Research Article Profit Optimization in SLA-Aware …downloads.hindawi.com/journals/mpe/2014/534510.pdfResearch Article Profit Optimization in SLA-Aware Cloud Services with a Finite

Research ArticleProfit Optimization in SLA-Aware Cloud Serviceswith a Finite Capacity Queuing Model

Yi-Ju Chiang and Yen-Chieh Ouyang

Department of Electrical Engineering National Chung Hsing University Taichung 40227 Taiwan

Correspondence should be addressed to Yen-Chieh Ouyang ycouyangnchuedutw

Received 26 February 2014 Accepted 24 April 2014 Published 5 June 2014

Academic Editor Her-Terng Yau

Copyright copy 2014 Y-J Chiang and Y-C Ouyang This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited

Cloud computing realizes a utility computing paradigm by offering shared resources through an Internet-based computingHowever how a system control can enhance profit and simultaneously satisfy the service level agreements (SLAs) has become oneof themajor interests for cloud providers In this paper a cloud server farm providedwith finite capacity ismodeled as anMM119877119870queuing system Revenue losses are estimated according to the system controls and impatient customer behaviorsThree importantissues are solved in this paper First a profit function is developed in which both the system blocking loss and the user abandonmentloss are evaluated in total revenue A tradeoff between meeting system performances and reducing operating costs is conductedSecond the effects of system capacity control and utilization on various performances of waiting time loss probability and finalarrival rate are demonstrated Finally the proposed optimal profit control (OPC) policy allows a cloud provider tomake the optimaldecision in the number of servers and system capacity so as tomaximize profit As compared to a systemwithout applying the OPCpolicy enhancing providersrsquo profit and improving system performances can be obtained

1 Introduction

Cloud computing is a popular service paradigm recently in aninformation technology (IT) industry that offers infrastruc-ture platform software and so forth as services Consumerscan eliminate the burden of expensive hardwaresoftwarespending and complex infrastructure management Serviceresources (eg networks servers storage applications andservices) are allowed to be dynamically rented with minimalmanagement effort based on consumerrsquos needs rather thantraditional ldquoown-and-userdquo patterns For example AmazonEC2 Googlersquos AppEngineMicrosoftrsquos Azure and so forth aresome typical cloud service providers

Before starting a service there has been a formal con-tract known as service level agreement (SLA) that needs tobe agreed and signed by consumers and cloud providersin advance Quality of service (QoS) is a part of a SLAnegotiation to specify consumerrsquos expectation in terms ofconstraints [1] Mean waiting time expected queuing lengthblocking probability mean time to failure (MTTF) and soforth are all important performance indicators In case of

violating the SLA contract a penalty or compensation isrequired in accordance with seriousness Hence it is crucialto do an accurate performance analysis due to the fact thata performance guarantee plays a crucial role in a service-oriented system

Since no system is accompanied by infinite waiting bufferspace an analytic queuing model with finite capacity issuitable for the study of various server configuration settings[2 3] There are two common situations in a system that willresult in revenue losses First ldquocustomer abandonmentrdquo canbe found from communication systems to network servicesin daily life no exception in cloud computing servicesAbandonment means that an arrival task will leave a systemwithout obtaining service due to long queuing length orlatency Second reducing waiting buffer in a system is oneof the simplest ways to mitigate system congestion andshorten queuing length However the overflowing tasks willbe rejected from the system [4] which will directly result inrevenue losses for a cloud provider

Maintaining performance is usually the prime objec-tive in system administration [5] Besides the relationship

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 534510 11 pageshttpdxdoiorg1011552014534510

2 Mathematical Problems in Engineering

between system control (eg server quantity and waitingbuffer) and user abandonment (eg reneging) on effectivearrival rate and performances must all be taken into accountIn this paper we discuss the problem of profit optimizationin an abandonment system provided with finite capacity Ourcontribution in this paper is threefold as follows

(i) Blocking loss and abandonment loss that are rarelyanalyzed in previous works are considered in ourcloud model to give a more practical analysis inrevenue estimation A profit function is developedby taking resources provisioning cost power con-sumption cost system congestion cost and expectedrevenue into account

(ii) The first presented optimal profit control (OPC)policy combined with a heuristic algorithm allowsa cloud provider to effectively address constrainedoptimization problems Besides the effects of systemcapacities and utilizations on system performancesand loss probabilities are demonstrated

(iii) Simulations show that the optimal decision makingin server quantity and system capacity to maxi-mize profit within a loss probability guarantee canbe obtained by applying the proposed OPC policyEnhancing providersrsquo profit and improving systemperformances can be verified as compared to a systemwithout applying our policy

The rest of the paper is organized as follows Section 2gives a brief overviewof existing researches related to queuingmodels and profit optimization in cloud systems A cloudservice model with system capacity and server provisioningcontrols is presented in Section 3 The effects of systemcapacity and utilization on various performances are alsodemonstrated In Section 4 a profit function is developedbased on the expected revenue analysis and costs estima-tion An algorithm (Algorithm 1) is presented to effectivelysolve constrained optimization problems Section 5 showsthe comparison of experimental results with respect to theoptimal system control within a loss probability constraintFinally some conclusions are discussed in Section 6

2 Related Work

21 Queuing Models with Finite Capacity Researches inqueuing models with finite capacity have a long history herewe refer to [6 7] In [6] the authors addressed a rate controlproblem associated with a single server The controller couldchoose a buffer size for the queuing system and dynamicallycontrol the service rate depending on the current state ofthe system An infinite horizon cost minimization problemwas considered An explicit optimal strategy for the limitingdiffusion control problem was obtained This solution wasthen used to construct an asymptotically optimal controlpolicy

An analytic cost model for MG1N queuing systemswas presented in [7] The costs of customer loss versuscustomer delays by varying buffer size and processor speedwere considered In their analytic study the authors explored

the interplay of queue size customer loss and mean servicetime for various service time distributions However thepossibility of adding multiple servers and some form ofprocessor sharingwere ignored in their work Although cloudcomputing has attracted many research attentions [8ndash10]based on queuing models for a performance analysis fewliteratures took buffer capacity restriction into consideration

In [8] the authors described a novel approximate ana-lytical model for performance evaluation of cloud serverfarms and solved it to obtain accurate estimation of completeprobability distribution request response time and otherimportant performance indicatorsThey also pointed out thataccommodating heterogeneous services in a cloud centermight impose longer waiting time for its clients as comparedto its homogeneous equivalent with the same system utiliza-tion However they only focused on performance analysisno cost analysis or profit evaluation was discussed in theirresearch

In [9] the authors presented algorithms for schedulingservice requests and prioritizing their data accesses in a cloudservice with the main objective of maximizing profit Theprocessor sharing (PS) was used for developing a pricingmodel for clouds and then the data service was modeled asMM1FCFSThey assumed that consumerrsquos service requestsmight pay the service provider to reduce the rate of incomingrequests in order to maintain a satisfactory response time In[10] a cloud center was modeled as an MMmm queuingsystem to conduct a preliminary study of the fault recoveryimpact on a cloud service performance

