Research Article Stochastic Simulation on System ...downloads.hindawi.com/journals/mpe/2015/456485.pdfused to measure the degree, known as road network system reliability. Objective

Research ArticleStochastic Simulation on System Reliability and ComponentProbabilistic Importance of Road Network

Xiangyu Zhao1 Dongwei Wang1 Yadan Yan1 and Ziyuan Gu2

1School of Civil Engineering Zhengzhou University Zhengzhou 450001 China2SEU-Monash Joint Graduate School Southeast University Suzhou 215123 China

Correspondence should be addressed to Yadan Yan yadanyan1986gmailcom

Received 29 September 2014 Revised 4 January 2015 Accepted 11 January 2015

Academic Editor Wai Yuen Szeto

Copyright copy 2015 Xiangyu Zhao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Because of the combination explosion problem it is difficult to use probability analytical method to calculate the system reliabilityof large networks The paper develops a stochastic simulation (Monte Carlo-based) method to study the system reliability andcomponent probabilistic importance of the road network The proposed method considers the characteristics of the practical roadnetwork as follows both link (roadway segment) and node (intersection) components are emphasized in the road network thereliability for a link or node component may be at the in-between state namely its reliability value is between 0 and 1 The methodis then implemented using the object-oriented programming language C++ and integrated into a RARN-MGG (reliability analysisof road network using Monte Carlo GIS and grid) system Finally two numerical examples based on a simple road network and alarge real road network respectively are carried out to characterize the feasibility and to demonstrate the strength of the stochasticsimulation method

1 Introduction

Reliability evaluation of a road network has been extensivelystudied in the literature While the initial impetus appears tohave been derived from the study of major natural eventsmdashsuch as earthquakes [1]mdashaffecting the connectivity of a roadnetwork it has had wider impacts on the way of thinking inwhich less severe but more frequency-occurring events mayaffect the operation of a road network [2] A reliable roadnetwork should consider everyday disturbances includingminor accidents on-street parking violations snow floodingroad maintenance and traffic signal failures all of whichwould lead to the performance deterioration of certainroadway segments or intersections in the network Reliabilityof the road network reflects the quality of service that it wouldnormally provide Thus the importance of a reliable roadnetwork cannot be overemphasized

Although there are some calculatingmethods [3ndash5] aboutthe system reliability of the road network the componentprobabilistic importance measure which implies proba-bilistic contribution of improving component reliability tothe system reliability [6] is neglected Besides most of

the existing studies deal with the connectivity capacity ortravel time reliability of the link while node (intersection inthe road network) reliability is not taken into account

Extending the definition of connectivity reliability systemreliability of a road network in this study is defined as ameasure of network reliability given the reliability valuesof each roadway (link) and intersection (node) componentReliability describes the ability of a system or componentto function under stated conditions for a specified periodof time A reliability value is theoretically defined as theprobability of an item to perform its required or intendedfunction under stated conditions for a specific period oftime The reliability of the road network reflects the ability ofthe road network in completing the traffic carrying functionin specific time and conditions Generally its probability isused to measure the degree known as road network systemreliability

Objective of this paper is to develop a stochastic sim-ulation (Monte Carlo-based) method to study the systemreliability and component probabilistic importance of theroad network First of all it introduces the concept ofsystem reliability and the Monte Carlo-based algorithm to

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 456485 5 pageshttpdxdoiorg1011552015456485

2 Mathematical Problems in Engineering

calculate the extended connectivity reliability for an O-Dpair and the system reliability for the road network Thenthe paper proposes the method to calculate the componentprobabilistic importance Finally two numerical examples(a small network and a large real network) are analyzedto illustrate the accuracy and feasibility of the proposedalgorithms The final section gives some concluding remarksand discussion of future research

2 System Reliability Measurement

Consider a road network with links and nodes Each link ornode is regarded as a component of the road network Let119901119894(119894 = 1 2 119899 0 lt 119901

119894lt 1) be the reliability value of

a component either a link or a node 119899 is the number ofcomponents in the road networkThe state or reliability valueof a component can be considered as the capacity underconditions of degradation because of certain disturbances

