrastructures to consumerson demand by utilizing the virtualization technology SPN isnot capable of adjustingmodels dynamically when the infras-tructure changes [13] (ii) Different workloads submitted byusers which are simultaneously running on cloud clustersmight have different QoS requirements such as responsetime execution time and data traffic [14] However SPN isincapable of representing the diversity of cloud computing(iii) The configurable shared resources in cloud computingare usually heterogeneous and geographically distributed sousing SPN to build upmodels will increase the computationalcomplexity and result in state explosion Because of theproblems mentioned above SPN does not adequately modeland analyze the performance of cloud computing in manysituations

To overcome those challenges we propose a novelextended form of SPN which is called Dynamic ScalableStochastic Petri Net (DSSPN) to conveniently model andanalyze the performance of cloud computing systems Inorder to support dynamic changes in cloud computing threekinds of functions are introduced to enhance the dynamicsof arcs transitions and places in DSSPN In addition manycloud service patterns under the same state can be com-pressed into a simple one by using DSSPN Therefore cloudcomputing systems can be easily modeled and analyzed byusing DSSPN without changing the original graph structureConsumers can easily evaluate performances only by settingthe three functions of the DSSPN model without spendingtoo much time on programming According to the feature ofSPN system decomposition and model compression can beapplied to reduce the complexity of the state space in DSSPNmodels

The main contributions of this paper include (1) propos-ing Dynamic Scalable Stochastic Petri Net (DSSPN) and thenfurther demonstrating firing rules and some properties of

DSSPN (2) presenting classified fair scheduling (CFS) takinginto consideration job diversity and resource heterogene-ity which can improve throughput and response time (3)validating the proposed approach and algorithm where weconduct and evaluate DSSPN models of fair scheduling andCFS algorithms by using Stochastic Petri Net Package (SPNP)[15 16] analysis and simulation

The remainder of the paper is organized as followsSection 2 describes the related works of this study Section 3specifies the novel analytical model called DSSPN and elab-orates its dynamics as well as some other properties In Sec-tion 4 we construct a DSSPN model of resource schedulingof fair scheduler and propose the classified fair scheduling(CFS) algorithm taking into consideration the workload andresources diversity In addition in order to alleviate the prob-lem of state explosion we adopt the multiuser multiservermodel [17] and analyze some parameters by using equivalentMarkov model to refine our original models in Section 4Section 5 demonstrates the experimental parameters setupand evaluates the system performance of the two schedulingalgorithms by using SPNP Finally conclusions and futureworks are given in Section 6

2 Related Works

Performance evaluation mainly focuses on relationshipsamong system configuration system load and performanceindicators and has drawn much research attention recentlyDue to the complexity of the problem most studies adoptmeasurement and simulation methods to quantitatively ana-lyze system performance [18]

By using some measuring devices or measuring pro-grams measurement and simulation methods can directlyobtain the performance indicators of systems or closelyrelated quantities and then work out performance indexes bythe corresponding calculation Ostermann et al analyze theperformance of theAmazonEC2 platformbased onmeasure-ment at the background of scientific computing [19] In addi-tion performance comparisons were made among EC2 andother platforms by using long-term traces of experimentaldata in some indicators such as resource acquisition releaseoverheads and system workload Calheiros et al proposeextensible simulation toolkit CloudSim which can modelsimulate and evaluate the performance of both cloud com-puting systems and application provisioning environments[20] CloudSim supports single and internetworked cloudscenarios and is used by several organizations to investigatecloud resource allocation and energy efficiency managementof data center resources Bautista et al present a perfor-mance measurement framework (PMF) for cloud computingsystems with integration software quality concepts fromISO 25010 [21] The PMF defines the requirements datatypes and evaluation criteria to measure ldquocluster behaviorrdquoperformance Mei et al study performance measurementof network IO applications in virtualized cloud [22] Thismeasurement is based on performance impact of coexistingapplications in a virtualized cloud such as throughput andresource sharing effectiveness In addition the measurement

Scientific Programming 3

can quantify the performance gains and losses and reveal theimportance of optimizing for application deployment

Measurement and simulation methods are the mostdirect and basic ones on performance evaluation which themodel method partly depends on However the two meth-ods are only applicable to existing and running systemsand there are a lot of insufficiencies and abuses in evaluatingthe performance of cloud systems which are under dynamicenvironments and involve lots of parameters such as timeconsuming low degree of simulation and quantitative diffi-culty In addition measurement and simulation methods arealso incapable of finding out performance bottlenecksThere-fore how to provide powerful mathematic tool intuitionaldescription method of models effective analysis methodand available analysis software is the urgent problem forperformance evaluation of cloud systems which is just thecore of analysis technology based on SPN However therehave been few studies on the application of SPN in cloudcomputing

Cao et al construct stochastic evaluation model basedon Queuing Petri Net for ldquoChinese Cloudrdquo of State KeyLaboratory of High-End Server amp Storage Technology [23]They still present three kinds of cloud system architecturesdistributed architecture centralized architecture and hybridarchitecture and thenmodel the three architectures based onQueuing Petri Net These models describe the relationshipsamong network CPU IO and request queue Finally systemthroughputs of the three architectures are compared indifferent task types and workloads with QPME tool [24]

Targeting the dynamic feature of cloud computing Fanet al propose a systematic method to describe the reliabilityrunning time and failure processing of resource scheduling[25] In this study resources scheduling process is abstractedas metaobject by using a reflection mechanism and PetriNet is introduced to model its components such as baselayer metalayer and metaobject protocol In addition theypresent an adaptive resource scheduling strategy describedby Computation Tree Logic (CTL) [26] which can realizedynamic reoptimization and distribution of system resourcesat runtime Finally Petri Net and its state space are used toverify the correctness and effectiveness of the proposed algo-rithm

In order to evaluate the performance of the Hadoop sys-tem Ruiz et al introduce Prioritised-Timed Coloured PetriNet (PTCPN) [27] to formally construct its stochastic modelof MapReduce paradigm [28] Tradeoffs are made betweenprocessing time and resource cost according to some perfor-mance evaluation parameters In addition state space andCPNTools [29] auxiliary software will execute the quanti-tative analysis of the system performance and the accuracyverification of the models

It is concluded that the above-mentioned methods ofperformance evaluation of cloud computing canwell describeand model the various properties of cloud computing butthere are difficulties in comparative analysis To overcomethese challenges we present a novel Dynamic ScalableStochastic Petri Net (DSSPN) to better depict the impor-tant properties of cloud systems Compared to other SPNsDSSPN has the following advantages (1) intuitive graphical

representation andmodel easy to understand (2) no require-ments for strong mathematical background (3) capability offlexibly depicting characteristics of cloud systems such asthe relationship between network topology and other com-ponents and (4) automatically deriving the steady-stateprobability of state transitions by using auxiliary softwaresuch as SPNP and SHLPNA

3 Dynamic Scalable Stochastic Petri Net

Cloud computing is a service-oriented computing modelwith the characteristics of large scale complexity resourceheterogeneity requirement of QoS diversity and scalabilityThose characteristics make the resource scheduling of cloudcomputing too complicated to be modeled and analyzed bythe traditional Stochastic PetriNet To overcome the problema novel Dynamic Scalable Stochastic Petri Net (DSSPN) isproposed in this study DSSPN is generated from SPN [9 30]and Stochastic Reward Net (SRN) [31] In later sections wewill further introduce the feasibility and applicability in bothmodeling and performance evaluation of cloud computingsystems To easily understand the definition of DSSPN wefirstly present some notations Let us suppose that 119878 is a setand 119890 is a number |119878| denotes the number of elements in119878 represents the power set of 119878 [119890] indicates the maximalinteger that is not larger than 119890N stands for the set of naturalnumbers that is N = 0 1 2 while N+ means the setof positive integers that is N+ = 1 2 Let 119872 119875 rarrN denote a marking of DSSPN 119877(119872) represents the set ofreachable marking of the marking 119872 For all 119909 isin 119875 cup 119879∙119909 = 119910 | (119910 119909) isin 119875 cup 119879 indicates the preset of 119909 while119909∙= 119910 | (119909 119910) isin 119875 cup 119879 means the postset of 119909 Φ is an

empty set and 120576 represents an empty element

31 Definitions of DSSPN

Definition 1 ADynamic Scalable Stochastic Petri Net is a 12-tuple (119875 119879 119865 119870119882 120582TS 119866 119864 119891 1198921198720) where

(1) 119875 = 1199011 1199012 119901

119899 is a finite set of places 119899 = |119875|

(2) 119879 = 119879119868cup 119879119879 is a finite set of transitions 119879

119868= 1199051198681

1199051198682 119905

119868119898 is a set of immediate transitions and119879

119879=

1199051198791 1199051198792 119905

119879119897 is a set of timed transitions119898 = |119879

119868|

119897 = |119879119879| note that 119879

119868cap 119879119879= Φ

(3) 119875 cup 119879 = Φ and 119875 cap 119879 = Φ(4) 119865 sube (119875 times 119879) cup (119879 times 119875) is a set of arcs(5) 119870 119875 rarr N+ cup infin is a capacity function where119870(119901)

denotes the capacity of the place 119901 let119870(119901) = 1198961and

1198791=∙119901 if 119872(119901) = 119896

1 for all 119905 isin 119879

1 119905 cannot be

enabled(6) let Pr(119872()) denote an expression of predicate logic

related to the marking of the set means a subsetof 119875

(7) 119882 119865 rarr Ncup120576cup119867cupPr(119872() rarr N) is a weightedfunction119882(119901 119905) and119882(119905 119901)denote theweight of thearc(119901 119905) and (119905 119901) respectively It may be a naturalinteger or a function depending on the marking of

4 Scientific Programming

the set if119882(119909 119910) = 120576 it can be viewed as the weightof (119909 119910) = 1 assume 119875

1is a subset of 119875 and 119873

1is a

positive integer if 119882(119909 119910) = Pr(119872(1198751)) rarr 119873

1 it

means when Pr(119872(1198751)) = true the weight of (119909 119910) is

1198731

(8) 119867 119872() rarr R+cup0 is a functionwhich indicates themapping from the marking of to a positive integerlet 1198751sube 119875 and 119885 = 119872(119901) | 119901 isin 119875

1 then119867(119885) isin N

(9) 120582 = 1205821 1205822 120582

119897 lowast (1 or 119867) is a finite set of average

transition enabling rates where 119897 = |119879119879|

(10) TS = TS1TS2 TS

1198991 is a finite set of types where

1198991 = |TS|

(11) 119866 119875 rarr TS is a function denoting the type assignedto place 119901

(12) 119864 119875 rarr Pr(119872()) rarr N cup 119867lowastN cupN is a functionindicating the values of types if 119866(119901) = TS

119894 then

119864(TS119894) stands for the value of tokens with type TS

119894

in place 119901 note that 119864(TS119894) may be time-variant it

generally denotes the value of current period of timewhen a process is executed

(13) 119891 119879 rarr Pr(119872()) cup 120576 is a function of enablingpredicate where 119891(119905) represents the enabling pred-icate of transition 119905 When 119891(119905) = 120576 it means thatthe enabling condition of transition 119905 is the same asin SPN

(14) 119892 (119879 rarr 120576 cup R+ cup 119867) cup (119875 rarr N+) is a functionof random switch where 119892(119905) denotes the enablingpriority of transition 119905 while 119892(119901)means the priorityof place 119901 if there is a transition without a randomswitch that is 119892(119905) = 120576 it represents that its enablingpriority is 1 and 119892(119901) = 120576 has the same meaning ofplace 119901

(15) 1198720 119875 rarr N is the initial marking which models the

initial status of a system and satisfies for all 119901 isin 1198751198720(119901) le 119870(119901)

As described above DSSPN is a novel extended form ofSPN The major differences lie in that the weight of an arc(or the transition enabling rate) not only is a constant butalso is a function depending on the marking of a subset of 119875The weights of arcs and the transition enabling rates can bedefined by customers In addition these values may actuallychange during the whole processThese features will increasethe dynamic flexibility of SPNand allow themodeling processto automatically adjust

Definition 2 Thetransition firing rule ofDSSPN is elaboratedas follows

(1) For all 119905 isin 119879 if forall119901 isin 119875

119872(119901) ge 119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) ge 119882(119901 119905)

if (119901 isin 119905∙ minus ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) le 119870 (119901)

if (119901 isin 119905∙ cap ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) otherwise(1)

It is said that transition 119905 with the marking119872 is enabledwhich is denoted as119872[119905⟩

(2) If 119872[119905⟩ and 119892(119905) = max119892(119904) | 119904 belongs to 119875and satisfies (1) then transition 119905 can fire After 119905 fired anew subsequent marking1198721015840 is generated from119872 which isdenoted as119872[119905⟩1198721015840 or119872 119905997888rarr 1198721015840 For all 119901 isin 119875

1198721015840

(119901)

=

119872(119901) minus119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) if (119901 isin 119905∙ minus ∙119905)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) if (119901 isin 119905∙ cap ∙119905)

119872 (119901) otherwise

(2)

In themarking119872 theremay bemultiple transitions beingenabled simultaneously In this case a transition is randomlychosen out from the set 1198791015840 to be fired where 1198791015840 = 119904 | 119892(119904) =max119892(119905) 119905 isin 11987910158401015840 and 11987910158401015840 = 119905 | 119872[119905⟩ 119905 isin 119879

In order to formalize the dynamics of DSSPN incidencematrix is introduced to depict its structure and behaviors

Definition 3 The structure of a DSSPN can be expressed byusing a matrix (called the incidence matrix of DSSPN) with 119899rows and119898 columns where 119899 = |119875| and119898 = |119879|

119860 = [119886+

119894119895minus 119886minus

119894119895]119899times119898

(3)

For 1 le 119894 le 119899 1 le 119895 le 119898

119886+

119894119895

=

119882(119905119895 119901119894) if ((119905

119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

1 if ((119905119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

0 otherwise

119886minus

119894119895

=

119882(119901119894 119905119895) if (119901

119894 (119905119895) isin 119865) and (119882(119901

119894 119905119895) = 120576)

1 if ((119901119894 119905119895) isin 119865) and (119901

119894119882 (119905

119895) = 120576)

0 otherwise

(4)

Because 119882(119905119895 119901119894) or 119882(119901

119894 119905119895) can be a constant or a

function depending on the marking of a subset of 119875 wefirstly divide the set of transitions into two subsets 119879

119888and

119879V Consider

119879119888= 119905 | forall119901 isin

∙

119905 cup 119905∙

119872 (119901) is related to 119886119894119895

119879V = 119879 minus 119879119888(5)

Scientific Programming 5

That is if any transition in 119879119888fired the incidence matrix

will be unchanged in current marking Otherwise a newmarking will be generated and the value(s) of some ele-ment(s) will change Suppose 120590 is a firing sequence of tran-sitions 120590 is firstly divided into two subsequences accordingto (6) 120590

119888and 120590V where 120590119888 (or 120590V) only includes transitions in

119879119888(or 119879V) and the orders of these transitions in 120590119888 and 120590V are

the same as that in 120590 Suppose 119862 (an119898-dimensional columnvector) only counts the firing number of the transitionsincluded in 120590

119888 and 120590 = 119905

11199052sdot sdot sdot 119905119896 Consider 119872

120590119888

997888rarr 1198721

1199051

997888rarr

1198722sdot sdot sdot119872119896

119905119896

997888rarr 119872119896+1

then a fundamental equation [30] isobtained The markings in the sequence change as follows

1198721= 119872 + 119860 sdot 119862

119872119895+1= 119872119895+ 119860lowast119895

(6)

where 1 le 119895 le 119896 119860lowast119895

denotes the 119895th column vector of 119860Note that if 119905 isin 119879V the values of these elements in incidencematrix 119860 which are related to 119872(119901) | 119901 isin 119905∙ cup ∙119905 should beupdated after 119905 fired

32 Properties of DSSPN The major motivation to modelsystems or processes by DSSPN is the simplicity and dynamicexpressions in representing systems with multiple users anddynamic environments In some situations there may beredundant transitions inDSSPNmodels In order to preciselyand concisely describe systems we offer the following theo-rems

Theorem 4 If there are some transitions with the samemeaning in a DSSPN model these transitions can be mergedinto one so that each transition is unique in a DSSPN modelthat is transition redundancy can be eliminated

Proof Assume transitions 1199052and 11990510158402have the same meaning

The preset and postset of 1199052are ∙1199052and 119905∙2 respectively Mean-

while the preset and postset of 11990510158402are ∙11990510158402and 1199051015840∙2Their enabling

predicates and random switches are 119891(1199052) 119891(11990510158402) 119892(1199052) and