When a user submitted a service request to the cloud therequest would first arrive at the cloud management system(CMS) which maintained a request queue If the queuewas not full the request would enter the queue otherwiseit would be dropped and the service fails Cloud serviceperformance was quantified by service response time whoseprobability density function was derived This was similarto our queuing model but their work only focused on faulttolerance analysis No system capacity control cost analysisor profit optimization was discussed in their work

22 Profit Optimization in Cloud Systems In [11] the authorshad presented a characterization of cloud federation aimed atenhancing providersrsquo profit and they characterized these deci-sions as a function of several parameters They studied theeffect of these decisions on the providerrsquos profit and evaluatedthe most appropriate providerrsquos configuration depending onthe environment conditions Evaluated parameters includedthe providerrsquos incomingworkload and the cost ofmaintainingthe providerrsquos resources operating Results demonstrated thatlocal resources were preferred over outsourced resourcesthough the latter could enhance the providerrsquos profit when theworkload could not be supported locally

In [12] the authors proposed policies that helped inthe decision-making process to increase resources utilizationand profit Since each provider had restricted amount ofcapacity increasing in load might overload a providerrsquosdata center and result in QoS violation or usersrsquo requestrejectionThiswork attempted to incorporate the outsourcing

Mathematical Problems in Engineering 3

Input Data(1) Arrival rate 120582(2) Potential abandonment index and expected revenue [119889(119904 119905)119873(119904 119905)](3) Cost matrix [119862

119867 119862119882 119862119880 119862119877]

(4) A given execution rate 120583 and a baseline rate 120583119861

(5) Loss probability threshold T(6) The upper bound parameters (119906 and 119887) for server

quantity and system capacity respectivelyOutput Data119877lowast and 119870lowast and 119865(119877lowast 119870lowast)

Step 1 For 119894 = 1 119894 = 119906 119894++Set 119877119894larr 119886 current server quantity

Step 2 For 119895 = 0 119895 = 119887 119895++Set 119870

119895larr 119886 current system capacity

Step 3 Calculate 120588119904 119875119870 119871119902 120582119889 120582lowast and loss probability using

(5)ndash(14) and (15) respectivelyStep 4 If loss probability lt 119879

Then record the current joint value of (119877119894 119870119895) and

identify it as an approved testparameter

ElseReturn to Step 1 and begin to test next

index of parametersEnd

Step 5 When all test parameters have been done119877119894+119886 119870119895+119886 119877

119906 119870119887 larr current approved

parametersBring all revenue and cost parameters into thedeveloped profit function and test all current approvedparameters

Step 6 If a joint value of (119877119894+119886 119870119895+119886

) obtains the maximumprofit value in all testsThen

Output (119877119894+119886 119870119895+119886

) and F(119877119894+119886 119870119895+119886

)Else

Return to Step 5 and begin to test next indexof the approved parameters

End

Algorithm 1 OPC algorithm

problem with option of terminating spot VMs within a datacenter Their objective was to maximize a providerrsquos profit byaccommodating as many on-demand requests as possible

In [13] the authors treated a multiserver system as anMMm queuing model such that the problem of optimalmultiserver configuration for profit maximization in a cloudcomputing environment could be formulated and solved ana-lyticallyTheir pricingmodel took the amount of a service theworkload of an application environment the configurationof a multiserver system the service level agreement and soforth into consideration They also considered two serverspeed and power consumption models namely the idle-speed model and the constant-speed model

In [14] the response time based on different allocationof resources was modeled and used for different servers Thetotal profit in their system was the total price gained fromthe clients subtracted by the cost of operating the activeservers in their systemThe problemwas formulized based on

Generalized Processor Sharing and an elaborate multistageheuristic algorithm was used to solve the problem Howeverprevious studies did not take customer abandonment behav-iors system capacity control loss probability estimation andso forth into consideration To the best of our knowledgeprofit and system control analysis in a cloud computingsystem with abandonment events has not been investigated

3 A Cloud Multiserver Model

31119872119872119877119870 Queuing System We consider a cloud serverfarm with finite waiting buffer and model it as an MM119877119870queuing system [15] The mathematical expressions arederived in detail as follows There are 119877 identical servers inoperation and at most 119870 tasks are allowed in the systemincluding those in services and in queue Task demands arrivefrom an infinite source with mean arrival rate 120582 of Poissondistribution Service times are assumed to be independently


and identically distributed and have an exponential distribu-tion with parameter 120583 The service discipline is first-come-first-served (FCFS) In the cloud server farm let the states 119899represent the number of tasks currently in the system Thevalues of the arrival rate and service rate are taken to be

120582119899=

120582 0 le 119899 le 119870 minus 1

0 119899 ge 119870

120583119899=

119899120583 1 le 119899 le 119877 minus 1

119877120583 119877 le 119899 le 119870

(1)

We denote the notation 119875119899by the probability of 119899 task

currently served in the system (119899 = 0 1 2 infin) and 119875119900

implies the probability that there is no task in system Fora steady-state case the state probability functions 119875

119899can be

obtained from the birth-and-death formula According tothe value 119899 (number of task services) that may happen twosegments are defined by the vector [Segment 1 Segment 2] =[1 le 119899 le 119877 minus 1 119877 le 119899 le 119870] With the expressions in (1) theinitial state probability functions 119875

119899can be derived in terms

of two segments as follows

Segment 1 1 le 119899 le 119877 minus 1

119875119899=1205820sdot 1205821sdot 1205822sdot sdot sdot 120582119899minus1

1205831sdot 1205832sdot 1205833sdot sdot sdot 120583119899

119875119900=120582119899

119899120583119899119875119900 (2)

Segment 2 119877 le 119899 le 119870

119875119899=

120582119899

119877119877119899minus119877120583119899119875119900 (3)

Equations (2) and (3) are the closed forms of the stateprobability functions 119875

119899in which the number of tasks in

service system may happen To obtain 119875119900 (2) and (3) are

brought into the normalizing equation sum119870119899=0119875119899= 1

119870

sum

119899=0

119875119899= 119875119900sdot (1 +

119877minus1

sum

119899=1

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877) (4)

Then the steady-state probability of zero service 119875119900can be

obtained as follows

119875119900= [

119877minus1

sum

119899=0

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877]

minus1

(5)

32 A Designed Cloud Control Model The proposed cloudservice model is provided with finite capacity (denoted by119870) as shown in Figure 1 The system blocking rate effectivearrival rate user abandonment rate and final arrival ratewill be analyzed according to the system capacity and serverprovisioning control Users are allowed to send task demandsinto the service system if there has been some waiting spaceleft (as long as the current number of tasks in buffer are lessthan119870minus119877) Otherwise theywould be rejected by prerejectingmechanism (PRM) and lost

PRM is used to control and block excess tasks in advancebefore they are sent into the system It is reasonable to

assume that if customers are rejected before they send taskdemands a service provider has no responsibility to give anycompensation Therefore rejection penalties can be avoided[16] in this proposed model with PRM A cloud systemcontroller must distinguish carefully between the originalarrival rate and the effective arrival rate denoted by 120582