Let 119903 = (119903119894 119894 = 1 2 119899) denote a random vector

to group the random numbers 119903119894(119894 = 1 2 119899) is a

random variable with uniform distribution between 0 and 1To calculate the system reliability we first use theMonteCarlosimulation approach to generate 119871 realizations of randomvariable 119903 denoted by 119903(119897) 119897 = 1 2 119871 If 119903

119894gt 119901119894

component 119894 fails otherwise component 119894 is reliable Since119903119894(119894 = 1 2 119899) is uniformly distributed the larger the119901119894is the smaller the probability that 119903

119894gt 119901119894is which

indicates that there is a smaller probability that component119894 fails Hence larger 119901

119894denotes a more reliable component

This is the judgment criterion for further using Monte Carlosimulation to calculate the system reliability According to thefailure state of each component a path search algorithmcouldbe used to judge whether the O-D pair119908

119896(119896 = 1 2 119870) is

still connected [7]This process is repeated 119871 times In each test a set of

random numbers is generated and the connectivity for eachO-D pair 119908

119896(119896 = 1 2 119870) is judged 119870 is the number of

O-Dpairs in the road networkDenoting the number of timesthat the O-D pair119908

119896(119898 = 1 2 119870) is connected by 1198711015840 the

connectivity reliability can be calculated as follows

119875119896=1198711015840

119871 (1)

when the sample size 119871 approaches infinity Mak et al gavestatistical analysis results on the relationship between qualityof the approximated solution and sample size [8] Theseresults can guide us to take an appropriate sample size for agiven instance

Then the system reliability of the road network could bedefined as

119875119878=

119870

sum

119896=1

120572119896119875119896 (2)

where 120572119896is the weighting value for O-D pair 119908

119896(119896 =

1 2 119870) in the road network The weighting value couldbe determined according to the degree of importance of eachO-D pair and sum119870

119896=1120572119896= 1

3 Component ProbabilisticImportance Analysis

In the road network system contribution of each componentto the system reliability is different Variation of the reliabilityvalue of some components may significantly affect the systemreliability while some others may just have small or even fewimpacts The component probabilistic importance could beused to evaluate probabilistic contribution of improving com-ponent reliability to the system reliabilityThismeasuremightbe helpful for better understanding the mechanism of systemreliability of road networks and thus gives more hints such asthe upgrading of a roadway and the strengthening of trafficmanagement to minimize the performance deterioration

Denote the component probabilistic importance of com-ponent 119894 by 119868prob119894 Δ119901119894 is the variation of the reliability valueof component 119894 Δ119875

119904is the variation of the system reliability

due to the variation of the reliability value of component 119894Therefore 119868prob119894 can be calculated using the Monte Carlo

simulation Assume the reliability value of component 119894 is 1and the new road network under this condition is denoted by119878119894 Then

119868prob119894 =Δ119875119878

Δ119901119894

=119875119878119894minus 119875119878

1 minus 119901119894

=sum119898

119896=1120572119896119901119896119894minus sum119898

119896=1120572119896119901119896

1 minus 119901119894

=sum119898

119896=1120572119896(119901119896119894minus 119901119896)

1 minus 119901119894

(3)

where 119875119896119894

is the connectivity of O-D pair 119908119896when the

reliability value of component 119894 is 1 119901119896is the connectivity of

O-D pair 119908119896in the original network 119878

Δ119901119896is defined as follows

Δ119901119896= 119901119896119894minus 119901119896=1

119873

119873

sum

119897=1

(119877119896119894119897minus 119877119896119897) (4)

where 119877119896119894119897= 1 if the O-D pair119908

119896is connected in the 119897th test

in network 119878119894 otherwise 119877

119896119894119897= 0 Similarly 119877

119896119897= 1 if the

O-D pair 119908119896is connected in network 119878 otherwise 119877

119896119897= 0

Both 119877119896119894119897

and 119877119896119897are binary variables

Besides in the 119897th test component 119894 in the network 119878119894is

reliable because the reliability value has been assumed to be 1Assume 1198781015840 is the sample network for 119878 in the 119897th test and 1198781015840

119894is

the sample network for 119878119894in the 119897th test Then the difference

between networks 1198781015840 and 1198781015840119894only depends on component 119894

Hence if in the 119897th test component 119894 in network 1198781015840 is reliable1198781015840 and 1198781015840

119894will become identical

119877119896119894119897minus 119877119896119897= 0 (5)

And if 119908119896is connected in the 119897th test in network 1198781015840 119908