119892(1199051015840

2) respectively Let us suppose 119905

1is a forerunner transition

of 1199052and 11990510158401is a forerunner transition of 1199051015840

2The two transitions

can be merged as follows

(a) Transitions 1199052and 11990510158402are merged into one transition 119905

(b) The preset of 119905 is ∙119905 = ∙1199052cup∙1199051015840

2 For all 119901 119904 isin ∙119905cup119905∙ 119901 =

119904 if their types and values are the same that is119866(119901) =119866(119904) and 119864(119901) = 119864(119904) then places 119901 and 119904 will bemerged into one place denoted by 1199011015840 Moreover thetype and the corresponding value remain the same

(c) The enabling predicate is119891(119905) = 119891(1199052)or119891(119905

1015840

2) and the

random switch is 119892(119905) = 119892(1199052) and 119892(119905

1015840

2)

(d) Assume 119901 and 119904 will be merged if 119901 isin ∙1199052and 119904 isin ∙1199051015840

2

or 119901 isin 119905∙2and 119904 isin 1199051015840∙

2 the weights of arcs relating to

merged transition 119905 and place 1199011015840 are set as follows

119882(1199011015840

119905) =

119891 (1199052) 997888rarr 119882(119901 119905

2)

119891 (1199051015840

2) 997888rarr 119882(119904 119905

1015840

2)

or 119882(119905 1199011015840) =

119891 (1199052) 997888rarr 119882(119905

2 119901)

119891 (1199051015840

2) 997888rarr 119882(119905

1015840

2 119904)

(7)

Figure 1 shows an example to merge transitions 1199052and 1199053

with the same meaning For places 1199012 1199013 1199014 and 119901

5 assume

119866(1199012) = 119866(119901

3) 119864(119901

2) = 119864(119901

3) 119866(119901

4) = 119866(119901

5) and 119864(119901

4) =

119864(1199015) Note that the weights of some arcs relating to merged

transitions and places will be changed where

1199081015840

1=

119891 (1199052) 997888rarr 119908

1

119891 (1199053) 997888rarr 119908

2

1199081015840

2=

119891 (1199052) 997888rarr 119908

3

119891 (1199053) 997888rarr 119908

4

1199081015840

3=

119891 (1199053) 997888rarr 119908

5

0

(8)

As illustrated in Theorem 4 a DSSPN model can elimi-nate redundant transitions InDSSPN each service or activityonly corresponds to one transition that models a dynamicprocess or a system including multiple customers on a moreconvenient way

Theorem 5 A DSSPN can be transformed into a simple net[17] such that for all 119909 119910 isin 119875 cup 119879 the preset of 119909 is equal tothat of 119910 while the postset of 119909 is equal to that of 119910 only if 119909equals 119910 that is

(∙

119909 =∙119910) and (119909

∙

= 119910∙

) 997888rarr 119909 = 119910 forall119909 119910 isin 119875 cup 119879 (9)

Proof First we consider the case of two places with the samepreset and postset as shown in Figure 2 If 119866(119901) = 119866(119904) and119864(119901) = 119864(119904) we can easily transform it into a simple net justas illustrated in Theorem 4 Otherwise we insert two newimmediate transitions and two new places into the originalmodel Then the original net transforms into a simple oneTwo things to note here are the settings of new arcs andplacesthat is 119882(119901

1 1198891) = 119882(119889

1 1199013) = 119882(119901

3 1199052) = 119882(119901

1 1199052)

and119882(1199012 1198892) = 119882(119889

2 1199014) = 119882(119901

4 1199052) = 119882(119901

1 1199052) while

the settings of 1199013and 119901

4are the same as those of 119901

1and 119901

2

Similarly the case of two transitions with the same preset andpostset can be proven just as shown in Figure 3

4 System Model Based on DSSPN

Nowadays numerous cloud computing platforms are com-mercially available such as EucalyptusHadoop andAmazonEC2 [31ndash33] In this study we take a typical cloud systemby adopting fair scheduling algorithm as an example to con-struct a DSSPN model Figure 4 illustrates the basic workingprocess of tasks on a cloud platform in the light of thecharacteristics of a typical cloud system architecture In thecloud system jobs submitted by different customersmay havedifferent QoS requirements on computing time memory

6 Scientific Programming

t1t1

t4

t4t3

t2

p1

p6

p2

p2

p3

p5

p4

w1

w2

w4

w3

w5

p4p1

p6

w9984002

w9984003

w9984001

t998400

Figure 1 Equivalent transformation ofmerging two transitionswiththe same meaning

t1 t1t2 t2

p1 p1 p3

p2p4p2

d1

d2

Figure 2 Equivalent transformation of two places with the samepreset and postset

space data traffic response time and so forth That is atypical cloud platform can be viewed as a multiuser multitasksystem involving multiple data sets with different types ofprocessing jobs at the same time [32] In a cloud platformtasks are the basic processing units in the executive processDispatchers firstly select tasks according to a certain rulefrom the waiting queues and then assign them to appropriateresources adopting some scheduling policies However theproperties of cloud computing such as large scale dynamicsheterogeneity and diversity present a range of challengesfor performance evaluation of cloud systems and cloudoptimization problem [34] In order to verify the applicabilityand feasibility of DSSPN we will model and analyze theperformance of a typical cloud system based on DSSPN inthis section

41 Modeling Abstract Without loss of generality let usmakethe following assumptions for a typical cloud system

(1) There are 119899 clients denoted by 119888119894 Client 119894 submits jobs

into a waiting queue (ie pool 119894) with a capacity of 119887119894

(2) The minimum share of pool 119894 is denoted by ms119894

(3) In fair scheduling the set of priorities of each pool isVERY HIGHHIGHNORMALLOWVERY LOWIn order to facilitate the analysis the set of prioritiesare set to 5 4 3 2 1

(4) The arrival process of tasks submitted by client 119894obeys the Poisson distribution with rate of 120582

119894 When

the number of tasks submitted by client 119894 exceeds 119887119894

the job submission is rejected(5) In each waiting queue the scheduling discipline is

First Come First Served (FCFS)(6) There are119898 servers (denoted by 119904

119894) each of which has

119903119894virtual machines (VMs) shared by 119899 clients

(7) The service rate of each VM on 119904119895is 120583119895with exponent

distribution In addition the service rates are gener-ally independent of each other Note that the sum ofms119894is equal to or smaller than the total number of

resources that is sum119899119894=1le sum119898

119895=1119903119895

42 DSSPN Model of Fair Scheduling Based on DSSPN wemodel a typical cloud system adopting fair scheduling as amultiserver multiqueue system with 119899 clients and 119898 serversThe DSSPN model and involved notations are shown inFigure 5 andNotations In order to simplify the description ofthe DSSPN model we would not show the shared structuresof servers

All the places and transitions included in Figure 5 aredescribed as follows (1 le 119894 le 119899 1 le 119895 le 119898)

(1) 119888119895 a timed transition denotes client 119894 submitting tasks

with the firing rate of 120582119894 The enabling predicate 119891

119894of 119888119894is

119891119894(119872) 119872 (119901

119894) le 119887119894 1 le 119894 le 119899 (10)

That is client 119894 can submit tasks when the number of tasks issmaller than its capacity

(2) 119901119894 a place indicates the pool storing these tasks

submitted by client 119894 and119870(119901119894) = 119887119894 In addition119866(119901

119894) = MS

119864(119901119894) = ms

119894 and119892(119901

119894) = pl

119894 wherems

119894means the guaranteed

minimum share of pool 119894 pl119894represents the priority of pool 119894

and pl119894isin 5 4 3 2 1 (just as elaborated in previous section)

(3) 119877119895 a place stands for the status of server 119895 for

simplicity it is not shown in Figure 5119872(119877119895) is the number

of idle VMs of server 119895 119870(119877119895) = 119903119895 which means the total

number of VMs on server 119895(4) 119889119894119895 an immediate transition indicates the execution

of some scheduling or decisionThe scheduling or decision isexpressed by the enabling predicate119891

119894119895and random switch 119892

119894119895

associated with 119889119894119895

119891119894119895= (((AR

119894lt 119864 (119901

119894)) or (dem

119894ge 119864 (119901

119894)))

and (

119898

sum

119895=1

119872(119902119894119895) = 0) and (|SIDS (119872)| gt 0))

or ((dem119894ge 119864 (119901

119894))

and (for forallℎ = 119894 119864 (119901ℎ) le 119864 (119901

119894))

and (|SIDS (119872)| gt 0))

119892119894119895=

5 times1

|UDLMS (119872)| if 119894 isin UDLMS (119872)

4 times1

|UDGMS (119872)| if 119894 isin UDGMS (119872)

3 times1

|ULMS (119872)| if 119894 isin DLMS (119872)

2 times1

|MMS (119872)| if 119894 isin MMS (119872)

0 otherwise

(11)

In this scheme the highest priority is firstly given tothe unallocated pools whose demand is smaller than its

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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2 Scientific Programming

computing systems In this study we only focus on the modelmethod

Model method is a kind of analysis method of perfor-mance evaluation by studying and describing the relation-ships among performance system and load based onmathe-matical theories To facilitate mathematical descriptions andcalculation it usually requires simplification of the systemmodel and making some rational assumptions about thestatus of the system Compared to the other two methodsmodel method is based on a mature theoretical foundationand can clearly describe the relationship among all factorswith a lower cost

Stochastic Petri Net (SPN) is a powerful tool of modelmethod and can be applied to graphic modeling and mathe-matical analysis of many systems and areas such as computerscience communication network andmultiprocessor system[8ndash13] It can not only easily describe the properties ofsystems that have concurrency and synchronization charac-teristics but also clearly depict dynamic behaviors of systemsin an intuitive and efficient way In this way it is easy todiscover performance deficiencies and bottlenecks by usingSPN in analysis

However SPN is still not entirely suitable for modelingand performance evaluation of cloud computing systems (i)cloud computing offers scalable infrastructures to consumerson demand by utilizing the virtualization technology SPN isnot capable of adjustingmodels dynamically when the infras-tructure changes [13] (ii) Different workloads submitted byusers which are simultaneously running on cloud clustersmight have different QoS requirements such as responsetime execution time and data traffic [14] However SPN isincapable of representing the diversity of cloud computing(iii) The configurable shared resources in cloud computingare usually heterogeneous and geographically distributed sousing SPN to build upmodels will increase the computationalcomplexity and result in state explosion Because of theproblems mentioned above SPN does not adequately modeland analyze the performance of cloud computing in manysituations

To overcome those challenges we propose a novelextended form of SPN which is called Dynamic ScalableStochastic Petri Net (DSSPN) to conveniently model andanalyze the performance of cloud computing systems Inorder to support dynamic changes in cloud computing threekinds of functions are introduced to enhance the dynamicsof arcs transitions and places in DSSPN In addition manycloud service patterns under the same state can be com-pressed into a simple one by using DSSPN Therefore cloudcomputing systems can be easily modeled and analyzed byusing DSSPN without changing the original graph structureConsumers can easily evaluate performances only by settingthe three functions of the DSSPN model without spendingtoo much time on programming According to the feature ofSPN system decomposition and model compression can beapplied to reduce the complexity of the state space in DSSPNmodels

The main contributions of this paper include (1) propos-ing Dynamic Scalable Stochastic Petri Net (DSSPN) and thenfurther demonstrating firing rules and some properties of

DSSPN (2) presenting classified fair scheduling (CFS) takinginto consideration job diversity and resource heterogene-ity which can improve throughput and response time (3)validating the proposed approach and algorithm where weconduct and evaluate DSSPN models of fair scheduling andCFS algorithms by using Stochastic Petri Net Package (SPNP)[15 16] analysis and simulation

The remainder of the paper is organized as followsSection 2 describes the related works of this study Section 3specifies the novel analytical model called DSSPN and elab-orates its dynamics as well as some other properties In Sec-tion 4 we construct a DSSPN model of resource schedulingof fair scheduler and propose the classified fair scheduling(CFS) algorithm taking into consideration the workload andresources diversity In addition in order to alleviate the prob-lem of state explosion we adopt the multiuser multiservermodel [17] and analyze some parameters by using equivalentMarkov model to refine our original models in Section 4Section 5 demonstrates the experimental parameters setupand evaluates the system performance of the two schedulingalgorithms by using SPNP Finally conclusions and futureworks are given in Section 6

2 Related Works

Performance evaluation mainly focuses on relationshipsamong system configuration system load and performanceindicators and has drawn much research attention recentlyDue to the complexity of the problem most studies adoptmeasurement and simulation methods to quantitatively ana-lyze system performance [18]

By using some measuring devices or measuring pro-grams measurement and simulation methods can directlyobtain the performance indicators of systems or closelyrelated quantities and then work out performance indexes bythe corresponding calculation Ostermann et al analyze theperformance of theAmazonEC2 platformbased onmeasure-ment at the background of scientific computing [19] In addi-tion performance comparisons were made among EC2 andother platforms by using long-term traces of experimentaldata in some indicators such as resource acquisition releaseoverheads and system workload Calheiros et al proposeextensible simulation toolkit CloudSim which can modelsimulate and evaluate the performance of both cloud com-puting systems and application provisioning environments[20] CloudSim supports single and internetworked cloudscenarios and is used by several organizations to investigatecloud resource allocation and energy efficiency managementof data center resources Bautista et al present a perfor-mance measurement framework (PMF) for cloud computingsystems with integration software quality concepts fromISO 25010 [21] The PMF defines the requirements datatypes and evaluation criteria to measure ldquocluster behaviorrdquoperformance Mei et al study performance measurementof network IO applications in virtualized cloud [22] Thismeasurement is based on performance impact of coexistingapplications in a virtualized cloud such as throughput andresource sharing effectiveness In addition the measurement

Scientific Programming 3

can quantify the performance gains and losses and reveal theimportance of optimizing for application deployment

Measurement and simulation methods are the mostdirect and basic ones on performance evaluation which themodel method partly depends on However the two meth-ods are only applicable to existing and running systemsand there are a lot of insufficiencies and abuses in evaluatingthe performance of cloud systems which are under dynamicenvironments and involve lots of parameters such as timeconsuming low degree of simulation and quantitative diffi-culty In addition measurement and simulation methods arealso incapable of finding out performance bottlenecksThere-fore how to provide powerful mathematic tool intuitionaldescription method of models effective analysis methodand available analysis software is the urgent problem forperformance evaluation of cloud systems which is just thecore of analysis technology based on SPN However therehave been few studies on the application of SPN in cloudcomputing

Cao et al construct stochastic evaluation model basedon Queuing Petri Net for ldquoChinese Cloudrdquo of State KeyLaboratory of High-End Server amp Storage Technology [23]They still present three kinds of cloud system architecturesdistributed architecture centralized architecture and hybridarchitecture and thenmodel the three architectures based onQueuing Petri Net These models describe the relationshipsamong network CPU IO and request queue Finally systemthroughputs of the three architectures are compared indifferent task types and workloads with QPME tool [24]

Targeting the dynamic feature of cloud computing Fanet al propose a systematic method to describe the reliabilityrunning time and failure processing of resource scheduling[25] In this study resources scheduling process is abstractedas metaobject by using a reflection mechanism and PetriNet is introduced to model its components such as baselayer metalayer and metaobject protocol In addition theypresent an adaptive resource scheduling strategy describedby Computation Tree Logic (CTL) [26] which can realizedynamic reoptimization and distribution of system resourcesat runtime Finally Petri Net and its state space are used toverify the correctness and effectiveness of the proposed algo-rithm

In order to evaluate the performance of the Hadoop sys-tem Ruiz et al introduce Prioritised-Timed Coloured PetriNet (PTCPN) [27] to formally construct its stochastic modelof MapReduce paradigm [28] Tradeoffs are made betweenprocessing time and resource cost according to some perfor-mance evaluation parameters In addition state space andCPNTools [29] auxiliary software will execute the quanti-tative analysis of the system performance and the accuracyverification of the models

It is concluded that the above-mentioned methods ofperformance evaluation of cloud computing canwell describeand model the various properties of cloud computing butthere are difficulties in comparative analysis To overcomethese challenges we present a novel Dynamic ScalableStochastic Petri Net (DSSPN) to better depict the impor-tant properties of cloud systems Compared to other SPNsDSSPN has the following advantages (1) intuitive graphical

representation andmodel easy to understand (2) no require-ments for strong mathematical background (3) capability offlexibly depicting characteristics of cloud systems such asthe relationship between network topology and other com-ponents and (4) automatically deriving the steady-stateprobability of state transitions by using auxiliary softwaresuch as SPNP and SHLPNA