120572 the

relevant notations are described in Notation SectionThe fraction of rejecting arrival tasks in a steady state are

referred to as the blocking probability denoted by 119875119870 This

can be obtained by giving 119899 = 119870 in (3) as follows

119875119870=

120582119870

119877 sdot 119877119870minus119877120583119870 119875

119900=

120588119870

119877 sdot 119877119870minus119877119875119900 (6)

The effective arrival rate can be obtained as

120582120572= 120582 (1 minus 119875

119870) (7)

Customers that are rejected still can come back later aftera random time and they will be treated as a new arrival insubsequent periods

33 Abandonment Rate When there are no idle serversavailable an accepted task will be forwarded to a queue andwait until all tasks in front have completed their requiredservices Hence using (2) and (3) the expected queuinglength 119871

119902can be obtained as

119871119902=

119870

sum

119899=119877

(119899 minus 119877) 119875119899 (8)

To find the predicted waiting time119882119902in queue we apply the

well-known Littlersquos law [15] which is a widely used formulain queuing theory It states that the average number of itemsin queue is equal to the average arrival rate multiplied by theexpected waiting time Historically it can be written as

119871119902= 120582119882119902 (9)

We get the expected waiting time in queue prior to service as

119882119902=

119871119902

120582 (1 minus 119875119870) (10)

ldquoBalkingrdquo and ldquorenegingrdquo are used to describe customersrsquoabandonment behavior in queuing systems Balking meansthat tasks leave a system without getting service due tolong queuing length Unlike balking case customers alwaysrenege after facing a long waiting time in a queue In ourservice system the waiting-time information is sent to eacharrival for the purpose of enhancing service quality [1718] After being notified some impatient customers mayfeel that waiting time is too long to endure and they willabandon the system without obtaining service There arevarious reneging rules presented by previous researchers [19]Waiting time is usually the main factor to affect customerrsquosdecisions In addition reneging events also depend on somepotential factors Therefore a cloud provider will record theabandonment rate and update the latest data per planning


Arrival rate 120582Throughput

rateW

System capacity

Rejection rate Waiting-timeinformationAbandonment rate

Prerejecting mechanism

System capacity K

Dispatcher

Server quantity

Effective arrivalrate

OPC policy

Figure 1 A designed cloud service model with the system capacity and server provisioning controls

period in this proposed model to analyze the severity ofabandonment rate The notation 120582

119889is defined as the average

abandonment rate for a certain service pattern 119904 (while 119904 isin 119878)at period 119905 (while 119905 isin 119879) Then taking 120582

119889divided by the

predicted waiting time and effective arrival rate the potentialabandonment index can be obtained as

119889 (119904 119905) =120582119889

119882119902times 120582120572

(11)

The abandonment probability can be calculated as theexpected waiting time119882

119902multiplying by 119889(119904 119905)

119875119889= 119882119902times 119889 (119904 119905) =

120582119889

120582120572

(12)

34 Loss Probability Users decide to accept the cloud servicewith probability 1 minus 119875

119889 or abandon this cloud service with

probability 119875119889 Then the expected abandonment rate 120582

119889can

be obtained as

120582119889= 120582120572times 119875119889= 120582 (1 minus 119875

119870) times 119882

119902times 119889 (119904 119905) (13)

According to historical records we obtain the mean finalarrival rate 120582lowast that really wants to receive service as

120582lowast= 120582120572minus 120582119889= 120582120572minus 120582120572119875119889

= 120582120572(1 minus 119875

119889) = 120582 (1 minus 119875

119870) (1 minus 119875

119889)

(14)

Finally the loss probability can be obtained as

Loss probability = 120582 minus 120582lowast

120582=120582 [1 minus (1 minus 119875

119870) (1 minus 119875

119889)]

120582

= 119875119870+ 119875119889minus 119875119870sdot 119875119889

(15)

To gain more insight into the designed system behaviorand compare system performances among different capaci-ties several experiments are demonstrated It is assumed that119877 = 20 120583 = 45 and abandonment index = 001 while

the system capacity 119870 is made a variable from 119870 = 15119877 to119870 = 18119877 (ranging from 30 32 and 34 to 36) in four stepsand different arrival rates are considered It is known that asystem provided with more waiting buffer can reduce systemblocking probability however it will result in long waitingtime as shown in Figure 2(a)

Final arrival rate distribution under various systemcapacities and arrival rates is shown in Figure 2(b) It is notedthat final arrival rate reduces as system capacity and arrivalrate increase This is due to the fact that an arriving task isnot very likely to obtain service immediately when a system iscongested instead it has to wait in queue In other words thesituation of final arrival rate decrease is caused by providingexcessive waiting buffer with fixed servers which will directlyincrease the user abandonment rate Figure 3 demonstratesthe final arrival rate when the number of servers is increasedto 24 as compared to Figure 2(b) It is noted that the finalarrival rates increase as arrival rate increases however itstarts to decrease as arrival rate further increases

Results show that although final arrival rate can beincreased by providing more servers it will result in higherserver provisioning cost Therefore comprehensive evalua-tions for system losses and operational cost are necessary Inthe following we compare the system loss probability in anabandonment system with a nonabandonment system Theeffects of various system utilizations and system capacitieson the loss probability are studied It is assumed that systemutilizations = 065 07 075 and 08 and a cloud server farmis configured with 64 servers The system capacity is madevariable from 119877 + 1 to 2119877 (ranging from 65 to 128) and 120582 =2500min Loss probability in a nonabandonment system isshown in Figure 4(a) It is noticeable that keeping system in ahigher utilization will lead to a higher loss probability whilethe loss probability decreases rapidly as the system capacityfurther increases

In an abandonment system a loss probability directlydepends on the system blocking loss and user abandonmentloss Figure 4(b) demonstrates the loss probability whenan abandonment index is assumed value of 001 Resultsshow that the loss probabilities become stable rather thandecreasing rapidly (see Figure 4(a)) as the system capacity


1100 1250 1400 1550 1700 1850 2000

04

05

06

07

08

09

1

Arrival rate

Wai

ting

time i

n qu

eue (

s)

K = 30

K = 32

K = 34

K = 36

(a)

1100 1250 1400 1550 1700 1850 2000

K = 30

K = 32

K = 34

K = 36

890

895

900

905

910

915

Arrival rate

Fina

l arr

ival

rate

(b)

Figure 2 Waiting time and final arrival rate under various system capacities

further increases It seems more real as compared to anonabandonment system since a loss probability in a servicesystem is less likely to approach zero Although reducingthe system capacity can effectively reduce waiting time (seeFigure 2(a)) it causes a higher loss probability when a systemis under a higher utilizationTherefore conducting a tradeoffanalysis in a cloud system is essential to achieve a successfulcontrol for different performance levels

4 Revenue and Cost Analysis

41 Revenue Function Profit is one of the most importantindicators of how well a business development is no excep-tion in a cloud computing industry A value of expectedrevenue per task demand is denoted by 119873(119904 119905) for a servicepattern 119904 (119878 = 1 2 3 119904) at period 119905 (119879 = 1 2 3 119905)It is known that expecting more profit comes from revenueexpansion cost reduction or both simultaneously The totalrevenue is the original expected revenue minus the blockingloss and user abandonment loss which is equivalent tothe value of final arrival rate multiplying 119873(119904 119905) Then anexpected total revenue 119877(119877119870) per unit time under 119877 serverswith capacity119870 can be written as