119896is

certainly connected in the 119897th test in network 1198781015840119894

119877119896119894119897minus 119877119896119897= 1 minus 1 = 0 (6)

if 119908119896is disconnected in the 119897th test in network 1198781015840 and 119908

119896is

disconnected in the 119897th test in network 1198781015840119894

119877119896119894119897minus 119877119896119897= 0 minus 0 = 0 (7)

Mathematical Problems in Engineering 3

Preprocessing module

(1) Road network input

(2) Component information input

(3) Display setting

Based on the GIS technology

Calculation module

(1) System reliability calculation

(2) Component probabilistic

importance calculation

Based on Monte Carlo andgrid computing

Postprocessing module

(1) Calculation results output

(2) Cloud image of component

probabilistic importance output


Figure 1 Modules of the RARN-MGG system

if 119908119896is disconnected in the 119897th test in network 1198781015840 while 119908

119896is

connected in the 119897th test in network 1198781015840119894

119877119896119894119897minus 119877119896119897= 1 minus 0 = 1 (8)

According to (5)ndash(8) we can have

119877119896119894119897minus 119877119896119897

=

1 if in the 119897th test component 119894 in network 1198781015840 fails119908119896is disconnected in 1198781015840

and 119908119896is connected in 1198781015840

119894

0 otherwise(9)

According to (4) and (9)

119868prob119894 =sum119898

119896=1120572119896(119901119896119894minus 119901119896)

1 minus 119901119894

=sum119898

119896=1120572119896[(1119873)sum

119873

119897=1(119877119896119894119897minus 119877119896119897)]

1 minus 119901119894

=sum119898

119896=1120572119896119873119896119899

119873(1 minus 119901119894)

(10)

where 119873119896119899

denotes the number of times when component 119894in network 1198781015840 fails 119908

119896is connected in network 1198781015840

119894 and 119908

119896is

disconnected in network 1198781015840Besides under the condition that 119901

119894= 1 there has been

no possibility for reliability improvement of component 119894namely 119868prob119894 = 0 Moreover 119901

119894is the reliability value of

a component (ie a link or an intersection) in the roadnetwork It is an indicator to measure ability of component119894 to complete the traffic carrying function in specific timeand conditions which is the result of many traffic supply anddemand factors Hence rounding the component reliability119901119894to three or four decimals is always accurate enough in

practical research which ensures that formulation (3) is valid

4 Numerical Examples

Based on the proposed Monte Carlo-based algorithms forcalculating the system reliability and component probabilisticimportance a simulation system called RARN-MGG (reli-ability analysis of road network using Monte Carlo GIS

1

2

3

4

5

61

2

3 5

6

4

78

9

Figure 2 RARN-MGG system interface with a simple network

and grid) is developed The system uses the object-orientedprogramming language C++ and includes three main mod-ules preprocessing module calculation module and post-processing module (shown in Figure 1) It applies to any roadnetwork for calculating the system reliability and componentprobabilistic importance The calculation module which isthe core of the system is based on the proposedMonte Carlo-based algorithms and the grid computing technology

In the proposed Monte Carlo-based simulation methodthe problem about how to judge whether an O-D pair isconnected or disconnected is very important Hence bidi-rectional search is designed in the RARN-MGG system Pathsearch is from the Origin node and Destination node simul-taneously When the O-D pair is connected bidirectionalsearch could significantly reduce the search time When theO-D pair is disconnected bidirectional search could obtaina set of nodes connected to the Origin node and a set ofnodes connected to the Destination node Hence when anew component (either a link or a node) is added into theroad network the problem will just become judging whetherthe new component will make the O-D pair connectedThis search algorithm will greatly reduce the complexity andimprove the efficiency

We applied the method to a simple road network given inFigure 2 The network consists of six nodes nine links andone O-D pair Reliability values for links 1 2 3 4 5 6 7 8and 9 are 095 095 080 090 090 090 080 090 and 095respectively Reliability values for nodes 1 2 3 4 5 and 6 are099 095 090 090 085 and 095 respectively

Table 1 shows the comparison between results of thedeveloped RARN-MGG simulation system and results cal-culated using the probability analytical method [5] 119877