3 Dynamic Scalable Stochastic Petri Net

Cloud computing is a service-oriented computing modelwith the characteristics of large scale complexity resourceheterogeneity requirement of QoS diversity and scalabilityThose characteristics make the resource scheduling of cloudcomputing too complicated to be modeled and analyzed bythe traditional Stochastic PetriNet To overcome the problema novel Dynamic Scalable Stochastic Petri Net (DSSPN) isproposed in this study DSSPN is generated from SPN [9 30]and Stochastic Reward Net (SRN) [31] In later sections wewill further introduce the feasibility and applicability in bothmodeling and performance evaluation of cloud computingsystems To easily understand the definition of DSSPN wefirstly present some notations Let us suppose that 119878 is a setand 119890 is a number |119878| denotes the number of elements in119878 represents the power set of 119878 [119890] indicates the maximalinteger that is not larger than 119890N stands for the set of naturalnumbers that is N = 0 1 2 while N+ means the setof positive integers that is N+ = 1 2 Let 119872 119875 rarrN denote a marking of DSSPN 119877(119872) represents the set ofreachable marking of the marking 119872 For all 119909 isin 119875 cup 119879∙119909 = 119910 | (119910 119909) isin 119875 cup 119879 indicates the preset of 119909 while119909∙= 119910 | (119909 119910) isin 119875 cup 119879 means the postset of 119909 Φ is an

empty set and 120576 represents an empty element

31 Definitions of DSSPN

Definition 1 ADynamic Scalable Stochastic Petri Net is a 12-tuple (119875 119879 119865 119870119882 120582TS 119866 119864 119891 1198921198720) where

(1) 119875 = 1199011 1199012 119901

119899 is a finite set of places 119899 = |119875|

(2) 119879 = 119879119868cup 119879119879 is a finite set of transitions 119879

119868= 1199051198681

1199051198682 119905

119868119898 is a set of immediate transitions and119879

119879=

1199051198791 1199051198792 119905

119879119897 is a set of timed transitions119898 = |119879

119868|

119897 = |119879119879| note that 119879

119868cap 119879119879= Φ

(3) 119875 cup 119879 = Φ and 119875 cap 119879 = Φ(4) 119865 sube (119875 times 119879) cup (119879 times 119875) is a set of arcs(5) 119870 119875 rarr N+ cup infin is a capacity function where119870(119901)

denotes the capacity of the place 119901 let119870(119901) = 1198961and

1198791=∙119901 if 119872(119901) = 119896

1 for all 119905 isin 119879

1 119905 cannot be

enabled(6) let Pr(119872()) denote an expression of predicate logic

related to the marking of the set means a subsetof 119875

(7) 119882 119865 rarr Ncup120576cup119867cupPr(119872() rarr N) is a weightedfunction119882(119901 119905) and119882(119905 119901)denote theweight of thearc(119901 119905) and (119905 119901) respectively It may be a naturalinteger or a function depending on the marking of

4 Scientific Programming

the set if119882(119909 119910) = 120576 it can be viewed as the weightof (119909 119910) = 1 assume 119875

1is a subset of 119875 and 119873

1is a

positive integer if 119882(119909 119910) = Pr(119872(1198751)) rarr 119873

1 it

means when Pr(119872(1198751)) = true the weight of (119909 119910) is

1198731

(8) 119867 119872() rarr R+cup0 is a functionwhich indicates themapping from the marking of to a positive integerlet 1198751sube 119875 and 119885 = 119872(119901) | 119901 isin 119875

1 then119867(119885) isin N

(9) 120582 = 1205821 1205822 120582

119897 lowast (1 or 119867) is a finite set of average

transition enabling rates where 119897 = |119879119879|

(10) TS = TS1TS2 TS

1198991 is a finite set of types where

1198991 = |TS|

(11) 119866 119875 rarr TS is a function denoting the type assignedto place 119901

(12) 119864 119875 rarr Pr(119872()) rarr N cup 119867lowastN cupN is a functionindicating the values of types if 119866(119901) = TS

119894 then

119864(TS119894) stands for the value of tokens with type TS

119894

in place 119901 note that 119864(TS119894) may be time-variant it

generally denotes the value of current period of timewhen a process is executed

(13) 119891 119879 rarr Pr(119872()) cup 120576 is a function of enablingpredicate where 119891(119905) represents the enabling pred-icate of transition 119905 When 119891(119905) = 120576 it means thatthe enabling condition of transition 119905 is the same asin SPN

(14) 119892 (119879 rarr 120576 cup R+ cup 119867) cup (119875 rarr N+) is a functionof random switch where 119892(119905) denotes the enablingpriority of transition 119905 while 119892(119901)means the priorityof place 119901 if there is a transition without a randomswitch that is 119892(119905) = 120576 it represents that its enablingpriority is 1 and 119892(119901) = 120576 has the same meaning ofplace 119901

(15) 1198720 119875 rarr N is the initial marking which models the

initial status of a system and satisfies for all 119901 isin 1198751198720(119901) le 119870(119901)

As described above DSSPN is a novel extended form ofSPN The major differences lie in that the weight of an arc(or the transition enabling rate) not only is a constant butalso is a function depending on the marking of a subset of 119875The weights of arcs and the transition enabling rates can bedefined by customers In addition these values may actuallychange during the whole processThese features will increasethe dynamic flexibility of SPNand allow themodeling processto automatically adjust

Definition 2 Thetransition firing rule ofDSSPN is elaboratedas follows

(1) For all 119905 isin 119879 if forall119901 isin 119875

119872(119901) ge 119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) ge 119882(119901 119905)

if (119901 isin 119905∙ minus ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) le 119870 (119901)

if (119901 isin 119905∙ cap ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) otherwise(1)

It is said that transition 119905 with the marking119872 is enabledwhich is denoted as119872[119905⟩

(2) If 119872[119905⟩ and 119892(119905) = max119892(119904) | 119904 belongs to 119875and satisfies (1) then transition 119905 can fire After 119905 fired anew subsequent marking1198721015840 is generated from119872 which isdenoted as119872[119905⟩1198721015840 or119872 119905997888rarr 1198721015840 For all 119901 isin 119875

1198721015840

(119901)

=

119872(119901) minus119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) if (119901 isin 119905∙ minus ∙119905)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) if (119901 isin 119905∙ cap ∙119905)

119872 (119901) otherwise

(2)

In themarking119872 theremay bemultiple transitions beingenabled simultaneously In this case a transition is randomlychosen out from the set 1198791015840 to be fired where 1198791015840 = 119904 | 119892(119904) =max119892(119905) 119905 isin 11987910158401015840 and 11987910158401015840 = 119905 | 119872[119905⟩ 119905 isin 119879

In order to formalize the dynamics of DSSPN incidencematrix is introduced to depict its structure and behaviors

Definition 3 The structure of a DSSPN can be expressed byusing a matrix (called the incidence matrix of DSSPN) with 119899rows and119898 columns where 119899 = |119875| and119898 = |119879|

119860 = [119886+

119894119895minus 119886minus

119894119895]119899times119898

(3)

For 1 le 119894 le 119899 1 le 119895 le 119898

119886+

119894119895

=

119882(119905119895 119901119894) if ((119905

119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

1 if ((119905119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

0 otherwise

119886minus

119894119895

=

119882(119901119894 119905119895) if (119901

119894 (119905119895) isin 119865) and (119882(119901

119894 119905119895) = 120576)

1 if ((119901119894 119905119895) isin 119865) and (119901

119894119882 (119905

119895) = 120576)

0 otherwise

(4)

Because 119882(119905119895 119901119894) or 119882(119901

119894 119905119895) can be a constant or a

function depending on the marking of a subset of 119875 wefirstly divide the set of transitions into two subsets 119879

119888and

119879V Consider

119879119888= 119905 | forall119901 isin

∙

119905 cup 119905∙

119872 (119901) is related to 119886119894119895

119879V = 119879 minus 119879119888(5)

Scientific Programming 5

That is if any transition in 119879119888fired the incidence matrix

will be unchanged in current marking Otherwise a newmarking will be generated and the value(s) of some ele-ment(s) will change Suppose 120590 is a firing sequence of tran-sitions 120590 is firstly divided into two subsequences accordingto (6) 120590

119888and 120590V where 120590119888 (or 120590V) only includes transitions in

119879119888(or 119879V) and the orders of these transitions in 120590119888 and 120590V are

the same as that in 120590 Suppose 119862 (an119898-dimensional columnvector) only counts the firing number of the transitionsincluded in 120590

119888 and 120590 = 119905

11199052sdot sdot sdot 119905119896 Consider 119872

120590119888

997888rarr 1198721

1199051

997888rarr

1198722sdot sdot sdot119872119896

119905119896

997888rarr 119872119896+1

then a fundamental equation [30] isobtained The markings in the sequence change as follows

1198721= 119872 + 119860 sdot 119862

119872119895+1= 119872119895+ 119860lowast119895

(6)

where 1 le 119895 le 119896 119860lowast119895

denotes the 119895th column vector of 119860Note that if 119905 isin 119879V the values of these elements in incidencematrix 119860 which are related to 119872(119901) | 119901 isin 119905∙ cup ∙119905 should beupdated after 119905 fired

32 Properties of DSSPN The major motivation to modelsystems or processes by DSSPN is the simplicity and dynamicexpressions in representing systems with multiple users anddynamic environments In some situations there may beredundant transitions inDSSPNmodels In order to preciselyand concisely describe systems we offer the following theo-rems

Theorem 4 If there are some transitions with the samemeaning in a DSSPN model these transitions can be mergedinto one so that each transition is unique in a DSSPN modelthat is transition redundancy can be eliminated

Proof Assume transitions 1199052and 11990510158402have the same meaning

The preset and postset of 1199052are ∙1199052and 119905∙2 respectively Mean-

while the preset and postset of 11990510158402are ∙11990510158402and 1199051015840∙2Their enabling

predicates and random switches are 119891(1199052) 119891(11990510158402) 119892(1199052) and

119892(1199051015840

2) respectively Let us suppose 119905

1is a forerunner transition

of 1199052and 11990510158401is a forerunner transition of 1199051015840

2The two transitions

can be merged as follows

(a) Transitions 1199052and 11990510158402are merged into one transition 119905

(b) The preset of 119905 is ∙119905 = ∙1199052cup∙1199051015840

2 For all 119901 119904 isin ∙119905cup119905∙ 119901 =

119904 if their types and values are the same that is119866(119901) =119866(119904) and 119864(119901) = 119864(119904) then places 119901 and 119904 will bemerged into one place denoted by 1199011015840 Moreover thetype and the corresponding value remain the same

(c) The enabling predicate is119891(119905) = 119891(1199052)or119891(119905

1015840

2) and the

random switch is 119892(119905) = 119892(1199052) and 119892(119905

1015840

2)

(d) Assume 119901 and 119904 will be merged if 119901 isin ∙1199052and 119904 isin ∙1199051015840

2

or 119901 isin 119905∙2and 119904 isin 1199051015840∙

2 the weights of arcs relating to

merged transition 119905 and place 1199011015840 are set as follows

119882(1199011015840

119905) =

119891 (1199052) 997888rarr 119882(119901 119905

2)

119891 (1199051015840

2) 997888rarr 119882(119904 119905

1015840

2)

or 119882(119905 1199011015840) =

119891 (1199052) 997888rarr 119882(119905

2 119901)

119891 (1199051015840

2) 997888rarr 119882(119905

1015840

2 119904)

(7)

Figure 1 shows an example to merge transitions 1199052and 1199053

with the same meaning For places 1199012 1199013 1199014 and 119901

5 assume

119866(1199012) = 119866(119901

3) 119864(119901

2) = 119864(119901

3) 119866(119901

4) = 119866(119901

5) and 119864(119901

4) =

119864(1199015) Note that the weights of some arcs relating to merged

transitions and places will be changed where

1199081015840

1=

119891 (1199052) 997888rarr 119908

1

119891 (1199053) 997888rarr 119908

2

1199081015840

2=

119891 (1199052) 997888rarr 119908

3

119891 (1199053) 997888rarr 119908

4

1199081015840

3=

119891 (1199053) 997888rarr 119908

5

0

(8)

As illustrated in Theorem 4 a DSSPN model can elimi-nate redundant transitions InDSSPN each service or activityonly corresponds to one transition that models a dynamicprocess or a system including multiple customers on a moreconvenient way

Theorem 5 A DSSPN can be transformed into a simple net[17] such that for all 119909 119910 isin 119875 cup 119879 the preset of 119909 is equal tothat of 119910 while the postset of 119909 is equal to that of 119910 only if 119909equals 119910 that is

(∙

119909 =∙119910) and (119909

∙

= 119910∙

) 997888rarr 119909 = 119910 forall119909 119910 isin 119875 cup 119879 (9)

Proof First we consider the case of two places with the samepreset and postset as shown in Figure 2 If 119866(119901) = 119866(119904) and119864(119901) = 119864(119904) we can easily transform it into a simple net justas illustrated in Theorem 4 Otherwise we insert two newimmediate transitions and two new places into the originalmodel Then the original net transforms into a simple oneTwo things to note here are the settings of new arcs andplacesthat is 119882(119901

1 1198891) = 119882(119889

1 1199013) = 119882(119901

3 1199052) = 119882(119901

1 1199052)

and119882(1199012 1198892) = 119882(119889

2 1199014) = 119882(119901

4 1199052) = 119882(119901

1 1199052) while

the settings of 1199013and 119901

4are the same as those of 119901

1and 119901

2

Similarly the case of two transitions with the same preset andpostset can be proven just as shown in Figure 3

4 System Model Based on DSSPN

Nowadays numerous cloud computing platforms are com-mercially available such as EucalyptusHadoop andAmazonEC2 [31ndash33] In this study we take a typical cloud systemby adopting fair scheduling algorithm as an example to con-struct a DSSPN model Figure 4 illustrates the basic workingprocess of tasks on a cloud platform in the light of thecharacteristics of a typical cloud system architecture In thecloud system jobs submitted by different customersmay havedifferent QoS requirements on computing time memory

6 Scientific Programming

t1t1

t4

t4t3

t2

p1

p6

p2

p2

p3

p5

p4

w1

w2

w4

w3

w5

p4p1

p6

w9984002

w9984003

w9984001

t998400

Figure 1 Equivalent transformation ofmerging two transitionswiththe same meaning

t1 t1t2 t2

p1 p1 p3

p2p4p2

d1

d2

Figure 2 Equivalent transformation of two places with the samepreset and postset

space data traffic response time and so forth That is atypical cloud platform can be viewed as a multiuser multitasksystem involving multiple data sets with different types ofprocessing jobs at the same time [32] In a cloud platformtasks are the basic processing units in the executive processDispatchers firstly select tasks according to a certain rulefrom the waiting queues and then assign them to appropriateresources adopting some scheduling policies However theproperties of cloud computing such as large scale dynamicsheterogeneity and diversity present a range of challengesfor performance evaluation of cloud systems and cloudoptimization problem [34] In order to verify the applicabilityand feasibility of DSSPN we will model and analyze theperformance of a typical cloud system based on DSSPN inthis section

41 Modeling Abstract Without loss of generality let usmakethe following assumptions for a typical cloud system

(1) There are 119899 clients denoted by 119888119894 Client 119894 submits jobs

into a waiting queue (ie pool 119894) with a capacity of 119887119894

(2) The minimum share of pool 119894 is denoted by ms119894

(3) In fair scheduling the set of priorities of each pool isVERY HIGHHIGHNORMALLOWVERY LOWIn order to facilitate the analysis the set of prioritiesare set to 5 4 3 2 1

(4) The arrival process of tasks submitted by client 119894obeys the Poisson distribution with rate of 120582

119894 When

the number of tasks submitted by client 119894 exceeds 119887119894

the job submission is rejected(5) In each waiting queue the scheduling discipline is

First Come First Served (FCFS)(6) There are119898 servers (denoted by 119904

119894) each of which has

119903119894virtual machines (VMs) shared by 119899 clients

(7) The service rate of each VM on 119904119895is 120583119895with exponent

distribution In addition the service rates are gener-ally independent of each other Note that the sum ofms119894is equal to or smaller than the total number of

resources that is sum119899119894=1le sum119898

119895=1119903119895

42 DSSPN Model of Fair Scheduling Based on DSSPN wemodel a typical cloud system adopting fair scheduling as amultiserver multiqueue system with 119899 clients and 119898 serversThe DSSPN model and involved notations are shown inFigure 5 andNotations In order to simplify the description ofthe DSSPN model we would not show the shared structuresof servers

All the places and transitions included in Figure 5 aredescribed as follows (1 le 119894 le 119899 1 le 119895 le 119898)

(1) 119888119895 a timed transition denotes client 119894 submitting tasks

with the firing rate of 120582119894 The enabling predicate 119891

119894of 119888119894is

119891119894(119872) 119872 (119901

119894) le 119887119894 1 le 119894 le 119899 (10)