119877 (119877119870) = 120582120572 (1 minus 119875119903)119873 (119904 119905)

= 120582 (1 minus 119875119870) (1 minus 119875

119903)119873 (119904 119905)

(16)

42 Power Consumption Cost Not only server provisioningbut also operating power consumption is major cost burdensin a cloud system Typically the power consumed by a CPUis approximately proportional to a CPU frequency 119891 and to

1200 1400 1600 1800 20001070

1080

1090

1100

1110

1120

1130

Arrival rate

Fina

l arr

ival

rate

K = 30K = 32

K = 34K = 36

Figure 3 Final arrival rate under various system capacities when119877 = 24

the square of a CPU voltage 119881 defined by 119875 = 119886 sdot 119862 sdot 1198812 sdot 119891where ldquo119862rdquo is the capacitance being switched per clock cycleSince most gates do not operate at every clock cycle theyare often accompanied by an activity factor ldquo119886rdquo [20ndash23] Ifa processor execution rate is not given in accordance with


System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)

Figure 4 Loss probabilities (a) in a nonabandonment system and (b) in an abandonment system under various 120588 and 119870 values

Table 1 Cost and system parameters

Parameter Description119862119877

Server provisioning cost per server per unit time120576 Power consumption cost per Watt per unit time120583119887

A baseline service rate120583 A given execution rate (120583 gt 120583

119887)

1205831015840 An increased execution rate (120583 minus 120583

119887)

119862119867

Cost incurred by holding tasks in buffer per unittime

119862119882

Cost incurred by tasks waiting in buffer per unit time

119862119880

Cost incurred by providing per waiting buffer spaceper unit time

a baseline rate 120583119887 the operating cost of accelerating rate 1205831015840

needs to be calculated It is known that the voltage and theclock frequency are related to 119881 prop 119891

119909 (119909 gt 0) for an idealcase Therefore both 1205831015840 = 119911119891 and 119881 = 119910119891

119909 (119909 119910 gt 0)are set to facilitate the discussion of execution rate powerconsumption [24] The power consumption 119875 can be writtenas119875 = 119886sdot119862sdot1198812 sdot119891 = 119886sdot119862sdot1199102 sdot1198912119909+1 = 119886sdot119862sdot1199102 sdot(1205831015840119911)2119909+1 = 1205931205831015840120592where120593 = 119886sdot119862sdot11991021199112119909+1 and 120592 = 2119909+1The related notationsare listed in Table 1

Notation 119861 is denoted by the power consumption ofbaseline execution rate and notation 120576 is denoted by theincurred cost per Watt per unit time as presented in Table 1Let 120583 denote the given execution rate (120583 gt 120583

119887) then

the expected power consumption cost 119881(119877 120583) per unit time

under 119877 servers with the given execution rate 120583 can beobtained as

119881 (119877 120583) = 120588lowastminus1120576119877 (120593(120583 minus 120583

119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)

Besides it is known that a systemutilization denoted by120588lowast isalso an important factor to affect resources provisioning cost[25 26]

43 System Congestion Cost Service applications for an on-demand service pattern are typically time sensitive There-fore cloud providers have the responsibility to make a com-pensation for differentiated levels of service As presented inTable 1 cost items that are considered in a system congestionfunction include 119862

119882(incurred by tasks waiting in a queue)

119862119867(incurred by holding tasks in a queue) and 119862

119880(incurred

by providing per waiting buffer space) Hence a systemcongestion cost denoted by 119878(119877119870) under 119877 servers withcapacity 119870 can be obtained as

119878 (119877119870) = 119862119882119882lowast+ 119862119867119871lowast+ 119862119880 (119870 minus 119877) (18)

44 Profit Function After constructing the revenue and costfunctions an expected profit function per unit time can bedeveloped Our objective is to determine an optimal jointvalue of 119877lowast and119870lowast under a given execution rate 120583 in a cloud


server farm so as tomaximize profitTheprofitmaximization(PM) problem can be presented mathematically as

Maximize PM

Subject to 0 le 120583119887lt 120583

Loss probability lt 119879

where PM = 119865 (119877119870)

= 119877 (119877119870) minus 120588lowastminus1119877119862119877minus 120588lowastminus1119881 (119877 120583) minus 119878 (119877119870)

= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877

minus 120588lowastminus1120576119877 (120593(120583 minus 120583

119887)120592+ 119861) minus 119862

119882119882lowast

minus 119862119867119871lowastminus 119862119880120572

(19)

It is known that a loss probability is one of the majorconcerns for customers since no one wants to be rejectedor leave because of facing long waiting time In this work aSLA is specified by the following relationship loss probabilityle 119879 where 119879 is the maximum threshold value It is extremelydifficult to obtain the analytical results for the optimal value(119877lowast 119870lowast) due to the fact that this profit function is nonlinearandhighly complex Instead the optimal profit control (OPC)algorithm is presented to find the optimal solution For theOPC policy satisfying the SLA constraint has the highestpriority in determining the optimal solution

5 Numerical Validation

Experiments are conducted to validate that (i) the optimalresources provisioning can be obtained by implementingthe proposed heuristic algorithms and show that the OPCpolicy is practical and (ii) more profit gaining and significantperformance improvement can be achieved by applying theOPC policy as compared to a general method which isgiven only by considering a performance guarantee Simula-tions are demonstrated by considering the following systemparameters and cost parameters 120576 = 01 120592 = 2 120593 = 2119861 = 8 120583

119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001

and all computational programs are coded by MATLABTheOPCheuristic algorithm is applied to search the optimal jointsolution of the number of servers and system capacity withinthe loss probability guarantee of119879 = 001 in a SLA constraint

Profit distribution under various number of servers andsystem capacities is shown in Figure 5 As can be seenprofit increases as the number of servers and system capacityincrease in the beginning However as the server quantity orsystem capacity further exceeds a certain value it will causeno more increasing in profit and begin to decrease graduallyFigure 6 shows the loss probability distribution under variousnumber of servers and system capacities As can be seenthe loss probability decreases obviously as the number ofservers increases The effect of server quantity on reducingloss probability is more obvious than system capacity It is

99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106

System capacityNumber of servers

Profi

t

F(103 119) = 1094322

109114119124129134139

Figure 5 Profit distribution under various system capacities and thenumber of servers

109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056

Figure 6 Loss probability constraint

also noted that only when the number of servers is larger than102 a system can satisfy the 119879 = 001 constraint Within the119879 = 001 constraint the maximum profit of 1094322 and lossprobability of 056 can be obtained at the optimal solution(119877lowast 119870lowast) = (103 119)

Next the proposed OPC policy is evaluated on the basisof comparisons with a general method It implies that a cloudresources provisioning is controlled based on a performanceguaranteethreshold (in most cloud system managementalgorithmsmethods [27ndash29]) For brevity here it is referredto as a non-OPC policy since no system loss evaluation and