119904is

the system reliability 119868(119894)(119894 = 1 2 6) is the component

probabilistic importance of node 119894 and 119868[119895](119895 = 1 2 9)

is the component probabilistic importance of link 119895 FromTable 1 we can find that results based on Monte Carlosimulation proposed by this paper and results based on


Rail station and

shopping area

Highway terminalarea

Figure 3 Real road network and land use of Zhengzhou City

Table 1 Comparison between calculation results of two methods

Index 119877119904

119868(1)

119868(2)

119868(3)

119868(4)

119868(5)

119868(6)

119868[1]

Analytical method 08690 08777 02786 00850 01533 02681 09147 01516Monte Carlo 08692 08786 02785 00850 01533 02682 09147 01515Index 119868

[2]119868[3]

119868[4]

119868[5]

119868[6]

119868[7]

119868[8]

119868[9]

Analytical method 00707 00081 01256 00290 00533 00056 01433 01650Monte Carlo 00707 00081 01256 00290 00532 00056 01433 01648

the probability analytical methodmatch very well which fur-thermore testifies the feasibility applicability of the simulationmethod

Moreover the method is applied to a large road networkof Zhengzhou City China (shown in Figure 3) The networkconsists of 475 links 280 nodes and 78120O-Dpairs Assumethat the reliability value for each link in the road network is05 and the reliability value for each node in the road networkis 09Theweighting value for eachO-Dpair is identical Eachlink and node is encoded with a number

The number of simulations or tests is set to be 200000Using the RARN-MGG method system reliability value ofthis road network is 005681 Figures 4 and 5 suggest the loca-tion of the top 60 links and nodeswith the highest component

probabilistic importance in the road network respectivelyThe redder the color is the greater the importance is

According to Figure 4 with the increase of the number oflinks connected to a node component probabilistic impor-tance of this node increases such as node 121 and node 110The main political economic and cultural activities are con-centrated in the northeast and southwest parts of ZhengzhouCity However due to the separation effects of the railwaylines (ie wide lines in black and white in Figure 3) thereare only six roads (links 261 227 322 and 379) connectingthese two parts From Figure 5 we can find that links 227and 322 are among the set of top 30 links with the highestcomponent probabilistic importance Besidesmost of the top30 links and nodes with the highest component probabilistic


Figure 4 Location of nodes with the highest 60-component prob-abilistic importance

Figure 5 Location of links with the highest 60-component proba-bilistic importance

importance distribute around the railway station highwayterminal and hospital regions which are usually crowdedin everyday life To a large extent calculation results couldreflect the actual road traffic conditions

5 Conclusions

This paper investigated the system reliability and componentprobabilistic importance of a road network Characteristics ofthe practical road network are taken into accountWith these

considerations the stochastic simulation methods based onMonte Carlo are proposed A RARN-MGG system whichincorporates this proposed method is then developed Thesystem can be employed as a useful and practical quantitativeanalysis tool to assist the decision-making for the roadmanagement departments such as evaluating the system reli-ability of different road network planning schemes findingthe key components that need to be upgraded or improved orpredicting the increased system reliability of a road networkwhen it adds a new link

Conflict of Interests

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

Acknowledgments

This paper is supported by the Project (no 51278468) of theNational Natural Science Foundation of China The authorsalso thank the anonymous reviewer for the helpful commentswhich are important for the improvement of the paper

References

[1] M G H Bell and Y Iida Transportation Network Analysis JohnWiley amp Sons Chichester UK 1997

[2] S Clark and D Watling ldquoModelling network travel time reli-ability under stochastic demandrdquo Transportation Research PartB Methodological vol 39 no 2 pp 119ndash140 2005

[3] A ChenH YangHK Lo andWH Tang ldquoCapacity reliabilityof a road network an assessment methodology and numericalresultsrdquo Transportation Research Part B Methodological vol 36no 3 pp 225ndash252 2002

[4] H K Lo and Y-K Tung ldquoNetwork with degradable linkscapacity analysis and designrdquo Transportation Research Part BMethodological vol 37 no 4 pp 345ndash363 2003

[5] A S Hasanuddin ldquoA simple technique for computing networkreliabilityrdquo IEEE Transactions on Reliability vol 31 no 1 pp 41ndash44 1982

[6] J E Henley and H Kumamoto Reliability Engineering and RiskAssessment Prentice Hall 1981