That is client 119894 can submit tasks when the number of tasks issmaller than its capacity

(2) 119901119894 a place indicates the pool storing these tasks

submitted by client 119894 and119870(119901119894) = 119887119894 In addition119866(119901

119894) = MS

119864(119901119894) = ms

119894 and119892(119901

119894) = pl

119894 wherems

119894means the guaranteed

minimum share of pool 119894 pl119894represents the priority of pool 119894

and pl119894isin 5 4 3 2 1 (just as elaborated in previous section)

(3) 119877119895 a place stands for the status of server 119895 for

simplicity it is not shown in Figure 5119872(119877119895) is the number

of idle VMs of server 119895 119870(119877119895) = 119903119895 which means the total

number of VMs on server 119895(4) 119889119894119895 an immediate transition indicates the execution

of some scheduling or decisionThe scheduling or decision isexpressed by the enabling predicate119891

119894119895and random switch 119892

119894119895

associated with 119889119894119895

119891119894119895= (((AR

119894lt 119864 (119901

119894)) or (dem

119894ge 119864 (119901

119894)))

and (

119898

sum

119895=1

119872(119902119894119895) = 0) and (|SIDS (119872)| gt 0))

or ((dem119894ge 119864 (119901

119894))

and (for forallℎ = 119894 119864 (119901ℎ) le 119864 (119901

119894))

and (|SIDS (119872)| gt 0))

119892119894119895=

5 times1

|UDLMS (119872)| if 119894 isin UDLMS (119872)

4 times1

|UDGMS (119872)| if 119894 isin UDGMS (119872)

3 times1

|ULMS (119872)| if 119894 isin DLMS (119872)

2 times1

|MMS (119872)| if 119894 isin MMS (119872)

0 otherwise

(11)

In this scheme the highest priority is firstly given tothe unallocated pools whose demand is smaller than its

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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Scientific Programming 3

can quantify the performance gains and losses and reveal theimportance of optimizing for application deployment

Measurement and simulation methods are the mostdirect and basic ones on performance evaluation which themodel method partly depends on However the two meth-ods are only applicable to existing and running systemsand there are a lot of insufficiencies and abuses in evaluatingthe performance of cloud systems which are under dynamicenvironments and involve lots of parameters such as timeconsuming low degree of simulation and quantitative diffi-culty In addition measurement and simulation methods arealso incapable of finding out performance bottlenecksThere-fore how to provide powerful mathematic tool intuitionaldescription method of models effective analysis methodand available analysis software is the urgent problem forperformance evaluation of cloud systems which is just thecore of analysis technology based on SPN However therehave been few studies on the application of SPN in cloudcomputing

Cao et al construct stochastic evaluation model basedon Queuing Petri Net for ldquoChinese Cloudrdquo of State KeyLaboratory of High-End Server amp Storage Technology [23]They still present three kinds of cloud system architecturesdistributed architecture centralized architecture and hybridarchitecture and thenmodel the three architectures based onQueuing Petri Net These models describe the relationshipsamong network CPU IO and request queue Finally systemthroughputs of the three architectures are compared indifferent task types and workloads with QPME tool [24]

Targeting the dynamic feature of cloud computing Fanet al propose a systematic method to describe the reliabilityrunning time and failure processing of resource scheduling[25] In this study resources scheduling process is abstractedas metaobject by using a reflection mechanism and PetriNet is introduced to model its components such as baselayer metalayer and metaobject protocol In addition theypresent an adaptive resource scheduling strategy describedby Computation Tree Logic (CTL) [26] which can realizedynamic reoptimization and distribution of system resourcesat runtime Finally Petri Net and its state space are used toverify the correctness and effectiveness of the proposed algo-rithm

In order to evaluate the performance of the Hadoop sys-tem Ruiz et al introduce Prioritised-Timed Coloured PetriNet (PTCPN) [27] to formally construct its stochastic modelof MapReduce paradigm [28] Tradeoffs are made betweenprocessing time and resource cost according to some perfor-mance evaluation parameters In addition state space andCPNTools [29] auxiliary software will execute the quanti-tative analysis of the system performance and the accuracyverification of the models

It is concluded that the above-mentioned methods ofperformance evaluation of cloud computing canwell describeand model the various properties of cloud computing butthere are difficulties in comparative analysis To overcomethese challenges we present a novel Dynamic ScalableStochastic Petri Net (DSSPN) to better depict the impor-tant properties of cloud systems Compared to other SPNsDSSPN has the following advantages (1) intuitive graphical

representation andmodel easy to understand (2) no require-ments for strong mathematical background (3) capability offlexibly depicting characteristics of cloud systems such asthe relationship between network topology and other com-ponents and (4) automatically deriving the steady-stateprobability of state transitions by using auxiliary softwaresuch as SPNP and SHLPNA

3 Dynamic Scalable Stochastic Petri Net

Cloud computing is a service-oriented computing modelwith the characteristics of large scale complexity resourceheterogeneity requirement of QoS diversity and scalabilityThose characteristics make the resource scheduling of cloudcomputing too complicated to be modeled and analyzed bythe traditional Stochastic PetriNet To overcome the problema novel Dynamic Scalable Stochastic Petri Net (DSSPN) isproposed in this study DSSPN is generated from SPN [9 30]and Stochastic Reward Net (SRN) [31] In later sections wewill further introduce the feasibility and applicability in bothmodeling and performance evaluation of cloud computingsystems To easily understand the definition of DSSPN wefirstly present some notations Let us suppose that 119878 is a setand 119890 is a number |119878| denotes the number of elements in119878 represents the power set of 119878 [119890] indicates the maximalinteger that is not larger than 119890N stands for the set of naturalnumbers that is N = 0 1 2 while N+ means the setof positive integers that is N+ = 1 2 Let 119872 119875 rarrN denote a marking of DSSPN 119877(119872) represents the set ofreachable marking of the marking 119872 For all 119909 isin 119875 cup 119879∙119909 = 119910 | (119910 119909) isin 119875 cup 119879 indicates the preset of 119909 while119909∙= 119910 | (119909 119910) isin 119875 cup 119879 means the postset of 119909 Φ is an

empty set and 120576 represents an empty element

31 Definitions of DSSPN

Definition 1 ADynamic Scalable Stochastic Petri Net is a 12-tuple (119875 119879 119865 119870119882 120582TS 119866 119864 119891 1198921198720) where

(1) 119875 = 1199011 1199012 119901

119899 is a finite set of places 119899 = |119875|

(2) 119879 = 119879119868cup 119879119879 is a finite set of transitions 119879

119868= 1199051198681

1199051198682 119905

119868119898 is a set of immediate transitions and119879

119879=

1199051198791 1199051198792 119905

119879119897 is a set of timed transitions119898 = |119879

119868|

119897 = |119879119879| note that 119879

119868cap 119879119879= Φ

(3) 119875 cup 119879 = Φ and 119875 cap 119879 = Φ(4) 119865 sube (119875 times 119879) cup (119879 times 119875) is a set of arcs(5) 119870 119875 rarr N+ cup infin is a capacity function where119870(119901)

denotes the capacity of the place 119901 let119870(119901) = 1198961and

1198791=∙119901 if 119872(119901) = 119896

1 for all 119905 isin 119879

1 119905 cannot be

enabled(6) let Pr(119872()) denote an expression of predicate logic

related to the marking of the set means a subsetof 119875

(7) 119882 119865 rarr Ncup120576cup119867cupPr(119872() rarr N) is a weightedfunction119882(119901 119905) and119882(119905 119901)denote theweight of thearc(119901 119905) and (119905 119901) respectively It may be a naturalinteger or a function depending on the marking of

4 Scientific Programming

the set if119882(119909 119910) = 120576 it can be viewed as the weightof (119909 119910) = 1 assume 119875

1is a subset of 119875 and 119873

1is a

positive integer if 119882(119909 119910) = Pr(119872(1198751)) rarr 119873

1 it

means when Pr(119872(1198751)) = true the weight of (119909 119910) is

1198731

(8) 119867 119872() rarr R+cup0 is a functionwhich indicates themapping from the marking of to a positive integerlet 1198751sube 119875 and 119885 = 119872(119901) | 119901 isin 119875

1 then119867(119885) isin N

(9) 120582 = 1205821 1205822 120582

119897 lowast (1 or 119867) is a finite set of average

transition enabling rates where 119897 = |119879119879|

(10) TS = TS1TS2 TS

1198991 is a finite set of types where

1198991 = |TS|

(11) 119866 119875 rarr TS is a function denoting the type assignedto place 119901

(12) 119864 119875 rarr Pr(119872()) rarr N cup 119867lowastN cupN is a functionindicating the values of types if 119866(119901) = TS

119894 then

119864(TS119894) stands for the value of tokens with type TS

119894

in place 119901 note that 119864(TS119894) may be time-variant it

generally denotes the value of current period of timewhen a process is executed

(13) 119891 119879 rarr Pr(119872()) cup 120576 is a function of enablingpredicate where 119891(119905) represents the enabling pred-icate of transition 119905 When 119891(119905) = 120576 it means thatthe enabling condition of transition 119905 is the same asin SPN

(14) 119892 (119879 rarr 120576 cup R+ cup 119867) cup (119875 rarr N+) is a functionof random switch where 119892(119905) denotes the enablingpriority of transition 119905 while 119892(119901)means the priorityof place 119901 if there is a transition without a randomswitch that is 119892(119905) = 120576 it represents that its enablingpriority is 1 and 119892(119901) = 120576 has the same meaning ofplace 119901

(15) 1198720 119875 rarr N is the initial marking which models the

initial status of a system and satisfies for all 119901 isin 1198751198720(119901) le 119870(119901)

As described above DSSPN is a novel extended form ofSPN The major differences lie in that the weight of an arc(or the transition enabling rate) not only is a constant butalso is a function depending on the marking of a subset of 119875The weights of arcs and the transition enabling rates can bedefined by customers In addition these values may actuallychange during the whole processThese features will increasethe dynamic flexibility of SPNand allow themodeling processto automatically adjust

Definition 2 Thetransition firing rule ofDSSPN is elaboratedas follows

(1) For all 119905 isin 119879 if forall119901 isin 119875

119872(119901) ge 119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) ge 119882(119901 119905)

if (119901 isin 119905∙ minus ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) le 119870 (119901)

if (119901 isin 119905∙ cap ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) otherwise(1)

It is said that transition 119905 with the marking119872 is enabledwhich is denoted as119872[119905⟩

(2) If 119872[119905⟩ and 119892(119905) = max119892(119904) | 119904 belongs to 119875and satisfies (1) then transition 119905 can fire After 119905 fired anew subsequent marking1198721015840 is generated from119872 which isdenoted as119872[119905⟩1198721015840 or119872 119905997888rarr 1198721015840 For all 119901 isin 119875

1198721015840

(119901)

=

119872(119901) minus119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) if (119901 isin 119905∙ minus ∙119905)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) if (119901 isin 119905∙ cap ∙119905)

119872 (119901) otherwise

(2)

In themarking119872 theremay bemultiple transitions beingenabled simultaneously In this case a transition is randomlychosen out from the set 1198791015840 to be fired where 1198791015840 = 119904 | 119892(119904) =max119892(119905) 119905 isin 11987910158401015840 and 11987910158401015840 = 119905 | 119872[119905⟩ 119905 isin 119879

In order to formalize the dynamics of DSSPN incidencematrix is introduced to depict its structure and behaviors

Definition 3 The structure of a DSSPN can be expressed byusing a matrix (called the incidence matrix of DSSPN) with 119899rows and119898 columns where 119899 = |119875| and119898 = |119879|

119860 = [119886+

119894119895minus 119886minus

119894119895]119899times119898

(3)

For 1 le 119894 le 119899 1 le 119895 le 119898

119886+

119894119895

=

119882(119905119895 119901119894) if ((119905

119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

1 if ((119905119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

0 otherwise

119886minus

119894119895

=

119882(119901119894 119905119895) if (119901

119894 (119905119895) isin 119865) and (119882(119901

119894 119905119895) = 120576)

1 if ((119901119894 119905119895) isin 119865) and (119901

119894119882 (119905

119895) = 120576)

0 otherwise

(4)

Because 119882(119905119895 119901119894) or 119882(119901

119894 119905119895) can be a constant or a

function depending on the marking of a subset of 119875 wefirstly divide the set of transitions into two subsets 119879

119888and

119879V Consider

119879119888= 119905 | forall119901 isin

∙

119905 cup 119905∙

119872 (119901) is related to 119886119894119895

119879V = 119879 minus 119879119888(5)

Scientific Programming 5

That is if any transition in 119879119888fired the incidence matrix

will be unchanged in current marking Otherwise a newmarking will be generated and the value(s) of some ele-ment(s) will change Suppose 120590 is a firing sequence of tran-sitions 120590 is firstly divided into two subsequences accordingto (6) 120590

119888and 120590V where 120590119888 (or 120590V) only includes transitions in

119879119888(or 119879V) and the orders of these transitions in 120590119888 and 120590V are

the same as that in 120590 Suppose 119862 (an119898-dimensional columnvector) only counts the firing number of the transitionsincluded in 120590

119888 and 120590 = 119905

11199052sdot sdot sdot 119905119896 Consider 119872

120590119888

997888rarr 1198721

1199051

997888rarr

1198722sdot sdot sdot119872119896

119905119896

997888rarr 119872119896+1

then a fundamental equation [30] isobtained The markings in the sequence change as follows

1198721= 119872 + 119860 sdot 119862

119872119895+1= 119872119895+ 119860lowast119895

(6)

where 1 le 119895 le 119896 119860lowast119895

denotes the 119895th column vector of 119860Note that if 119905 isin 119879V the values of these elements in incidencematrix 119860 which are related to 119872(119901) | 119901 isin 119905∙ cup ∙119905 should beupdated after 119905 fired

32 Properties of DSSPN The major motivation to modelsystems or processes by DSSPN is the simplicity and dynamicexpressions in representing systems with multiple users anddynamic environments In some situations there may beredundant transitions inDSSPNmodels In order to preciselyand concisely describe systems we offer the following theo-rems

Theorem 4 If there are some transitions with the samemeaning in a DSSPN model these transitions can be mergedinto one so that each transition is unique in a DSSPN modelthat is transition redundancy can be eliminated

Proof Assume transitions 1199052and 11990510158402have the same meaning

The preset and postset of 1199052are ∙1199052and 119905∙2 respectively Mean-

while the preset and postset of 11990510158402are ∙11990510158402and 1199051015840∙2Their enabling

predicates and random switches are 119891(1199052) 119891(11990510158402) 119892(1199052) and

119892(1199051015840

2) respectively Let us suppose 119905

1is a forerunner transition

of 1199052and 11990510158401is a forerunner transition of 1199051015840

2The two transitions

can be merged as follows

(a) Transitions 1199052and 11990510158402are merged into one transition 119905

(b) The preset of 119905 is ∙119905 = ∙1199052cup∙1199051015840

2 For all 119901 119904 isin ∙119905cup119905∙ 119901 =

119904 if their types and values are the same that is119866(119901) =119866(119904) and 119864(119901) = 119864(119904) then places 119901 and 119904 will bemerged into one place denoted by 1199011015840 Moreover thetype and the corresponding value remain the same

(c) The enabling predicate is119891(119905) = 119891(1199052)or119891(119905

1015840

2) and the

random switch is 119892(119905) = 119892(1199052) and 119892(119905

1015840

2)

(d) Assume 119901 and 119904 will be merged if 119901 isin ∙1199052and 119904 isin ∙1199051015840

2

or 119901 isin 119905∙2and 119904 isin 1199051015840∙

2 the weights of arcs relating to

merged transition 119905 and place 1199011015840 are set as follows

119882(1199011015840

119905) =

119891 (1199052) 997888rarr 119882(119901 119905

2)

119891 (1199051015840

2) 997888rarr 119882(119904 119905

1015840

2)

or 119882(119905 1199011015840) =

119891 (1199052) 997888rarr 119882(119905

2 119901)

119891 (1199051015840

2) 997888rarr 119882(119905

1015840

2 119904)

(7)

Figure 1 shows an example to merge transitions 1199052and 1199053

with the same meaning For places 1199012 1199013 1199014 and 119901

5 assume

119866(1199012) = 119866(119901

3) 119864(119901

2) = 119864(119901

3) 119866(119901

4) = 119866(119901

5) and 119864(119901

4) =

119864(1199015) Note that the weights of some arcs relating to merged

transitions and places will be changed where

1199081015840

1=

119891 (1199052) 997888rarr 119908

1

119891 (1199053) 997888rarr 119908

2

1199081015840

2=

119891 (1199052) 997888rarr 119908

3

119891 (1199053) 997888rarr 119908

4

1199081015840

3=

119891 (1199053) 997888rarr 119908

5

0

(8)