0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30

Figure 7 Loss probability comparisons between the OPC and non-OPC policies

120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373

Figure 8 Abandonment rate comparisons between the OPC and non-OPC policies

profit optimality analysis are considered Here simulationsdiffer from previous investigations since we try to verify thatthe OPC policy is also applicable if a system is providedwith a fixed capacity A system with different arrival ratesof 980 990 and 1000 is considered and others use the sameparameters as previous simulations

A system capacity is fixed by 36 and both policies needto comply with the same constraint of 119879 = 01 Lossprobability and abandonment rate comparisons are shownin Figures 7 and 8 respectively The solutions determinedby a non-OPC policy are 0099 0069 and 0063 to meet119879 = 01 constraint respectively It is noted that the number


25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298

Figure 9 Comparisons of profit between the OPC and non-OPCpolicies

of servers determined by a non-OPC policy is 25 26 and 26respectively which is fewer than the OPC policy since thegeneral solution is determined only based on its performanceguarantee for the purpose of reducing server provisioningcost and power consumption cost Although the number ofservers controlled by the OPC policy is larger than a non-OPC policy it can obtain lower loss probability and alleviateuser abandonment rate

The abandonment rates are 734 829 and 935 respec-tively for the OPC policy and 7417 4982 and 4373respectively for the non-OPC policy Besides more profitalso can be achieved as shown in Figure 9 The profit valuesare 258383 261179 and 263905 respectively for the OPCpolicy and 238298 251846 and 252123 respectively for anon-OPC policy Finally the QoS improvement rates aremeasured which calculate the relative value of improvementsto an original value instead of an absolute value the results areshown in Figure 10 As can be seen applying the OPC policynot only can obtain more profit but also has the potentialto greatly improve various performances including waitingtime abandonment rate blocking rate and loss probability

6 Conclusions

Developing a successful cloud service system depends onaccurate performance evaluations and effective system con-trols Enhancing profit and simultaneously satisfying the SLAconstraint have become one of the major interests for acloud provider In this paper the relationship between systemcontrols and user abandonment on the loss probability is

Waiting time Reneging rate Blocking rate Loss probability0

10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000

Figure 10 QoS improvement rates by applying the OPC policy

estimated in the designed model The effects of varioussystem capacities and utilizations on the waiting time finalarrival rate and system loss probability are studied Ourmain goal is to maximize profit under a loss probabilityguarantee Revenue is evaluated by taking system blockingloss abandonment loss and final arrival rate into con-sideration The first proposed OPC policy combined witha heuristic algorithm allows cloud providers to effectivelyconduct the server quantity and the system capacity controlsIt also contributes to addressing the tradeoffproblembetweenmaintaining systemperformances and enhancing profit Sim-ulation results show that the effectiveness of the OPC policycan be validated The benefits of enhancing providersrsquo profitand improving performances can be achieved as compared toa general method

Notations

119875119870 Blocking probability while system capacity is119870

120582120572 Efficient arrival rate where tasks originally look

forward to receiving service119889(119904 119905) Potential abandonment index depended on

historical records at period 119905 (while 119905 isin 119879) for aservice type 119904 (while 119904 isin 119878)

119875119889 Potential abandonment probability which

would be expressed as a function of 119889(119904 119905) and119882119902

120582119889 Abandonment rate which would be expressed

as a function of 119889(119904 119905)119882119902 and 120582

120572

120582lowast Expected final arrival rate119871lowast Mean final queuing length119882lowast Mean final waiting time

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper


References

[1] I Foster Y Zhao I Raicu and S Lu ldquoCloud computing andgrid computing 360-degree comparedrdquo in Proceedings of theGrid Computing Environments Workshop (GCE rsquo08) pp 1ndash10November 2008

[2] A Chydzinski and B Adamczyk ldquoTransient and stationarylosses in a finite-buffer queue with batch arrivalsrdquoMathematicalProblems in Engineering vol 2012 Article ID 326830 17 pages2012

[3] B Dragovic N-K Park N Zrnic and R Mestrovic ldquoMath-ematical models of multiserver queuing system for dynamicperformance evaluation in portrdquo Mathematical Problems inEngineering vol 2012 Article ID 710834 19 pages 2012

[4] P Fitzpatrick C S Lee and B Warfield ldquoTeletraffic perfor-mance of mobile radio networks with hierarchical cells andoverflowrdquo IEEE Journal on Selected Areas in Communicationsvol 15 no 8 pp 1549ndash1557 1997

[5] F E Sandnes ldquoSecure distributed configuration managementwith randomised scheduling of system-administration tasksrdquoIEICE Transactions on Information and Systems vol 86 no 9pp 1601ndash1610 2003

[6] A P Ghosh and A P Weerasinghe ldquoOptimal buffer size anddynamic rate control for a queueing system with impatientcustomers in heavy trafficrdquo Stochastic Processes and TheirApplications vol 120 no 11 pp 2103ndash2141 2010

[7] D Doran L Lipsky and S Thompson ldquoCost-based optimiza-tion of buffer size in MG1N systems under different service-time distributionsrdquo in Proceedings of the 9th IEEE InternationalSymposium on Network Computing and Applications (NCA rsquo10)pp 28ndash35 July 2010

[8] H Khazaei J Misic and V B Misic ldquoPerformance analysis ofcloud computing centers using MGmm+r queuing systemsrdquoIEEE Transactions on Parallel and Distributed Systems vol 23no 5 pp 936ndash943 2012

[9] Y C Lee CWang A Y Zomaya and B B Zhou ldquoProfit-drivenscheduling for cloud services with data access awarenessrdquoJournal of Parallel and Distributed Computing vol 72 no 4 pp591ndash602 2012

[10] B Yang F Tan Y Dai and S Guo ldquoPerformance evaluation ofcloud service considering fault recoveryrdquo in Proceedings of the1st International Conference on Cloud Computing (CloudComrsquo09) pp 571ndash576 2009

[11] I Goiri J Guitart and J Torres ldquoCharacterizing cloud federa-tion for enhancing providersrsquo profitrdquo in Proceedings of the 3rdIEEE International Conference on Cloud Computing (CLOUDrsquo10) pp 123ndash130 July 2010

[12] A N Toosi R N Calheiros R K Thulasiram and R BuyyaldquoResource provisioning policies to increase IaaS providerrsquosprofit in a federated cloud environmentrdquo in Proceedings ofthe 13th IEEE International Conference on High PerformanceComputing and Communications (HPCC rsquo11) pp 279ndash287September 2011

[13] J Cao KHwang K Li andA Y Zomaya ldquoOptimalmultiserverconfiguration for profit maximization in cloud computingrdquoIEEE Transactions on Parallel and Distributed Systems vol 24no 6 pp 1087ndash1096 2013

[14] H Goudarzi and M Pedram ldquoMaximizing profit in cloudcomputing system via resource allocationrdquo in Proceedings of the31st International Conference on Distributed Computing SystemsWorkshops (ICDCSW rsquo11) pp 1ndash6 June 2011

[15] D Gross J F Shortle J M Thompson and C M HarrisFundamentals of Queueing Theory Wiley Series in Probabilityand Statistics John Wiley amp Sons Hoboken NJ USA 4thedition 2008