[7] W Xu S He R Song and S S Chaudhry ldquoFinding the Kshortest paths in a schedule-based transit networkrdquo Computersand Operations Research vol 39 no 8 pp 1812ndash1826 2012

[8] W-K Mak D P Morton and R K Wood ldquoMonte Carlobounding techniques for determining solution quality instochastic programsrdquo Operations Research Letters vol 24 no1-2 pp 47ndash56 1999

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Algebra

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Stochastic AnalysisInternational Journal of


calculate the extended connectivity reliability for an O-Dpair and the system reliability for the road network Thenthe paper proposes the method to calculate the componentprobabilistic importance Finally two numerical examples(a small network and a large real network) are analyzedto illustrate the accuracy and feasibility of the proposedalgorithms The final section gives some concluding remarksand discussion of future research

2 System Reliability Measurement

Consider a road network with links and nodes Each link ornode is regarded as a component of the road network Let119901119894(119894 = 1 2 119899 0 lt 119901

119894lt 1) be the reliability value of

a component either a link or a node 119899 is the number ofcomponents in the road networkThe state or reliability valueof a component can be considered as the capacity underconditions of degradation because of certain disturbances

Let 119903 = (119903119894 119894 = 1 2 119899) denote a random vector

to group the random numbers 119903119894(119894 = 1 2 119899) is a

random variable with uniform distribution between 0 and 1To calculate the system reliability we first use theMonteCarlosimulation approach to generate 119871 realizations of randomvariable 119903 denoted by 119903(119897) 119897 = 1 2 119871 If 119903

119894gt 119901119894

component 119894 fails otherwise component 119894 is reliable Since119903119894(119894 = 1 2 119899) is uniformly distributed the larger the119901119894is the smaller the probability that 119903

119894gt 119901119894is which

indicates that there is a smaller probability that component119894 fails Hence larger 119901

119894denotes a more reliable component

This is the judgment criterion for further using Monte Carlosimulation to calculate the system reliability According to thefailure state of each component a path search algorithmcouldbe used to judge whether the O-D pair119908

119896(119896 = 1 2 119870) is

still connected [7]This process is repeated 119871 times In each test a set of

random numbers is generated and the connectivity for eachO-D pair 119908

119896(119896 = 1 2 119870) is judged 119870 is the number of

O-Dpairs in the road networkDenoting the number of timesthat the O-D pair119908

119896(119898 = 1 2 119870) is connected by 1198711015840 the

connectivity reliability can be calculated as follows

119875119896=1198711015840

119871 (1)

when the sample size 119871 approaches infinity Mak et al gavestatistical analysis results on the relationship between qualityof the approximated solution and sample size [8] Theseresults can guide us to take an appropriate sample size for agiven instance

Then the system reliability of the road network could bedefined as

119875119878=

119870

sum

119896=1

120572119896119875119896 (2)

where 120572119896is the weighting value for O-D pair 119908

119896(119896 =

1 2 119870) in the road network The weighting value couldbe determined according to the degree of importance of eachO-D pair and sum119870

119896=1120572119896= 1

3 Component ProbabilisticImportance Analysis

In the road network system contribution of each componentto the system reliability is different Variation of the reliabilityvalue of some components may significantly affect the systemreliability while some others may just have small or even fewimpacts The component probabilistic importance could beused to evaluate probabilistic contribution of improving com-ponent reliability to the system reliabilityThismeasuremightbe helpful for better understanding the mechanism of systemreliability of road networks and thus gives more hints such asthe upgrading of a roadway and the strengthening of trafficmanagement to minimize the performance deterioration

Denote the component probabilistic importance of com-ponent 119894 by 119868prob119894 Δ119901119894 is the variation of the reliability valueof component 119894 Δ119875

119904is the variation of the system reliability

due to the variation of the reliability value of component 119894Therefore 119868prob119894 can be calculated using the Monte Carlo

simulation Assume the reliability value of component 119894 is 1and the new road network under this condition is denoted by119878119894 Then

119868prob119894 =Δ119875119878

Δ119901119894

=119875119878119894minus 119875119878

1 minus 119901119894

=sum119898

119896=1120572119896119901119896119894minus sum119898

119896=1120572119896119901119896

1 minus 119901119894

=sum119898

119896=1120572119896(119901119896119894minus 119901119896)