As illustrated in Theorem 4 a DSSPN model can elimi-nate redundant transitions InDSSPN each service or activityonly corresponds to one transition that models a dynamicprocess or a system including multiple customers on a moreconvenient way

Theorem 5 A DSSPN can be transformed into a simple net[17] such that for all 119909 119910 isin 119875 cup 119879 the preset of 119909 is equal tothat of 119910 while the postset of 119909 is equal to that of 119910 only if 119909equals 119910 that is

(∙

119909 =∙119910) and (119909

∙

= 119910∙

) 997888rarr 119909 = 119910 forall119909 119910 isin 119875 cup 119879 (9)

Proof First we consider the case of two places with the samepreset and postset as shown in Figure 2 If 119866(119901) = 119866(119904) and119864(119901) = 119864(119904) we can easily transform it into a simple net justas illustrated in Theorem 4 Otherwise we insert two newimmediate transitions and two new places into the originalmodel Then the original net transforms into a simple oneTwo things to note here are the settings of new arcs andplacesthat is 119882(119901

1 1198891) = 119882(119889

1 1199013) = 119882(119901

3 1199052) = 119882(119901

1 1199052)

and119882(1199012 1198892) = 119882(119889

2 1199014) = 119882(119901

4 1199052) = 119882(119901

1 1199052) while

the settings of 1199013and 119901

4are the same as those of 119901

1and 119901

2

Similarly the case of two transitions with the same preset andpostset can be proven just as shown in Figure 3

4 System Model Based on DSSPN

Nowadays numerous cloud computing platforms are com-mercially available such as EucalyptusHadoop andAmazonEC2 [31ndash33] In this study we take a typical cloud systemby adopting fair scheduling algorithm as an example to con-struct a DSSPN model Figure 4 illustrates the basic workingprocess of tasks on a cloud platform in the light of thecharacteristics of a typical cloud system architecture In thecloud system jobs submitted by different customersmay havedifferent QoS requirements on computing time memory

6 Scientific Programming

t1t1

t4

t4t3

t2

p1

p6

p2

p2

p3

p5

p4

w1

w2

w4

w3

w5

p4p1

p6

w9984002

w9984003

w9984001

t998400

Figure 1 Equivalent transformation ofmerging two transitionswiththe same meaning

t1 t1t2 t2

p1 p1 p3

p2p4p2

d1

d2

Figure 2 Equivalent transformation of two places with the samepreset and postset

space data traffic response time and so forth That is atypical cloud platform can be viewed as a multiuser multitasksystem involving multiple data sets with different types ofprocessing jobs at the same time [32] In a cloud platformtasks are the basic processing units in the executive processDispatchers firstly select tasks according to a certain rulefrom the waiting queues and then assign them to appropriateresources adopting some scheduling policies However theproperties of cloud computing such as large scale dynamicsheterogeneity and diversity present a range of challengesfor performance evaluation of cloud systems and cloudoptimization problem [34] In order to verify the applicabilityand feasibility of DSSPN we will model and analyze theperformance of a typical cloud system based on DSSPN inthis section

41 Modeling Abstract Without loss of generality let usmakethe following assumptions for a typical cloud system

(1) There are 119899 clients denoted by 119888119894 Client 119894 submits jobs

into a waiting queue (ie pool 119894) with a capacity of 119887119894

(2) The minimum share of pool 119894 is denoted by ms119894

(3) In fair scheduling the set of priorities of each pool isVERY HIGHHIGHNORMALLOWVERY LOWIn order to facilitate the analysis the set of prioritiesare set to 5 4 3 2 1

(4) The arrival process of tasks submitted by client 119894obeys the Poisson distribution with rate of 120582

119894 When

the number of tasks submitted by client 119894 exceeds 119887119894

the job submission is rejected(5) In each waiting queue the scheduling discipline is

First Come First Served (FCFS)(6) There are119898 servers (denoted by 119904

119894) each of which has

119903119894virtual machines (VMs) shared by 119899 clients

(7) The service rate of each VM on 119904119895is 120583119895with exponent

distribution In addition the service rates are gener-ally independent of each other Note that the sum ofms119894is equal to or smaller than the total number of

resources that is sum119899119894=1le sum119898

119895=1119903119895

42 DSSPN Model of Fair Scheduling Based on DSSPN wemodel a typical cloud system adopting fair scheduling as amultiserver multiqueue system with 119899 clients and 119898 serversThe DSSPN model and involved notations are shown inFigure 5 andNotations In order to simplify the description ofthe DSSPN model we would not show the shared structuresof servers

All the places and transitions included in Figure 5 aredescribed as follows (1 le 119894 le 119899 1 le 119895 le 119898)

(1) 119888119895 a timed transition denotes client 119894 submitting tasks

with the firing rate of 120582119894 The enabling predicate 119891

119894of 119888119894is

119891119894(119872) 119872 (119901

119894) le 119887119894 1 le 119894 le 119899 (10)

That is client 119894 can submit tasks when the number of tasks issmaller than its capacity

(2) 119901119894 a place indicates the pool storing these tasks

submitted by client 119894 and119870(119901119894) = 119887119894 In addition119866(119901

119894) = MS

119864(119901119894) = ms

119894 and119892(119901

119894) = pl

119894 wherems

119894means the guaranteed

minimum share of pool 119894 pl119894represents the priority of pool 119894

and pl119894isin 5 4 3 2 1 (just as elaborated in previous section)

(3) 119877119895 a place stands for the status of server 119895 for

simplicity it is not shown in Figure 5119872(119877119895) is the number

of idle VMs of server 119895 119870(119877119895) = 119903119895 which means the total

number of VMs on server 119895(4) 119889119894119895 an immediate transition indicates the execution

of some scheduling or decisionThe scheduling or decision isexpressed by the enabling predicate119891

119894119895and random switch 119892

119894119895

associated with 119889119894119895

119891119894119895= (((AR

119894lt 119864 (119901

119894)) or (dem

119894ge 119864 (119901

119894)))

and (

119898

sum

119895=1

119872(119902119894119895) = 0) and (|SIDS (119872)| gt 0))

or ((dem119894ge 119864 (119901

119894))

and (for forallℎ = 119894 119864 (119901ℎ) le 119864 (119901

119894))

and (|SIDS (119872)| gt 0))

119892119894119895=

5 times1

|UDLMS (119872)| if 119894 isin UDLMS (119872)

4 times1

|UDGMS (119872)| if 119894 isin UDGMS (119872)

3 times1

|ULMS (119872)| if 119894 isin DLMS (119872)

2 times1

|MMS (119872)| if 119894 isin MMS (119872)

0 otherwise

(11)

In this scheme the highest priority is firstly given tothe unallocated pools whose demand is smaller than its

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

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4 Scientific Programming

the set if119882(119909 119910) = 120576 it can be viewed as the weightof (119909 119910) = 1 assume 119875

1is a subset of 119875 and 119873

1is a

positive integer if 119882(119909 119910) = Pr(119872(1198751)) rarr 119873

1 it

means when Pr(119872(1198751)) = true the weight of (119909 119910) is

1198731

(8) 119867 119872() rarr R+cup0 is a functionwhich indicates themapping from the marking of to a positive integerlet 1198751sube 119875 and 119885 = 119872(119901) | 119901 isin 119875

1 then119867(119885) isin N

(9) 120582 = 1205821 1205822 120582

119897 lowast (1 or 119867) is a finite set of average

transition enabling rates where 119897 = |119879119879|

(10) TS = TS1TS2 TS

1198991 is a finite set of types where

1198991 = |TS|

(11) 119866 119875 rarr TS is a function denoting the type assignedto place 119901

(12) 119864 119875 rarr Pr(119872()) rarr N cup 119867lowastN cupN is a functionindicating the values of types if 119866(119901) = TS

119894 then

119864(TS119894) stands for the value of tokens with type TS

119894

in place 119901 note that 119864(TS119894) may be time-variant it

generally denotes the value of current period of timewhen a process is executed

(13) 119891 119879 rarr Pr(119872()) cup 120576 is a function of enablingpredicate where 119891(119905) represents the enabling pred-icate of transition 119905 When 119891(119905) = 120576 it means thatthe enabling condition of transition 119905 is the same asin SPN

(14) 119892 (119879 rarr 120576 cup R+ cup 119867) cup (119875 rarr N+) is a functionof random switch where 119892(119905) denotes the enablingpriority of transition 119905 while 119892(119901)means the priorityof place 119901 if there is a transition without a randomswitch that is 119892(119905) = 120576 it represents that its enablingpriority is 1 and 119892(119901) = 120576 has the same meaning ofplace 119901

(15) 1198720 119875 rarr N is the initial marking which models the

initial status of a system and satisfies for all 119901 isin 1198751198720(119901) le 119870(119901)

As described above DSSPN is a novel extended form ofSPN The major differences lie in that the weight of an arc(or the transition enabling rate) not only is a constant butalso is a function depending on the marking of a subset of 119875The weights of arcs and the transition enabling rates can bedefined by customers In addition these values may actuallychange during the whole processThese features will increasethe dynamic flexibility of SPNand allow themodeling processto automatically adjust

Definition 2 Thetransition firing rule ofDSSPN is elaboratedas follows

(1) For all 119905 isin 119879 if forall119901 isin 119875

119872(119901) ge 119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) ge 119882(119901 119905)

if (119901 isin 119905∙ minus ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) le 119870 (119901)

if (119901 isin 119905∙ cap ∙119905) and (119891 (119905) isin true 120576)

119872 (119901) otherwise(1)

It is said that transition 119905 with the marking119872 is enabledwhich is denoted as119872[119905⟩

(2) If 119872[119905⟩ and 119892(119905) = max119892(119904) | 119904 belongs to 119875and satisfies (1) then transition 119905 can fire After 119905 fired anew subsequent marking1198721015840 is generated from119872 which isdenoted as119872[119905⟩1198721015840 or119872 119905997888rarr 1198721015840 For all 119901 isin 119875

1198721015840

(119901)

=

119872(119901) minus119882(119901 119905) if (119901 isin ∙119905) and true 120576

119872 (119901) +119882(119905 119901) if (119901 isin 119905∙ minus ∙119905)

119872 (119901) +119882(119905 119901) minus 119882(119901 119905) if (119901 isin 119905∙ cap ∙119905)

119872 (119901) otherwise

(2)

In themarking119872 theremay bemultiple transitions beingenabled simultaneously In this case a transition is randomlychosen out from the set 1198791015840 to be fired where 1198791015840 = 119904 | 119892(119904) =max119892(119905) 119905 isin 11987910158401015840 and 11987910158401015840 = 119905 | 119872[119905⟩ 119905 isin 119879

In order to formalize the dynamics of DSSPN incidencematrix is introduced to depict its structure and behaviors

Definition 3 The structure of a DSSPN can be expressed byusing a matrix (called the incidence matrix of DSSPN) with 119899rows and119898 columns where 119899 = |119875| and119898 = |119879|

119860 = [119886+

119894119895minus 119886minus

119894119895]119899times119898

(3)

For 1 le 119894 le 119899 1 le 119895 le 119898

119886+

119894119895

=

119882(119905119895 119901119894) if ((119905

119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

1 if ((119905119895 119901119894) isin 119865) and (119882(119905

119895 119901119894) = 120576)

0 otherwise

119886minus

119894119895

=

119882(119901119894 119905119895) if (119901

119894 (119905119895) isin 119865) and (119882(119901

119894 119905119895) = 120576)

1 if ((119901119894 119905119895) isin 119865) and (119901

119894119882 (119905

119895) = 120576)

0 otherwise

(4)

Because 119882(119905119895 119901119894) or 119882(119901

119894 119905119895) can be a constant or a

function depending on the marking of a subset of 119875 wefirstly divide the set of transitions into two subsets 119879

119888and

119879V Consider

119879119888= 119905 | forall119901 isin

∙

119905 cup 119905∙

119872 (119901) is related to 119886119894119895

119879V = 119879 minus 119879119888(5)

Scientific Programming 5

That is if any transition in 119879119888fired the incidence matrix

will be unchanged in current marking Otherwise a newmarking will be generated and the value(s) of some ele-ment(s) will change Suppose 120590 is a firing sequence of tran-sitions 120590 is firstly divided into two subsequences accordingto (6) 120590

119888and 120590V where 120590119888 (or 120590V) only includes transitions in

119879119888(or 119879V) and the orders of these transitions in 120590119888 and 120590V are

the same as that in 120590 Suppose 119862 (an119898-dimensional columnvector) only counts the firing number of the transitionsincluded in 120590

119888 and 120590 = 119905

11199052sdot sdot sdot 119905119896 Consider 119872

120590119888

997888rarr 1198721

1199051

997888rarr

1198722sdot sdot sdot119872119896

119905119896

997888rarr 119872119896+1

then a fundamental equation [30] isobtained The markings in the sequence change as follows

1198721= 119872 + 119860 sdot 119862

119872119895+1= 119872119895+ 119860lowast119895

(6)

where 1 le 119895 le 119896 119860lowast119895

denotes the 119895th column vector of 119860Note that if 119905 isin 119879V the values of these elements in incidencematrix 119860 which are related to 119872(119901) | 119901 isin 119905∙ cup ∙119905 should beupdated after 119905 fired

32 Properties of DSSPN The major motivation to modelsystems or processes by DSSPN is the simplicity and dynamicexpressions in representing systems with multiple users anddynamic environments In some situations there may beredundant transitions inDSSPNmodels In order to preciselyand concisely describe systems we offer the following theo-rems

Theorem 4 If there are some transitions with the samemeaning in a DSSPN model these transitions can be mergedinto one so that each transition is unique in a DSSPN modelthat is transition redundancy can be eliminated

Proof Assume transitions 1199052and 11990510158402have the same meaning

The preset and postset of 1199052are ∙1199052and 119905∙2 respectively Mean-

while the preset and postset of 11990510158402are ∙11990510158402and 1199051015840∙2Their enabling

predicates and random switches are 119891(1199052) 119891(11990510158402) 119892(1199052) and

119892(1199051015840

2) respectively Let us suppose 119905

1is a forerunner transition

of 1199052and 11990510158401is a forerunner transition of 1199051015840

2The two transitions

can be merged as follows

(a) Transitions 1199052and 11990510158402are merged into one transition 119905

(b) The preset of 119905 is ∙119905 = ∙1199052cup∙1199051015840

2 For all 119901 119904 isin ∙119905cup119905∙ 119901 =

119904 if their types and values are the same that is119866(119901) =119866(119904) and 119864(119901) = 119864(119904) then places 119901 and 119904 will bemerged into one place denoted by 1199011015840 Moreover thetype and the corresponding value remain the same

(c) The enabling predicate is119891(119905) = 119891(1199052)or119891(119905

1015840

2) and the

random switch is 119892(119905) = 119892(1199052) and 119892(119905

1015840

2)

(d) Assume 119901 and 119904 will be merged if 119901 isin ∙1199052and 119904 isin ∙1199051015840

2

or 119901 isin 119905∙2and 119904 isin 1199051015840∙

2 the weights of arcs relating to

merged transition 119905 and place 1199011015840 are set as follows

119882(1199011015840

119905) =

119891 (1199052) 997888rarr 119882(119901 119905

2)

119891 (1199051015840

2) 997888rarr 119882(119904 119905

1015840

2)

or 119882(119905 1199011015840) =

119891 (1199052) 997888rarr 119882(119905

2 119901)

119891 (1199051015840

2) 997888rarr 119882(119905

1015840

2 119904)

(7)

Figure 1 shows an example to merge transitions 1199052and 1199053

with the same meaning For places 1199012 1199013 1199014 and 119901

5 assume

119866(1199012) = 119866(119901

3) 119864(119901

2) = 119864(119901

3) 119866(119901

4) = 119866(119901

5) and 119864(119901

4) =

119864(1199015) Note that the weights of some arcs relating to merged

transitions and places will be changed where

1199081015840

1=

119891 (1199052) 997888rarr 119908

1

119891 (1199053) 997888rarr 119908

2

1199081015840

2=

119891 (1199052) 997888rarr 119908

3

119891 (1199053) 997888rarr 119908

4

1199081015840

3=

119891 (1199053) 997888rarr 119908

5

0

(8)

As illustrated in Theorem 4 a DSSPN model can elimi-nate redundant transitions InDSSPN each service or activityonly corresponds to one transition that models a dynamicprocess or a system including multiple customers on a moreconvenient way

Theorem 5 A DSSPN can be transformed into a simple net[17] such that for all 119909 119910 isin 119875 cup 119879 the preset of 119909 is equal tothat of 119910 while the postset of 119909 is equal to that of 119910 only if 119909equals 119910 that is