[16] Y Bartal S Leonardi A Marchetti-Spaccamela J Sgall andL Stougie ldquoMultiprocessor scheduling with rejectionrdquo SIAMJournal on Discrete Mathematics vol 13 no 1 pp 64ndash78 2000

[17] P Afeche and H Mendelson ldquoPricing and priority auctionsin queueing systems with a generalized delay cost structurerdquoManagement Science vol 50 no 7 pp 869ndash882 2004

[18] P Guo and P Zipkin ldquoAnalysis and comparison of queues withdifferent levels of delay informationrdquoManagement Science vol53 no 6 pp 962ndash970 2007

[19] J Ou and B M Rao ldquoBenefits of providing amenities to impa-tient waiting customersrdquoComputers ampOperations Research vol30 no 14 pp 2211ndash2225 2003

[20] A P Chandrakasan S Sheng and R W Brodersen ldquoLow-power CMOS digital designrdquo IEEE Journal of Solid-State Cir-cuits vol 27 no 4 pp 473ndash484 1992

[21] B Zhai D Blaauw D Sylvester and K Flautner ldquoTheoreticaland practical limits of dynamic voltage scalingrdquo in Proceedingsof the 41st Design Automation Conference pp 868ndash873 June2004

[22] CPU Power Dissipation httpenwikipediaorgwikiCPUpower dissipation

[23] Central Processing Unit httpenwikipediaorgwikiCentralprocessing unit


[25] 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 149ndash1672010

[26] Y C Lee and A Y Zomaya ldquoEnergy efficient utilization ofresources in cloud computing systemsrdquo Journal of Supercomput-ing vol 60 no 2 pp 268ndash280 2012

[27] L Mao Y Yang H Xu and Y Chen ldquoService selectionalgorithm based on constraint for cloud workflow systemrdquoJournal of Software vol 8 no 5 pp 1124ndash1131 2013

[28] G Le K Xu and J Song ldquoDynamic resource provisioningand scheduling with deadline constraint in elastic cloudrdquo inProceedings of the International Conference on Service Science(ICSS rsquo13) pp 113ndash117 April 2013

[29] B Li A M Song and J Song ldquoA distributed QoS-constrainttask scheduling scheme in cloud computing environmentmodel and algorithmrdquo Advances in Information Sciences andService Sciences vol 4 no 5 pp 283ndash291 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


between system control (eg server quantity and waitingbuffer) and user abandonment (eg reneging) on effectivearrival rate and performances must all be taken into accountIn this paper we discuss the problem of profit optimizationin an abandonment system provided with finite capacity Ourcontribution in this paper is threefold as follows

(i) Blocking loss and abandonment loss that are rarelyanalyzed in previous works are considered in ourcloud model to give a more practical analysis inrevenue estimation A profit function is developedby taking resources provisioning cost power con-sumption cost system congestion cost and expectedrevenue into account

(ii) The first presented optimal profit control (OPC)policy combined with a heuristic algorithm allowsa cloud provider to effectively address constrainedoptimization problems Besides the effects of systemcapacities and utilizations on system performancesand loss probabilities are demonstrated

(iii) Simulations show that the optimal decision makingin server quantity and system capacity to maxi-mize profit within a loss probability guarantee canbe obtained by applying the proposed OPC policyEnhancing providersrsquo profit and improving systemperformances can be verified as compared to a systemwithout applying our policy

The rest of the paper is organized as follows Section 2gives a brief overviewof existing researches related to queuingmodels and profit optimization in cloud systems A cloudservice model with system capacity and server provisioningcontrols is presented in Section 3 The effects of systemcapacity and utilization on various performances are alsodemonstrated In Section 4 a profit function is developedbased on the expected revenue analysis and costs estima-tion An algorithm (Algorithm 1) is presented to effectivelysolve constrained optimization problems Section 5 showsthe comparison of experimental results with respect to theoptimal system control within a loss probability constraintFinally some conclusions are discussed in Section 6

2 Related Work

21 Queuing Models with Finite Capacity Researches inqueuing models with finite capacity have a long history herewe refer to [6 7] In [6] the authors addressed a rate controlproblem associated with a single server The controller couldchoose a buffer size for the queuing system and dynamicallycontrol the service rate depending on the current state ofthe system An infinite horizon cost minimization problemwas considered An explicit optimal strategy for the limitingdiffusion control problem was obtained This solution wasthen used to construct an asymptotically optimal controlpolicy

An analytic cost model for MG1N queuing systemswas presented in [7] The costs of customer loss versuscustomer delays by varying buffer size and processor speedwere considered In their analytic study the authors explored

the interplay of queue size customer loss and mean servicetime for various service time distributions However thepossibility of adding multiple servers and some form ofprocessor sharingwere ignored in their work Although cloudcomputing has attracted many research attentions [8ndash10]based on queuing models for a performance analysis fewliteratures took buffer capacity restriction into consideration

In [8] the authors described a novel approximate ana-lytical model for performance evaluation of cloud serverfarms and solved it to obtain accurate estimation of completeprobability distribution request response time and otherimportant performance indicatorsThey also pointed out thataccommodating heterogeneous services in a cloud centermight impose longer waiting time for its clients as comparedto its homogeneous equivalent with the same system utiliza-tion However they only focused on performance analysisno cost analysis or profit evaluation was discussed in theirresearch

In [9] the authors presented algorithms for schedulingservice requests and prioritizing their data accesses in a cloudservice with the main objective of maximizing profit Theprocessor sharing (PS) was used for developing a pricingmodel for clouds and then the data service was modeled asMM1FCFSThey assumed that consumerrsquos service requestsmight pay the service provider to reduce the rate of incomingrequests in order to maintain a satisfactory response time In[10] a cloud center was modeled as an MMmm queuingsystem to conduct a preliminary study of the fault recoveryimpact on a cloud service performance

When a user submitted a service request to the cloud therequest would first arrive at the cloud management system(CMS) which maintained a request queue If the queuewas not full the request would enter the queue otherwiseit would be dropped and the service fails Cloud serviceperformance was quantified by service response time whoseprobability density function was derived This was similarto our queuing model but their work only focused on faulttolerance analysis No system capacity control cost analysisor profit optimization was discussed in their work

22 Profit Optimization in Cloud Systems In [11] the authorshad presented a characterization of cloud federation aimed atenhancing providersrsquo profit and they characterized these deci-sions as a function of several parameters They studied theeffect of these decisions on the providerrsquos profit and evaluatedthe most appropriate providerrsquos configuration depending onthe environment conditions Evaluated parameters includedthe providerrsquos incomingworkload and the cost ofmaintainingthe providerrsquos resources operating Results demonstrated thatlocal resources were preferred over outsourced resourcesthough the latter could enhance the providerrsquos profit when theworkload could not be supported locally

In [12] the authors proposed policies that helped inthe decision-making process to increase resources utilizationand profit Since each provider had restricted amount ofcapacity increasing in load might overload a providerrsquosdata center and result in QoS violation or usersrsquo requestrejectionThiswork attempted to incorporate the outsourcing