1 minus 119901119894

(3)

where 119875119896119894

is the connectivity of O-D pair 119908119896when the

reliability value of component 119894 is 1 119901119896is the connectivity of

O-D pair 119908119896in the original network 119878

Δ119901119896is defined as follows

Δ119901119896= 119901119896119894minus 119901119896=1

119873

119873

sum

119897=1

(119877119896119894119897minus 119877119896119897) (4)

where 119877119896119894119897= 1 if the O-D pair119908

119896is connected in the 119897th test

in network 119878119894 otherwise 119877

119896119894119897= 0 Similarly 119877

119896119897= 1 if the

O-D pair 119908119896is connected in network 119878 otherwise 119877

119896119897= 0

Both 119877119896119894119897

and 119877119896119897are binary variables

Besides in the 119897th test component 119894 in the network 119878119894is

reliable because the reliability value has been assumed to be 1Assume 1198781015840 is the sample network for 119878 in the 119897th test and 1198781015840

119894is

the sample network for 119878119894in the 119897th test Then the difference

between networks 1198781015840 and 1198781015840119894only depends on component 119894

Hence if in the 119897th test component 119894 in network 1198781015840 is reliable1198781015840 and 1198781015840

119894will become identical

119877119896119894119897minus 119877119896119897= 0 (5)

And if 119908119896is connected in the 119897th test in network 1198781015840 119908

119896is

certainly connected in the 119897th test in network 1198781015840119894

119877119896119894119897minus 119877119896119897= 1 minus 1 = 0 (6)

if 119908119896is disconnected in the 119897th test in network 1198781015840 and 119908

119896is

disconnected in the 119897th test in network 1198781015840119894

119877119896119894119897minus 119877119896119897= 0 minus 0 = 0 (7)





(3) Display setting


Calculation module












119896is


119877119896119894119897minus 119877119896119897= 1 minus 0 = 1 (8)


119877119896119894119897minus 119877119896119897

=



119894

0 otherwise(9)


119868prob119894 =sum119898

119896=1120572119896(119901119896119894minus 119901119896)

1 minus 119901119894

=sum119898

119896=1120572119896[(1119873)sum

119873

119897=1(119877119896119894119897minus 119877119896119897)]

1 minus 119901119894

=sum119898

119896=1120572119896119873119896119899

119873(1 minus 119901119894)

(10)

where 119873119896119899



119894 and 119908

119896is









1

2

3

4

5

61

2

3 5

6

4

78

9






119904is





Rail station and

shopping area




Index 119877119904

119868(1)

119868(2)

119868(3)

119868(4)

119868(5)

119868(6)

119868[1]


[2]119868[3]

119868[4]

119868[5]

119868[6]

119868[7]

119868[8]

119868[9]











5 Conclusions





Acknowledgments


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













(3) Display setting


Calculation module












119896is


119877119896119894119897minus 119877119896119897= 1 minus 0 = 1 (8)


119877119896119894119897minus 119877119896119897

=



119894

0 otherwise(9)


119868prob119894 =sum119898

119896=1120572119896(119901119896119894minus 119901119896)

1 minus 119901119894

=sum119898

119896=1120572119896[(1119873)sum

119873

119897=1(119877119896119894119897minus 119877119896119897)]

1 minus 119901119894

=sum119898

119896=1120572119896119873119896119899

119873(1 minus 119901119894)

(10)

where 119873119896119899



119894 and 119908

119896is









1

2

3

4

5

61

2

3 5

6

4

78

9






119904is





Rail station and

shopping area




Index 119877119904

119868(1)

119868(2)

119868(3)

119868(4)

119868(5)

119868(6)

119868[1]


[2]119868[3]

119868[4]

119868[5]

119868[6]

119868[7]

119868[8]

119868[9]











5 Conclusions





Acknowledgments


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Rail station and

shopping area




Index 119877119904

119868(1)

119868(2)

119868(3)

119868(4)

119868(5)

119868(6)

119868[1]


[2]119868[3]

119868[4]

119868[5]

119868[6]

119868[7]

119868[8]

119868[9]











5 Conclusions





Acknowledgments


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 Conclusions





Acknowledgments


References
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


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Documents

Research Article Stochastic Simulation on System ...downloads.hindawi.com/journals/mpe/2015/456485.pdfused to measure the degree, known as road network system reliability. Objective