(∙

119909 =∙119910) and (119909

∙

= 119910∙

) 997888rarr 119909 = 119910 forall119909 119910 isin 119875 cup 119879 (9)

Proof First we consider the case of two places with the samepreset and postset as shown in Figure 2 If 119866(119901) = 119866(119904) and119864(119901) = 119864(119904) we can easily transform it into a simple net justas illustrated in Theorem 4 Otherwise we insert two newimmediate transitions and two new places into the originalmodel Then the original net transforms into a simple oneTwo things to note here are the settings of new arcs andplacesthat is 119882(119901

1 1198891) = 119882(119889

1 1199013) = 119882(119901

3 1199052) = 119882(119901

1 1199052)

and119882(1199012 1198892) = 119882(119889

2 1199014) = 119882(119901

4 1199052) = 119882(119901

1 1199052) while

the settings of 1199013and 119901

4are the same as those of 119901

1and 119901

2

Similarly the case of two transitions with the same preset andpostset can be proven just as shown in Figure 3

4 System Model Based on DSSPN

Nowadays numerous cloud computing platforms are com-mercially available such as EucalyptusHadoop andAmazonEC2 [31ndash33] In this study we take a typical cloud systemby adopting fair scheduling algorithm as an example to con-struct a DSSPN model Figure 4 illustrates the basic workingprocess of tasks on a cloud platform in the light of thecharacteristics of a typical cloud system architecture In thecloud system jobs submitted by different customersmay havedifferent QoS requirements on computing time memory

6 Scientific Programming

t1t1

t4

t4t3

t2

p1

p6

p2

p2

p3

p5

p4

w1

w2

w4

w3

w5

p4p1

p6

w9984002

w9984003

w9984001

t998400

Figure 1 Equivalent transformation ofmerging two transitionswiththe same meaning

t1 t1t2 t2

p1 p1 p3

p2p4p2

d1

d2

Figure 2 Equivalent transformation of two places with the samepreset and postset

space data traffic response time and so forth That is atypical cloud platform can be viewed as a multiuser multitasksystem involving multiple data sets with different types ofprocessing jobs at the same time [32] In a cloud platformtasks are the basic processing units in the executive processDispatchers firstly select tasks according to a certain rulefrom the waiting queues and then assign them to appropriateresources adopting some scheduling policies However theproperties of cloud computing such as large scale dynamicsheterogeneity and diversity present a range of challengesfor performance evaluation of cloud systems and cloudoptimization problem [34] In order to verify the applicabilityand feasibility of DSSPN we will model and analyze theperformance of a typical cloud system based on DSSPN inthis section

41 Modeling Abstract Without loss of generality let usmakethe following assumptions for a typical cloud system

(1) There are 119899 clients denoted by 119888119894 Client 119894 submits jobs

into a waiting queue (ie pool 119894) with a capacity of 119887119894

(2) The minimum share of pool 119894 is denoted by ms119894

(3) In fair scheduling the set of priorities of each pool isVERY HIGHHIGHNORMALLOWVERY LOWIn order to facilitate the analysis the set of prioritiesare set to 5 4 3 2 1

(4) The arrival process of tasks submitted by client 119894obeys the Poisson distribution with rate of 120582

119894 When

the number of tasks submitted by client 119894 exceeds 119887119894

the job submission is rejected(5) In each waiting queue the scheduling discipline is

First Come First Served (FCFS)(6) There are119898 servers (denoted by 119904

119894) each of which has

119903119894virtual machines (VMs) shared by 119899 clients

(7) The service rate of each VM on 119904119895is 120583119895with exponent

distribution In addition the service rates are gener-ally independent of each other Note that the sum ofms119894is equal to or smaller than the total number of

resources that is sum119899119894=1le sum119898

119895=1119903119895

42 DSSPN Model of Fair Scheduling Based on DSSPN wemodel a typical cloud system adopting fair scheduling as amultiserver multiqueue system with 119899 clients and 119898 serversThe DSSPN model and involved notations are shown inFigure 5 andNotations In order to simplify the description ofthe DSSPN model we would not show the shared structuresof servers

All the places and transitions included in Figure 5 aredescribed as follows (1 le 119894 le 119899 1 le 119895 le 119898)

(1) 119888119895 a timed transition denotes client 119894 submitting tasks

with the firing rate of 120582119894 The enabling predicate 119891

119894of 119888119894is

119891119894(119872) 119872 (119901

119894) le 119887119894 1 le 119894 le 119899 (10)

That is client 119894 can submit tasks when the number of tasks issmaller than its capacity

(2) 119901119894 a place indicates the pool storing these tasks

submitted by client 119894 and119870(119901119894) = 119887119894 In addition119866(119901

119894) = MS

119864(119901119894) = ms

119894 and119892(119901

119894) = pl

119894 wherems

119894means the guaranteed

minimum share of pool 119894 pl119894represents the priority of pool 119894

and pl119894isin 5 4 3 2 1 (just as elaborated in previous section)

(3) 119877119895 a place stands for the status of server 119895 for

simplicity it is not shown in Figure 5119872(119877119895) is the number

of idle VMs of server 119895 119870(119877119895) = 119903119895 which means the total

number of VMs on server 119895(4) 119889119894119895 an immediate transition indicates the execution

of some scheduling or decisionThe scheduling or decision isexpressed by the enabling predicate119891

119894119895and random switch 119892

119894119895

associated with 119889119894119895

119891119894119895= (((AR

119894lt 119864 (119901

119894)) or (dem

119894ge 119864 (119901

119894)))

and (

119898

sum

119895=1

119872(119902119894119895) = 0) and (|SIDS (119872)| gt 0))

or ((dem119894ge 119864 (119901

119894))

and (for forallℎ = 119894 119864 (119901ℎ) le 119864 (119901

119894))

and (|SIDS (119872)| gt 0))

119892119894119895=

5 times1

|UDLMS (119872)| if 119894 isin UDLMS (119872)

4 times1

|UDGMS (119872)| if 119894 isin UDGMS (119872)

3 times1

|ULMS (119872)| if 119894 isin DLMS (119872)

2 times1

|MMS (119872)| if 119894 isin MMS (119872)

0 otherwise

(11)

In this scheme the highest priority is firstly given tothe unallocated pools whose demand is smaller than its

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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Electrical and Computer Engineering

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

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Scientific Programming 5

That is if any transition in 119879119888fired the incidence matrix

will be unchanged in current marking Otherwise a newmarking will be generated and the value(s) of some ele-ment(s) will change Suppose 120590 is a firing sequence of tran-sitions 120590 is firstly divided into two subsequences accordingto (6) 120590

119888and 120590V where 120590119888 (or 120590V) only includes transitions in

119879119888(or 119879V) and the orders of these transitions in 120590119888 and 120590V are

the same as that in 120590 Suppose 119862 (an119898-dimensional columnvector) only counts the firing number of the transitionsincluded in 120590

119888 and 120590 = 119905

11199052sdot sdot sdot 119905119896 Consider 119872

120590119888

997888rarr 1198721

1199051

997888rarr

1198722sdot sdot sdot119872119896

119905119896

997888rarr 119872119896+1

then a fundamental equation [30] isobtained The markings in the sequence change as follows

1198721= 119872 + 119860 sdot 119862

119872119895+1= 119872119895+ 119860lowast119895

(6)

where 1 le 119895 le 119896 119860lowast119895

denotes the 119895th column vector of 119860Note that if 119905 isin 119879V the values of these elements in incidencematrix 119860 which are related to 119872(119901) | 119901 isin 119905∙ cup ∙119905 should beupdated after 119905 fired

32 Properties of DSSPN The major motivation to modelsystems or processes by DSSPN is the simplicity and dynamicexpressions in representing systems with multiple users anddynamic environments In some situations there may beredundant transitions inDSSPNmodels In order to preciselyand concisely describe systems we offer the following theo-rems

Theorem 4 If there are some transitions with the samemeaning in a DSSPN model these transitions can be mergedinto one so that each transition is unique in a DSSPN modelthat is transition redundancy can be eliminated

Proof Assume transitions 1199052and 11990510158402have the same meaning

The preset and postset of 1199052are ∙1199052and 119905∙2 respectively Mean-

while the preset and postset of 11990510158402are ∙11990510158402and 1199051015840∙2Their enabling

predicates and random switches are 119891(1199052) 119891(11990510158402) 119892(1199052) and

119892(1199051015840

2) respectively Let us suppose 119905

1is a forerunner transition

of 1199052and 11990510158401is a forerunner transition of 1199051015840

2The two transitions

can be merged as follows

(a) Transitions 1199052and 11990510158402are merged into one transition 119905

(b) The preset of 119905 is ∙119905 = ∙1199052cup∙1199051015840

2 For all 119901 119904 isin ∙119905cup119905∙ 119901 =

119904 if their types and values are the same that is119866(119901) =119866(119904) and 119864(119901) = 119864(119904) then places 119901 and 119904 will bemerged into one place denoted by 1199011015840 Moreover thetype and the corresponding value remain the same

(c) The enabling predicate is119891(119905) = 119891(1199052)or119891(119905

1015840

2) and the

random switch is 119892(119905) = 119892(1199052) and 119892(119905

1015840

2)

(d) Assume 119901 and 119904 will be merged if 119901 isin ∙1199052and 119904 isin ∙1199051015840

2

or 119901 isin 119905∙2and 119904 isin 1199051015840∙

2 the weights of arcs relating to

merged transition 119905 and place 1199011015840 are set as follows

119882(1199011015840

119905) =

119891 (1199052) 997888rarr 119882(119901 119905

2)

119891 (1199051015840

2) 997888rarr 119882(119904 119905

1015840

2)

or 119882(119905 1199011015840) =

119891 (1199052) 997888rarr 119882(119905

2 119901)

119891 (1199051015840

2) 997888rarr 119882(119905

1015840

2 119904)

(7)

Figure 1 shows an example to merge transitions 1199052and 1199053

with the same meaning For places 1199012 1199013 1199014 and 119901

5 assume

119866(1199012) = 119866(119901

3) 119864(119901

2) = 119864(119901

3) 119866(119901

4) = 119866(119901

5) and 119864(119901

4) =

119864(1199015) Note that the weights of some arcs relating to merged

transitions and places will be changed where

1199081015840

1=

119891 (1199052) 997888rarr 119908

1

119891 (1199053) 997888rarr 119908

2

1199081015840

2=

119891 (1199052) 997888rarr 119908

3

119891 (1199053) 997888rarr 119908

4

1199081015840

3=

119891 (1199053) 997888rarr 119908

5

0

(8)

As illustrated in Theorem 4 a DSSPN model can elimi-nate redundant transitions InDSSPN each service or activityonly corresponds to one transition that models a dynamicprocess or a system including multiple customers on a moreconvenient way

Theorem 5 A DSSPN can be transformed into a simple net[17] such that for all 119909 119910 isin 119875 cup 119879 the preset of 119909 is equal tothat of 119910 while the postset of 119909 is equal to that of 119910 only if 119909equals 119910 that is

(∙

119909 =∙119910) and (119909

∙

= 119910∙

) 997888rarr 119909 = 119910 forall119909 119910 isin 119875 cup 119879 (9)

Proof First we consider the case of two places with the samepreset and postset as shown in Figure 2 If 119866(119901) = 119866(119904) and119864(119901) = 119864(119904) we can easily transform it into a simple net justas illustrated in Theorem 4 Otherwise we insert two newimmediate transitions and two new places into the originalmodel Then the original net transforms into a simple oneTwo things to note here are the settings of new arcs andplacesthat is 119882(119901

1 1198891) = 119882(119889

1 1199013) = 119882(119901

3 1199052) = 119882(119901

1 1199052)

and119882(1199012 1198892) = 119882(119889

2 1199014) = 119882(119901

4 1199052) = 119882(119901

1 1199052) while

the settings of 1199013and 119901

4are the same as those of 119901

1and 119901

2

Similarly the case of two transitions with the same preset andpostset can be proven just as shown in Figure 3

4 System Model Based on DSSPN

Nowadays numerous cloud computing platforms are com-mercially available such as EucalyptusHadoop andAmazonEC2 [31ndash33] In this study we take a typical cloud systemby adopting fair scheduling algorithm as an example to con-struct a DSSPN model Figure 4 illustrates the basic workingprocess of tasks on a cloud platform in the light of thecharacteristics of a typical cloud system architecture In thecloud system jobs submitted by different customersmay havedifferent QoS requirements on computing time memory

6 Scientific Programming

t1t1

t4

t4t3

t2

p1

p6

p2

p2

p3

p5

p4

w1

w2

w4

w3

w5

p4p1

p6

w9984002

w9984003

w9984001

t998400

Figure 1 Equivalent transformation ofmerging two transitionswiththe same meaning

t1 t1t2 t2

p1 p1 p3

p2p4p2

d1

d2

Figure 2 Equivalent transformation of two places with the samepreset and postset

space data traffic response time and so forth That is atypical cloud platform can be viewed as a multiuser multitasksystem involving multiple data sets with different types ofprocessing jobs at the same time [32] In a cloud platformtasks are the basic processing units in the executive processDispatchers firstly select tasks according to a certain rulefrom the waiting queues and then assign them to appropriateresources adopting some scheduling policies However theproperties of cloud computing such as large scale dynamicsheterogeneity and diversity present a range of challengesfor performance evaluation of cloud systems and cloudoptimization problem [34] In order to verify the applicabilityand feasibility of DSSPN we will model and analyze theperformance of a typical cloud system based on DSSPN inthis section

41 Modeling Abstract Without loss of generality let usmakethe following assumptions for a typical cloud system

(1) There are 119899 clients denoted by 119888119894 Client 119894 submits jobs

into a waiting queue (ie pool 119894) with a capacity of 119887119894

(2) The minimum share of pool 119894 is denoted by ms119894

(3) In fair scheduling the set of priorities of each pool isVERY HIGHHIGHNORMALLOWVERY LOWIn order to facilitate the analysis the set of prioritiesare set to 5 4 3 2 1

(4) The arrival process of tasks submitted by client 119894obeys the Poisson distribution with rate of 120582

119894 When

the number of tasks submitted by client 119894 exceeds 119887119894

the job submission is rejected(5) In each waiting queue the scheduling discipline is

First Come First Served (FCFS)(6) There are119898 servers (denoted by 119904

119894) each of which has

119903119894virtual machines (VMs) shared by 119899 clients

(7) The service rate of each VM on 119904119895is 120583119895with exponent

distribution In addition the service rates are gener-ally independent of each other Note that the sum ofms119894is equal to or smaller than the total number of

resources that is sum119899119894=1le sum119898

119895=1119903119895

42 DSSPN Model of Fair Scheduling Based on DSSPN wemodel a typical cloud system adopting fair scheduling as amultiserver multiqueue system with 119899 clients and 119898 serversThe DSSPN model and involved notations are shown inFigure 5 andNotations In order to simplify the description ofthe DSSPN model we would not show the shared structuresof servers

All the places and transitions included in Figure 5 aredescribed as follows (1 le 119894 le 119899 1 le 119895 le 119898)

(1) 119888119895 a timed transition denotes client 119894 submitting tasks

with the firing rate of 120582119894 The enabling predicate 119891

119894of 119888119894is

119891119894(119872) 119872 (119901

119894) le 119887119894 1 le 119894 le 119899 (10)

That is client 119894 can submit tasks when the number of tasks issmaller than its capacity

(2) 119901119894 a place indicates the pool storing these tasks

submitted by client 119894 and119870(119901119894) = 119887119894 In addition119866(119901

119894) = MS

119864(119901119894) = ms

119894 and119892(119901

119894) = pl

119894 wherems

119894means the guaranteed

minimum share of pool 119894 pl119894represents the priority of pool 119894

and pl119894isin 5 4 3 2 1 (just as elaborated in previous section)

(3) 119877119895 a place stands for the status of server 119895 for

simplicity it is not shown in Figure 5119872(119877119895) is the number

of idle VMs of server 119895 119870(119877119895) = 119903119895 which means the total

number of VMs on server 119895(4) 119889119894119895 an immediate transition indicates the execution

of some scheduling or decisionThe scheduling or decision isexpressed by the enabling predicate119891

119894119895and random switch 119892

119894119895

associated with 119889119894119895

119891119894119895= (((AR

119894lt 119864 (119901

119894)) or (dem

119894ge 119864 (119901

119894)))

and (

119898

sum

119895=1

119872(119902119894119895) = 0) and (|SIDS (119872)| gt 0))

or ((dem119894ge 119864 (119901

119894))

and (for forallℎ = 119894 119864 (119901ℎ) le 119864 (119901

119894))

and (|SIDS (119872)| gt 0))