119867 119862119882 119862119880 119862119877]





Step 2 For 119895 = 0 119895 = 119887 119895++Set 119870













Output (119877119894+119886 119870119895+119886

) and F(119877119894+119886 119870119895+119886

)Else


End










120582119899=

120582 0 le 119899 le 119870 minus 1

0 119899 ge 119870

120583119899=

119899120583 1 le 119899 le 119877 minus 1

119877120583 119877 le 119899 le 119870

(1)




119899can be







119875119900=120582119899

119899120583119899119875119900 (2)

Segment 2 119877 le 119899 le 119870

119875119899=

120582119899

119877119877119899minus119877120583119899119875119900 (3)





119870

sum

119899=0

119875119899= 119875119900sdot (1 +

119877minus1

sum

119899=1

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877) (4)


obtained as follows

119875119900= [

119877minus1

sum

119899=0

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877]

minus1

(5)




120572 the




119875119870=

120582119870

119877 sdot 119877119870minus119877120583119870 119875

119900=

120588119870

119877 sdot 119877119870minus119877119875119900 (6)


120582120572= 120582 (1 minus 119875

119870) (7)




119871119902=

119870

sum

119899=119877

(119899 minus 119877) 119875119899 (8)



119871119902= 120582119882119902 (9)


119882119902=

119871119902

120582 (1 minus 119875119870) (10)




rateW

System capacity



System capacity K

Dispatcher

Server quantity


OPC policy







119889 (119904 119905) =120582119889

119882119902times 120582120572

(11)


119902multiplying by 119889(119904 119905)

119875119889= 119882119902times 119889 (119904 119905) =

120582119889

120582120572

(12)




119889can

be obtained as

120582119889= 120582120572times 119875119889= 120582 (1 minus 119875

119870) times 119882

119902times 119889 (119904 119905) (13)



= 120582120572(1 minus 119875

119889) = 120582 (1 minus 119875

119870) (1 minus 119875

119889)

(14)



120582=120582 [1 minus (1 minus 119875

119870) (1 minus 119875

119889)]

120582

= 119875119870+ 119875119889minus 119875119870sdot 119875119889

(15)







1100 1250 1400 1550 1700 1850 2000

04

05

06

07

08

09

1

Arrival rate

Wai

ting

time i

n qu

eue (

s)

K = 30

K = 32

K = 34

K = 36

(a)

1100 1250 1400 1550 1700 1850 2000

K = 30

K = 32

K = 34

K = 36

890

895

900

905

910

915

Arrival rate

Fina

l arr

ival

rate

(b)





119877 (119877119870) = 120582120572 (1 minus 119875119903)119873 (119904 119905)

= 120582 (1 minus 119875119870) (1 minus 119875

119903)119873 (119904 119905)

(16)


1200 1400 1600 1800 20001070

1080

1090

1100

1110

1120

1130

Arrival rate

Fina

l arr

ival

rate

K = 30K = 32

K = 34K = 36




System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)






119887)


119887)

119862119867


119862119882


119862119880







119887) then




119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)





119880(incurred






Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119867 119862119882 119862119880 119862119877]





Step 2 For 119895 = 0 119895 = 119887 119895++Set 119870













Output (119877119894+119886 119870119895+119886

) and F(119877119894+119886 119870119895+119886

)Else


End










120582119899=

120582 0 le 119899 le 119870 minus 1

0 119899 ge 119870

120583119899=

119899120583 1 le 119899 le 119877 minus 1

119877120583 119877 le 119899 le 119870

(1)




119899can be







119875119900=120582119899

119899120583119899119875119900 (2)

Segment 2 119877 le 119899 le 119870

119875119899=

120582119899

119877119877119899minus119877120583119899119875119900 (3)





119870

sum

119899=0

119875119899= 119875119900sdot (1 +

119877minus1

sum

119899=1

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877) (4)


obtained as follows

119875119900= [

119877minus1

sum

119899=0

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877]

minus1

(5)




120572 the




119875119870=

120582119870

119877 sdot 119877119870minus119877120583119870 119875

119900=

120588119870

119877 sdot 119877119870minus119877119875119900 (6)


120582120572= 120582 (1 minus 119875

119870) (7)




119871119902=

119870

sum

119899=119877

(119899 minus 119877) 119875119899 (8)



119871119902= 120582119882119902 (9)


119882119902=

119871119902

120582 (1 minus 119875119870) (10)




rateW

System capacity



System capacity K

Dispatcher

Server quantity


OPC policy







119889 (119904 119905) =120582119889

119882119902times 120582120572

(11)


119902multiplying by 119889(119904 119905)

119875119889= 119882119902times 119889 (119904 119905) =

120582119889

120582120572

(12)




119889can

be obtained as

120582119889= 120582120572times 119875119889= 120582 (1 minus 119875

119870) times 119882

119902times 119889 (119904 119905) (13)



= 120582120572(1 minus 119875

119889) = 120582 (1 minus 119875

119870) (1 minus 119875

119889)

(14)



120582=120582 [1 minus (1 minus 119875

119870) (1 minus 119875

119889)]

120582

= 119875119870+ 119875119889minus 119875119870sdot 119875119889

(15)







1100 1250 1400 1550 1700 1850 2000

04

05

06

07

08

09

1

Arrival rate

Wai

ting

time i

n qu

eue (

s)

K = 30

K = 32

K = 34

K = 36

(a)

1100 1250 1400 1550 1700 1850 2000

K = 30

K = 32

K = 34

K = 36

890

895

900

905

910

915

Arrival rate

Fina

l arr

ival

rate

(b)





119877 (119877119870) = 120582120572 (1 minus 119875119903)119873 (119904 119905)

= 120582 (1 minus 119875119870) (1 minus 119875

119903)119873 (119904 119905)

(16)


1200 1400 1600 1800 20001070

1080

1090

1100

1110

1120

1130

Arrival rate

Fina

l arr

ival

rate

K = 30K = 32

K = 34K = 36




System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)






119887)


119887)

119862119867


119862119882


119862119880







119887) then




119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)





119880(incurred






Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120582119899=

120582 0 le 119899 le 119870 minus 1

0 119899 ge 119870

120583119899=

119899120583 1 le 119899 le 119877 minus 1

119877120583 119877 le 119899 le 119870

(1)




119899can be







119875119900=120582119899

119899120583119899119875119900 (2)

Segment 2 119877 le 119899 le 119870

119875119899=

120582119899

119877119877119899minus119877120583119899119875119900 (3)





119870

sum

119899=0

119875119899= 119875119900sdot (1 +

119877minus1

sum

119899=1

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877) (4)


obtained as follows

119875119900= [

119877minus1

sum

119899=0

120582119899

120583119899119899+

119870

sum

119899=119877

120582119899

120583119899119877119877119899minus119877]

minus1

(5)




120572 the




119875119870=

120582119870

119877 sdot 119877119870minus119877120583119870 119875

119900=

120588119870

119877 sdot 119877119870minus119877119875119900 (6)


120582120572= 120582 (1 minus 119875

119870) (7)




119871119902=

119870

sum

119899=119877

(119899 minus 119877) 119875119899 (8)