119892119894119895=

5 times1

|UDLMS (119872)| if 119894 isin UDLMS (119872)

4 times1

|UDGMS (119872)| if 119894 isin UDGMS (119872)

3 times1

|ULMS (119872)| if 119894 isin DLMS (119872)

2 times1

|MMS (119872)| if 119894 isin MMS (119872)

0 otherwise

(11)

In this scheme the highest priority is firstly given tothe unallocated pools whose demand is smaller than its

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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Distributed Sensor Networks

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

6 Scientific Programming

t1t1

t4

t4t3

t2

p1

p6

p2

p2

p3

p5

p4

w1

w2

w4

w3

w5

p4p1

p6

w9984002

w9984003

w9984001

t998400

Figure 1 Equivalent transformation ofmerging two transitionswiththe same meaning

t1 t1t2 t2

p1 p1 p3

p2p4p2

d1

d2

Figure 2 Equivalent transformation of two places with the samepreset and postset

space data traffic response time and so forth That is atypical cloud platform can be viewed as a multiuser multitasksystem involving multiple data sets with different types ofprocessing jobs at the same time [32] In a cloud platformtasks are the basic processing units in the executive processDispatchers firstly select tasks according to a certain rulefrom the waiting queues and then assign them to appropriateresources adopting some scheduling policies However theproperties of cloud computing such as large scale dynamicsheterogeneity and diversity present a range of challengesfor performance evaluation of cloud systems and cloudoptimization problem [34] In order to verify the applicabilityand feasibility of DSSPN we will model and analyze theperformance of a typical cloud system based on DSSPN inthis section

41 Modeling Abstract Without loss of generality let usmakethe following assumptions for a typical cloud system

(1) There are 119899 clients denoted by 119888119894 Client 119894 submits jobs

into a waiting queue (ie pool 119894) with a capacity of 119887119894

(2) The minimum share of pool 119894 is denoted by ms119894

(3) In fair scheduling the set of priorities of each pool isVERY HIGHHIGHNORMALLOWVERY LOWIn order to facilitate the analysis the set of prioritiesare set to 5 4 3 2 1

(4) The arrival process of tasks submitted by client 119894obeys the Poisson distribution with rate of 120582

119894 When

the number of tasks submitted by client 119894 exceeds 119887119894

the job submission is rejected(5) In each waiting queue the scheduling discipline is

First Come First Served (FCFS)(6) There are119898 servers (denoted by 119904

119894) each of which has

119903119894virtual machines (VMs) shared by 119899 clients

(7) The service rate of each VM on 119904119895is 120583119895with exponent

distribution In addition the service rates are gener-ally independent of each other Note that the sum ofms119894is equal to or smaller than the total number of

resources that is sum119899119894=1le sum119898

119895=1119903119895

42 DSSPN Model of Fair Scheduling Based on DSSPN wemodel a typical cloud system adopting fair scheduling as amultiserver multiqueue system with 119899 clients and 119898 serversThe DSSPN model and involved notations are shown inFigure 5 andNotations In order to simplify the description ofthe DSSPN model we would not show the shared structuresof servers

All the places and transitions included in Figure 5 aredescribed as follows (1 le 119894 le 119899 1 le 119895 le 119898)

(1) 119888119895 a timed transition denotes client 119894 submitting tasks

with the firing rate of 120582119894 The enabling predicate 119891

119894of 119888119894is

119891119894(119872) 119872 (119901

119894) le 119887119894 1 le 119894 le 119899 (10)

That is client 119894 can submit tasks when the number of tasks issmaller than its capacity

(2) 119901119894 a place indicates the pool storing these tasks

submitted by client 119894 and119870(119901119894) = 119887119894 In addition119866(119901

119894) = MS

119864(119901119894) = ms

119894 and119892(119901

119894) = pl

119894 wherems

119894means the guaranteed

minimum share of pool 119894 pl119894represents the priority of pool 119894

and pl119894isin 5 4 3 2 1 (just as elaborated in previous section)

(3) 119877119895 a place stands for the status of server 119895 for

simplicity it is not shown in Figure 5119872(119877119895) is the number

of idle VMs of server 119895 119870(119877119895) = 119903119895 which means the total

number of VMs on server 119895(4) 119889119894119895 an immediate transition indicates the execution

of some scheduling or decisionThe scheduling or decision isexpressed by the enabling predicate119891

119894119895and random switch 119892

119894119895

associated with 119889119894119895

119891119894119895= (((AR

119894lt 119864 (119901

119894)) or (dem

119894ge 119864 (119901

119894)))

and (

119898

sum

119895=1

119872(119902119894119895) = 0) and (|SIDS (119872)| gt 0))

or ((dem119894ge 119864 (119901

119894))

and (for forallℎ = 119894 119864 (119901ℎ) le 119864 (119901

119894))

and (|SIDS (119872)| gt 0))

119892119894119895=

5 times1

|UDLMS (119872)| if 119894 isin UDLMS (119872)

4 times1

|UDGMS (119872)| if 119894 isin UDGMS (119872)

3 times1

|ULMS (119872)| if 119894 isin DLMS (119872)

2 times1

|MMS (119872)| if 119894 isin MMS (119872)

0 otherwise

(11)

In this scheme the highest priority is firstly given tothe unallocated pools whose demand is smaller than its

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Scientific Programming 7

t1t1

t2t2

p1

p1 p7

p8

p4

p6

p4

p6p2p2

d3

d4

Figure 3 Equivalent transformation of two transitions with the same preset and postset

Requests

Customers

Request dispatcher

data

Service-oriented intermediate layer

Cloud shared resource pool

Top layer components

Figure 4 Basic working process of tasks on a cloud platform

cn

c1 p1

pn

d11 q11 s11

d1m q1m s1m

dn1 qn1 sn1

dnmqnm snm

middot middot middotmiddot middot middotmiddot middot middotmiddot middot middot

Figure 5The refinedDSSPNmodel of a typical cloud system adopt-ing fair scheduling

minimum share Secondly a higher priority is assigned tothe unallocated pools whose demands are equal to or greaterthan its minimum share Then a normal priority is given toallocated pools included inDLMS(119872) Finally if there are anyunallocated VMs these idle resources will be assigned to thepools included in MMS(119872)

(5) 119902119894119895 a place indicates the queue receiving tasks with the

capacity of 119903119894119895 that is 119870(119902

119894119895) = 119903119894119895

(6) 119904119894119895 a timed transition stands for a VMon server 119895with

the firing rate of 120583119894119895 The server 119895 is shared by VM 119904

119894119895 where

1 le 119894 le 119899 and 1 le 119895 le 119898

43 DSSPNModel of Classified Fair Scheduling Although fairscheduling can share a cluster among different users as fair aspossible it does not make good use of resources without con-sidering variousworkload types or resource diversity Varioustypes of workload with different requirements of resourcesconsequently launch different kinds of tasks usually includ-ing CPU intensive tasks and IO intensive tasks Hence it isbeneficial for improving hardware utilization to distinguishtypes of tasks and resources For example the processing timeof a CPU intensive task in resources with stronger computingpower would be shorter than that in other resources Let 119889

119894119896

denote the demand with type of 119896 and 119877119896represent the total

number of VMs with type of 119896 Because of limited space weonly illustrate the improved part in classified fair scheduling(CFS) algorithm shown in Algorithm 1 The remaining part

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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Distributed Sensor Networks

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Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

8 Scientific Programming

(1) Initialize the classification of all available resources(2) Initialize the classification of tasks when they are submitted to pools(3) for each pool i whose demand le its minimum share do(3) for each type k do(4) if 119889

119894119896le 119877119896then

(5) allocate the 119889119894119896resources with type of 119896

(6) 119877119896minus = 119889

119894119896

(6) else(7) allocate the 119877

119896resources with the type of 119896

(8) 119889119894119896minus = 119877

119896

(9) allocate 119889119894119896resources with other types while satisfying 119877

119895ge 119889119894119896 119895 isin 1 2 119897

(10) end if(11) end for(12) end for(13) for (each pool i whose demand gt its minimum share) and (remaining idle unallocated VMs) do(14) add the similar process as described above in light of the assigning decision of each pool(15) end for

Algorithm 1 The improved part of fair scheduling in CFS

of CFS is similar to that of fair scheduling presented byZaharia et al [35]

The descriptions of places and transitions in Figure 6 aresimilar to that in Figure 5 We will not reiterate them hereIn order to facilitate understanding we only emphasize themeaning of the subscripts for places and transitions Thesubscript 119894 denotes client 119894 the subscript 119896 represents taskswith type 119896 and the subscript 119895 describes server 119895 There aresome differences on the values of some notations betweenFigures 5 and 6 The enabling rate of 119888

119894119896is 120582119894119896 and 119870(119902

119894119896119895) =

119887119894119896119895 where sum119899

119894=1sum119897

119896=1119887119894119896119895= 119887119895 The enabling rate of 119904

119894119896119895is

120583119894119896119895 where sum119899

119894=1sum119897

119896=1120583119894119896119895= 120583119895 In addition the servers are

classified that is 119892(119901119894119896119895) isin 1 2 119897 The differences on the

values between Figures 5 and 6 are described as follows

AR119894=

119897

sum

119896=1

119898

sum

119895=1

119872(119902119894119896119895) times Z

dem119894=

119897

sum

119896=1

119872(119901119894119896) + AR

119894

SIDS (119872) = ℎ |119899

sum

119894=1

119897

sum

119896=1

119872(119902119894119896ℎ) le 119887ℎ

(12)

Let 119910119894119896119895

denote the service rate of 119904119894119896119895

provided for thetasks in queue 119902

119894119896119895

119910119894119896119895=

pl times 120583119894119896119895 if 119892 (119901

119894119896119895) = 119896

pl1015840 times 120583119894119896119895 otherwise

(13)

Note that pl gt pl1015840 The scheme would ensure tasks whosetypes are the same as that of servers served at a higher priority

The major difference between fair scheduling (FS) andCFS is that tasks and resources diversity are taken into

account Without loss of generality assume tasks andresources can be divided into 119897 categoriesThe refinedDSSPNmodel of CFS is shown in Figure 6Note that Algorithm 1 onlydescribes the improved part of FS [35] that is the decisionprocedure to allocate resources with various types to differentkinds of tasks

44 Analysis and Solution of DSSPN Models Although theproblem of state explosion is improved to some extent inDSSPN compared to other forms of Petri Nets it is stilldifficult to analyze the performance of large-scale cloud sys-tems Model refinement techniques elaborated by Lin [17]can develop compact models and expose the independenceas well as the interdependent relations between submodels ofan original model Model refinement can lay a foundationfor the decomposition and analysis of models Consequentlythe refinement of models has become a necessary step of themodel design The refinement methods have been appliedto the performance evaluation of high speed network andshared resources systems [17 36]

441 Equivalent Refinement Model and Markov Model Inthis section we will make further use of enabling predicatesand random switches of transitions to refine the model pro-posed above Figure 7 shows the equivalent model for modelsin Figures 5 and 6 while Figure 8 describes the equivalentMarkov model of Figure 7

Comparing Figure 7 with Figures 5 and 6 it can be foundthat the refined model is easier to understand and signifi-cantly reduces the state space by deleting any unnecessaryvanishing states In addition refined model greatly decreasesthe complexity in performance evaluation because of struc-tural similarities of submodels

In Figure 7 immediate transitions and place 119901119894(or 119901119894119896)

and related arcs are removed from Figure 5 (or Figure 6)where 1 le 119894 le 119899 and 1 le 119896 le 119897 The enabling predicates

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

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International Journal of

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Distributed Sensor Networks

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Hindawi Publishing Corporationhttpwwwhindawicom

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International Journal of

ReconfigurableComputing

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Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

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Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

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International Journal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

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RoboticsJournal of

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Scientific Programming 9

c11 p11

d111 q111 s111

d11mq11m s11m

d1l1 q1l1s1l1

d1lm

q1lm

cn1 pn1

dn11 qn11 sn11

dn1m

qn1m sn1m

dnl1

s1lm

c1l p1lcnl pnl

dnlmqnlm snlm

qnl1snl1

middot middot middot

middot middot middot

middot middot middot

Figure 6 The refined DSSPN model of a typical cloud system adopting CFS algorithm

cijcikj

pijpikj

sijsikj

Figure 7 The refined DSSPN model of Figures 5 and 6

and random switches associated with 119889119894119895and 119888119894119895(or 119889119894119896119895

and119888119894119896119895) have changed while others are remaining the same The

random switch of transition 119888119894119895is defined as follows

119892119894119895(119872) 120582

119894times 119892119894119895(119872) (14)

The enabling switch of transition 119888119894119896119895

is

119892119894119896119895(119872) 120582

119894119896times 119892119894119896119895(119872) (15)

442 Parameters Analysis In order to obtain the steady-stateprobabilities of all states a state transition matrix can be con-structed based on the state transition rate and Markov chainillustrated in Figure 8 Then the performance parameters ofthe modeled cloud system can be discussed Let 119875[119872] denotethe steady-state probability of119872

The throughput of transition 119905 is denoted as 119879119905

119879119905= sum

119872isin119867

119875 (119872) times 120582119905 (16)

where 119867 is a set of all markings under which transition 119905 isenabled with the enabling rate of 120582

119905in marking119872

The average number of tokens in place119901 is denoted as119873119901

119873119901= sum119895 times 119875 [119872(119901) = 119895] (17)

The throughput is a crucial indicator of the systemperformance Let 119879

119894119895(or 119879119894119896119895) indicate the throughput of

subsystem 119860119894119895(or 119860119894119896119895) According to the illustration in [16]

the throughput of the model can be calculated as follows

119879 =

119899

sum

119894=1

119898

sum

119895=1

119879119904119894119895

or 119879 =119899

sum

119894=1

119897

sum

119896=1

119898

sum

119895=1

119879119904119894119896119895

(18)

Another important indicator is response time 119877119894119895(or

119877119894119896119895) 119877119894 and 119877 denote the response time of subsystem 119860

119894119895

(or 119860119894119896119895) client 119894 and the system respectively

119877119894119895=119863119902119894119895

119879119904119894119895

119877119894=

119898

sum

119895=1

(119879119904119894119895times 119877119894119895

119898

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119898

sum

ℎ=1

119879119904119894ℎ

119879)

119877119894119896119895=119863119902119894119896119895

119879119904119894119896119895

119877119894119896=

119898

sum

119895=1

(119879119904119894119896119895times 119877119894119896119895

119898

sum

ℎ=1

119879119904119894119896ℎ)

119877119894=

119897

sum

119896=1

(119879119904119894119896times 119877119894119896

119897

sum

ℎ=1

119879119904119894ℎ)

119877 =

119899

sum

119894=1

(119877119894times

119897

sum

ℎ=1

119879119904119894ℎ

119879)

(19)

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

10 Scientific Programming

120582i times gij(M)120582ik times gikj(M)

Enabling condition gijgikj Enabling condition M(qij)M(qikj)

xij times 120583ijyikj times 120583ikj

middot middot middot

M[x11 xij xnm]

M[x111 xikj xnlm]

M[x11 xij + 1 xnm]

M[x111 xikj + 1 xnlm]M[

[

x11 xij minus 1 xnm]

M x111 xikj minus 1 xnlm]

Figure 8 The equivalent Markov model of Figure 7

The average rejection rate of tasks in the cloud systemwith FS at time 119905 is expressed by AER(119905)

AER (119905) =sum119899

119894=1(sum119898

119895=1119875 (119872(119901

119894119895)) gt 119887

119894)

119899 times 119905 (20)

The average rejection rate of tasks in the cloud systemwith CFS at time 119905 is expressed by AER1015840(119905)

AER1015840 (119905) =sum119899

119894=1(sum119897

119896=1sum119898

119895=1119875 (119872(119901

119894119896119895)) gt 119887

119894)

119899 times 119905 (21)

The average idle rate of servers in the cloud system withFS at time 119905 is expressed by AUR(119905)

AUR (119905)

=

sum119898

119895=1(sum119899

119894=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119895(119910)))))

119898 times 119905

(22)

where 119875(enabled(119904119894119895(119910))) means the probability that transi-

tion 119904119894119895(119910) can fire at time 119910

The average idle rate of servers in the cloud system withCFS at time 119905 is expressed by AUR1015840(119905)

AUR1015840 (119905)

=

sum119898

119895=1(sum119899

119894=1sum119897

119896=1sum119905

119910=0(1 minus 119875 (enabled (119904

119894119896119895(119910)))))

119898 times 119905

(23)

where 119875(enabled(119904119894119896119895(119910))) means the probability that transi-

tion 119904119894119896119895(119910) can fire at time 119910

In amultiusermultiserver cloud system the performanceparameters include the state changes of waiting queues andthe service rates of shared servers The improvement ofthroughput and the decrease of response time can be realizedby furthest parallelizing the operations of 119899 servers In otherwords load balance should be maintained

5 Case Study and Evaluation

In this section we provide a case to study the performanceof the DSSPN model based on steady-state probabilitiesTo verify the applicability and feasibility of DSSPN we

Table 1 Number of states and fired transitions

1 machine 2 machines 3 machines 4 machinesReachable states 283 569 1088 1594

Fired transitions 923 1977 3928 5842

only study some performance indicators of FS and CFS bymeans of the above method In addition Stochastic Petri NetPackage (SPNP) is applied to automatically derive the analyticsolution of performance for the DSSPN model This is bene-ficial in modeling and evaluating the performance of cloudsystems because the number of states might reach thousandseven only including few machines shown in Table 1Table 2describes the parameter settings in the simulation

The simulation was conducted to the cloud system con-sisting of 3 servers 2 customers and 2 categories That isthere are 4 waiting queues in FS while 8 waiting queues areexisting in CFS Assume 119892(1199041) = 1 and 119892(1199042) = 2 The tasksubmitted by each client can be classified into 2 groups In thesimulation scenario there are 4 VMs that can be running onserver 1 simultaneously while 5 VMs are running on server2

As shown in Figure 9 when the configuration parametersare identical the values of system average throughput insteady state of CFS are significantly greater than that of fairscheduling Figure 10 describes the average delay which isdepicted by average response time in DSSPN models insteady state of CFS and FS Apparently the average delay ofCFS is prominently smaller than that of fair scheduling Thatis CFS is a powerful way to decrease waiting time for usersAs can be seen from Figure 9 the difference of averagethroughput between CFS and FS can reach 148 when 120582

1=

6sec while the maximal difference of average delay betweenCFS and FS is 575 sec when 120582

1= 6sec

Figure 11 illustrates that average completion time of CFSis significantly better than that of FS The simulation resultspresent that the novel scheme (CFS) can efficiently increasethe average system throughput and thus can improve utiliza-tion of resources This means that it can realize economicbenefits in the commercial cloud services

Moreover Figures 9 10 and 11 also show that the perfor-mance of CFS is generally better than that of fair scheduling

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Scientific Programming 11

Table 2 Parameter settings in simulation

Algorithm 12058221205821

ms1

ms2

1198871119895

1198872119895

1198871119896119895

1198872119896119895

1205831119895

1205832119895

1205831119896119895

1205832119896119895

pl pl1015840 119887

FS 23 3 4 10 8 3 2 30

CFS 23 3 4 10 8 3 2 2 1 30

FSCFS

5

10

15

20

25

30

35

Aver

age t

hrou

ghpu

t

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 9 Average throughput when 1205821= 3 4 5 6 7

FSCFS

6

8

10

12

14

Aver

age r

espo

nse t

ime

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 10 Average response time when 1205821= 3 4 5 6 7

across all circumstances especially at heavy load Howeverqueues cannot be simulated efficiently because these schemesare only based on the current state of queues but ignore thedynamics of task in the queues The simulation results aredifferent by setting different input rates due to incapability ofpredicting the future state of the waiting queues

Figure 12 shows how the average rejection rate of thecloud system changes as service time goes on When the taskrequest in one waiting pool is up to 30 the system will rejectnew requests submitted by the corresponding user When1 le 119905 le 10 the average rejection rate of FS is higher than thatof CFS The differences between FS and CFS in the averagerejection rate are up to 4008 at service time of 5 secondsIn addition Figure 12 also illustrates that along with theoperation of the cloud system the average reject rate increaseswith the accumulation of backlogs in waiting queues

Figure 13 illustrates how the scheduling strategies affectthe average resource utilization of the system When 0 le 119905 le10 the average idle rate of servers in FS is lower than that

FSCFS

141618

222242628

Aver

age c

ompl

etio

n tim

e (se

c)

35 4 45 5 55 6 65 73Input rate of 1205821 (sec)

Figure 11 Average completion time when 1205821= 3 4 5 6 7

0

005

01

015

02

025

Aver

age r

ejec

tion

rate

2 3 4 5 6 7 8 9 101Service time t (sec)

FSCFS

Figure 12 Average rejection rate at different service time 119905

in CFS The maximal differences between FS and CFS in theaverage idle rate of servers at different service times are 4 Itmeans that there is potential to achieve higher utilization ratewith CFS algorithm by increasing the system throughput

6 Conclusion

In this paper we propose the definition of DSSPN thatcan easily describe the multiple clients systems based oncloud services such as a typical cloud platform The majormotivation to model systems or processes by DSSPN is itssimplicity and dynamic expressions to represent systems withmultiple users and dynamic environments Moreover wefurther elaborate dynamic property of DSSPN and analyzesome properties of DSSPN In the following section for someshortcomings of fair scheduling the classified fair scheduling(CFS) algorithm is proposed taking into consideration jobsand resources diversity

In the real world a typical cloud system is shared by mul-tiple applications including production applications batch

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

12 Scientific Programming

2 3 4 5 6 7 8 9 101016

018

02

022

024

026

Aver

age i

dle r

ate o

f ser

vers

Service time t (sec)

FSCFS

Figure 13 Average idle rate of servers at different service time 119905

jobs and interactive jobs Meanwhile different applicationshave different requirements on hardware resources and QoSparameters Therefore we adopt the multiuser multiservermodel to analyze the performance analysis and designDSSPN models for FS and CFS In order to avoid thestate space explosion the analysis techniques and modelrefinement techniques are applied to performance evaluationof their DSSPNmodels Finally SPNP is used to obtain somekey indicators of QoS that is system average throughputresponse time and average completion time are comparedbetween the two schemes Just as shown from Figures 9ndash11the performance of CFS is generally better than that of fairscheduling across all circumstances especially at heavy load

The following topics are of high interest for future work

(1) Other quality metrics such as energy consumptionand cost should be analyzed

(2) The proposed model is without considering local taskmigrations among servers in the same data center

(3) The theoretical derivations between simulationresults and actual cloud systems will be studied

Notations

Involved Notations and Equations in Figure 5

AR119894 The VMs allocated to pool 119894 AR

119894= sum119898

119895=1119872(119902119894119895)

sms The smallest minimum share among somepools sms = min119864(119901

ℎ) ℎ isin DGMS(119872)

dem119894 The demand of pool 119894 dem

119894= 119872(119901

119894) + AR

119894

sdem The smallest demand among some poolssdem = mindem

ℎ ℎ isin DGMS(119872)

def119894 The deficit between dem

119894and ms

119894

def119894= 119864(119901

119894) minus AR

119894

SIDS The set of all servers that has idle slot waiting tobe assigned SIDS(119872) = ℎ | sum119899

119894=1119872(119902119894ℎ) le 119887ℎ

DLMS The set of all pools whose demand is less thanits minimum share DLMS(119872) = ℎ | dem

ℎlt

119864(119901119894) 1 le ℎ le 119899

UDLMS The set of all unallocated pools whose demandis less than its minimum share UDLMS(119872) =119894 | sum119898

119894=1119872(119901119894119895) 119894 isin DLMS(119872)

DGMS The set of all pools whose demand is equal to orlarger than its minimum share DGMS(119872) =ℎ | dem

ℎge 119864(119901

119894) 1 le ℎ le 119899

UDGMS The set of all pools in DGMS without anyallocated resources at the current statusUDGMS(119872) = 119894 | sum119898

119895=1119872(119902119894119895) = 0 119894 isin

DGMS(119872)MMS The set of pools with the smallest minimum

share in DGMS MMS(119872) = ℎ | 119864(119901119894) =

sms ℎ isin DGMS

Competing Interests

The authors declare that they have no competing interests

Acknowledgments

This work was partially supported by the National NaturalScience Foundation of China (nos 61172063 61272093 and61572523) and special fund project for work method inno-vation of Ministry of Science and Technology of China (no2015IM010300)

References

[1] P Mell and T Grance The NIST Definition of Cloud Com-puting Recommendations of the National Institute Standardsand Technology-Special Publication 800-145 NIST Wash-ington DC USA httpnvlpubsnistgovnistpubsLegacySPnistspecialpublication800

[2] S Singh and I Chana ldquoQRSF QoS-aware resource schedulingframework in cloud computingrdquo Journal of Supercomputing vol71 no 1 pp 241ndash292 2014

[3] J Baliga R W A Ayre K Hinton and R S Tucker ldquoGreencloud computing balancing energy in processing storage andtransportrdquo Proceedings of the IEEE vol 99 no 1 pp 149ndash1672011

[4] B P Rimal A Jukan D Katsaros and Y Goeleven ldquoArchitec-tural requirements for cloud computing systems an enterprisecloud approachrdquo Journal of Grid Computing vol 9 no 1 pp 3ndash26 2011

[5] A L Bardsiri and S M Hashemi ldquoA review of workflowscheduling in cloud computing environmentrdquo InternationalJournal of Computer Science and Management Research vol 1no 3 pp 348ndash351 2012

[6] Y Chawla and M Bhonsle ldquoA study on scheduling methods incloud computingrdquo International Journal of Emerging Trends andTechnology in Computer Science vol 1 no 3 pp 12ndash17 2012

[7] L Chuang Stochastic Petri Net and System Performance Evalu-ation Tsinghua University Press Beijing China 2005

[8] M K Molloy ldquoDiscrete time stochastic Petri netsrdquo IEEE Trans-actions on Software Engineering vol 11 no 4 pp 417ndash423 1985

[9] M A Marsan G Balbo G Conte S Donatelli and G Frances-chinis ldquoModelling with generalized stochastic petri netsrdquo ACMSIGMETRICS Performance Evaluation Review vol 26 no 2 p2 1998

[10] WM P van derAalst ldquoThe application of Petri nets toworkflowmanagementrdquo Journal of Circuits Systems and Computers vol8 no 1 pp 21ndash66 1998

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Scientific Programming 13

[11] K JensenColoured Petri Nets Basic Concepts Analysis Methodsand Practical Use Springer New York NY USA 2013

[12] K Jensen and G Rozenberg High-Level Petri Nets Theory andApplication Springer Science and Business Media BerlinGermany 2012

[13] N Ferry A Rossini F Chauvel B Morin and A SolbergldquoTowards model-driven provisioning deployment monitor-ing and adaptation of multi-cloud systemsrdquo in Proceedingsof the IEEE 6th International Conference on Cloud Computing(CLOUD rsquo13) pp 887ndash894 IEEE Santa Clara Calif USA June2013

[14] B P Rimal E Choi and I Lumb ldquoA taxonomy and survey ofcloud computing systemsrdquo in Proceedings of the 5th Interna-tional Joint Conference on INC IMS and IDC pp 44ndash51 SeoulRepublic of Korea August 2009

[15] M Llorens and J Oliver ldquoMarked-controlled reconfigurableworkflow netsrdquo in Proceedings of the 8th International Sympo-sium on Symbolic andNumeric Algorithms for Scientific Comput-ing (SYNASC rsquo06) pp 407ndash413 Timisoara Romania September2006

[16] L Lei C Lin J Cai and X Shen ldquoPerformance analysis ofwireless opportunistic schedulers using stochastic Petri netsrdquoIEEE Transactions onWireless Communications vol 8 no 4 pp2076ndash2087 2009

[17] C Lin ldquoOn refinement of model structure for stochastic PetriNetsrdquo Journal of Software vol 1 p 017 2000

[18] Y Xia M Zhou X Luo S Pang and Q Zhu ldquoStochastic mod-eling and performance analysis ofmigration-enabled and error-prone cloudsrdquo IEEE Transactions on Industrial Informatics vol11 no 2 pp 495ndash504 2015

[19] S Ostermann A Iosup N Yigitbasi R Prodan T Fahringerand D Epema ldquoA performance analysis of EC2 cloud comput-ing services for scientific computingrdquo in Cloud Computing DR Avresky M Diaz A Bode B Ciciani and E Dekel Eds vol34 of Lecture Notes of the Institute for Computer Sciences Social-Informatics and Telecommunications Engineering pp 115ndash131Springer Berlin Germany 2010

[20] R N Calheiros R Ranjan A Beloglazov C A F De Rose andR Buyya ldquoCloudSim a toolkit for modeling and simulationof cloud computing environments and evaluation of resourceprovisioning algorithmsrdquo Software Practice and Experience vol41 no 1 pp 23ndash50 2011

[21] L Bautista A Abran and A April ldquoDesign of a performancemeasurement framework for cloud computingrdquo Journal ofSoftware Engineering and Applications vol 5 no 2 pp 69ndash752012

[22] Y Mei L Liu X Pu and S Sivathanu ldquoPerformance measure-ments and analysis of network IO applications in virtualizedcloudrdquo in Proceedings of the IEEE 3rd International Conferenceon Cloud Computing pp 59ndash66 Miami Fla USA July 2010

[23] Y Cao H Lu X Shi and P Duan ldquoEvaluation model of thecloud systems based on Queuing Petri netrdquo in Algorithms andArchitectures for Parallel Processing pp 413ndash423 Springer Inter-national Cham Switzerland 2015

[24] S Kounev and C Dutz ldquoQPME a performance modeling toolbased on queueing Petri NetsrdquoACMSIGMETRICS PerformanceEvaluation Review vol 36 no 4 pp 46ndash51 2009

[25] G Fan H Yu and L Chen ldquoA formal aspect-oriented methodfor modeling and analyzing adaptive resource scheduling incloud computingrdquo IEEE Transactions on Network and ServiceManagement vol 13 no 2 pp 281ndash294 2016

[26] M Reynolds ldquoAn axiomatization of full computation tree logicrdquoThe Journal of Symbolic Logic vol 66 no 3 pp 1011ndash1057 2001

[27] K Jensen and L M Kristensen Colored Petri Nets Modellingand Validation of Concurrent Systems Springer 2009

[28] M C Ruiz J Calleja and D Cazorla ldquoPetri nets formalizationof mapreduce paradigm to optimise the performance-costtradeordquo in Proceedings of the IEEE TrustcomBigDataSEISPAvol 3 pp 92ndash99 2015

[29] A V Ratzer LWells HM Lassen et al ldquoCPN tools for editingsimulating and analysing coloured Petri netsrdquo in Applicationsand Theory of Petri Nets 2003 pp 450ndash462 Springer 2003

[30] C Lin andDCMarinescu ldquoStochastic high-level Petri nets andapplicationsrdquo in High-Level Petri Nets pp 459ndash469 SpringerBerlin Germany 1991

[31] D Nurmi R Wolski C Grzegorczyk et al ldquoThe eucalyptusopen-source cloud-computing systemrdquo in Proceedings of the 9thIEEEACM International Symposium on Cluster Computing andtheGrid (CCGRID rsquo09) pp 124ndash131 Shanghai ChinaMay 2009

[32] T White Hadoop The Definitive Guide OrsquoReilly Media 2012[33] J Peng X Zhang Z Lei B ZhangW Zhang and Q Li ldquoCom-

parison of several cloud computing platformsrdquo in Proceedingsof the 2nd International Symposium on Information Science andEngineering pp 23ndash27 IEEE Shanghai China December 2009

[34] J Xu J Tang K Kwiat W Zhang and G Xue ldquoEnhancing sur-vivability in virtualized data centers a service-aware approachrdquoIEEE Journal on Selected Areas in Communications vol 31 no12 pp 2610ndash2619 2013

[35] M Zaharia D Borthakur J S Sarma et al ldquoJob schedulingformultiusermapreduce clustersrdquo Tech RepUCBEECS-2009-55 EECS Department University of California Berkeley CalifUSA 2009

[36] C Lin ldquoA model of systems with shared resources and analysisof approximate performancerdquo Chinese Journal of Computersvol 20 pp 865ndash871 1997

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Submit your manuscripts athttpwwwhindawicom

Computer Games Technology

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Distributed Sensor Networks

International Journal of

Advances in

FuzzySystems

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014

International Journal of

ReconfigurableComputing

Hindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Applied Computational Intelligence and Soft Computing

thinspAdvancesthinspinthinsp

Artificial Intelligence

HindawithinspPublishingthinspCorporationhttpwwwhindawicom Volumethinsp2014

Advances inSoftware EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Electrical and Computer Engineering

Journal of

Journal of

Computer Networks and Communications

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporation

httpwwwhindawicom Volume 2014

Advances in

Multimedia

International Journal of

Biomedical Imaging

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ArtificialNeural Systems

Advances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Computational Intelligence and Neuroscience

Industrial EngineeringJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Human-ComputerInteraction

Advances in

Computer EngineeringAdvances in

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014