119871119902= 120582119882119902 (9)


119882119902=

119871119902

120582 (1 minus 119875119870) (10)




rateW

System capacity



System capacity K

Dispatcher

Server quantity


OPC policy







119889 (119904 119905) =120582119889

119882119902times 120582120572

(11)


119902multiplying by 119889(119904 119905)

119875119889= 119882119902times 119889 (119904 119905) =

120582119889

120582120572

(12)




119889can

be obtained as

120582119889= 120582120572times 119875119889= 120582 (1 minus 119875

119870) times 119882

119902times 119889 (119904 119905) (13)



= 120582120572(1 minus 119875

119889) = 120582 (1 minus 119875

119870) (1 minus 119875

119889)

(14)



120582=120582 [1 minus (1 minus 119875

119870) (1 minus 119875

119889)]

120582

= 119875119870+ 119875119889minus 119875119870sdot 119875119889

(15)







1100 1250 1400 1550 1700 1850 2000

04

05

06

07

08

09

1

Arrival rate

Wai

ting

time i

n qu

eue (

s)

K = 30

K = 32

K = 34

K = 36

(a)

1100 1250 1400 1550 1700 1850 2000

K = 30

K = 32

K = 34

K = 36

890

895

900

905

910

915

Arrival rate

Fina

l arr

ival

rate

(b)





119877 (119877119870) = 120582120572 (1 minus 119875119903)119873 (119904 119905)

= 120582 (1 minus 119875119870) (1 minus 119875

119903)119873 (119904 119905)

(16)


1200 1400 1600 1800 20001070

1080

1090

1100

1110

1120

1130

Arrival rate

Fina

l arr

ival

rate

K = 30K = 32

K = 34K = 36




System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)






119887)


119887)

119862119867


119862119882


119862119880







119887) then




119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)





119880(incurred






Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











rateW

System capacity



System capacity K

Dispatcher

Server quantity


OPC policy







119889 (119904 119905) =120582119889

119882119902times 120582120572

(11)


119902multiplying by 119889(119904 119905)

119875119889= 119882119902times 119889 (119904 119905) =

120582119889

120582120572

(12)




119889can

be obtained as

120582119889= 120582120572times 119875119889= 120582 (1 minus 119875

119870) times 119882

119902times 119889 (119904 119905) (13)



= 120582120572(1 minus 119875

119889) = 120582 (1 minus 119875

119870) (1 minus 119875

119889)

(14)



120582=120582 [1 minus (1 minus 119875

119870) (1 minus 119875

119889)]

120582

= 119875119870+ 119875119889minus 119875119870sdot 119875119889

(15)







1100 1250 1400 1550 1700 1850 2000

04

05

06

07

08

09

1

Arrival rate

Wai

ting

time i

n qu

eue (

s)

K = 30

K = 32

K = 34

K = 36

(a)

1100 1250 1400 1550 1700 1850 2000

K = 30

K = 32

K = 34

K = 36

890

895

900

905

910

915

Arrival rate

Fina

l arr

ival

rate

(b)





119877 (119877119870) = 120582120572 (1 minus 119875119903)119873 (119904 119905)

= 120582 (1 minus 119875119870) (1 minus 119875

119903)119873 (119904 119905)

(16)


1200 1400 1600 1800 20001070

1080

1090

1100

1110

1120

1130

Arrival rate

Fina

l arr

ival

rate

K = 30K = 32

K = 34K = 36




System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)






119887)


119887)

119862119867


119862119882


119862119880







119887) then




119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)





119880(incurred






Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1100 1250 1400 1550 1700 1850 2000

04

05

06

07

08

09

1

Arrival rate

Wai

ting

time i

n qu

eue (

s)

K = 30

K = 32

K = 34

K = 36

(a)

1100 1250 1400 1550 1700 1850 2000

K = 30

K = 32

K = 34

K = 36

890

895

900

905

910

915

Arrival rate

Fina

l arr

ival

rate

(b)





119877 (119877119870) = 120582120572 (1 minus 119875119903)119873 (119904 119905)

= 120582 (1 minus 119875119870) (1 minus 119875

119903)119873 (119904 119905)

(16)


1200 1400 1600 1800 20001070

1080

1090

1100

1110

1120

1130

Arrival rate

Fina

l arr

ival

rate

K = 30K = 32

K = 34K = 36




System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)






119887)


119887)

119862119867


119862119882


119862119880







119887) then




119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)





119880(incurred






Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










System capacity

120588 = 065

120588 = 07

120588 = 075

120588 = 08

64 72 80 88 96 104 112 120 128minus36

minus32

minus28

minus24

minus20

minus16

minus12

minus8

minus4Lo

g 10(lo

ss p

roba

bilit

y)

(a)

64 72 80 88 96 104 112 120 128minus16

minus14

minus12

minus10

minus8

minus6

minus4

120588 = 065

120588 = 07

120588 = 075

120588 = 08

System capacity

Log 1

0(lo

ss p

roba

bilit

y)

(b)






119887)


119887)

119862119867


119862119882


119862119880







119887) then




119887)120592+ 119861)

= 120588lowastminus1120576119877 (120593120583

1015840120592+ 119861)

(17)





119880(incurred






Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Maximize PM

Subject to 0 le 120583119887lt 120583


where PM = 119865 (119877119870)


= 120582 (1 minus 119875119870) (1 minus 119875

119889)119873 (119904 119905) minus 120588

lowastminus1119877119862119877


119887)120592+ 119861) minus 119862

119882119882lowast


(19)




119887= 40sec 120583 = 45sec 119862

119867= 60 119862

119882= 60

119862119880= 30 119862

119877= 800 120582 = 4000min and 119889(119904 119905) = 001



99 101 103 105 107 109

10825

1085

10875

109

10925

1095

times106


Profi

t

F(103 119) = 1094322

109114119124129134139


109114119124129134139

102 103 104 105 106 107 108 109

0

0002

0004

0006

0008

001

0012

0014

0016

0018

002


Loss

pro

babi

lity

cons

trai

nt

SLA constraint

056





0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

005

01

015

02

025

03

035

04

045

Number of servers

Loss

pro

babi

lity

120582 = 980

120582 = 990120582 = 1000

Optimal solutions

00630069

0099

SLA constraint

24 25 26 27 28 29 30


120582 = 980120582 = 990120582 = 1000

24 25 26 27 28 29 300

50

100

150

200

250

300

350

Number of servers

Aban

donm

ent r

ate

7417

Optimal solutions

4982 4373





25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










25 26 27 28 29 30 31 32 33 34 35 36

235

24

245

25

255

26

265

times105

Number of servers

Profi

t

120582 = 980

120582 = 990

120582 = 1000

Optimal solutions

251864

252123

238298




6 Conclusions



10

20

30

40

50

Quality of service

Impr

ovem

ent r

ate (

)

120582 = 980

120582 = 990

120582 = 1000



Notations









120572





References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










References





































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Profit Optimization in SLA-Aware …downloads.hindawi.com/journals/mpe/2014/534510.pdfResearch Article Profit Optimization in SLA-Aware Cloud Services with a Finite