Research ArticleModeling and Analysis of Train Rear-End Collision AccidentsBased on Stochastic Petri Nets

Chao Wu Chengxuan Cao Yahua Sun and Keping Li

State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Chengxuan Cao cxcaobjtueducn

Received 25 September 2014 Revised 28 December 2014 Accepted 2 February 2015

Academic Editor Zhiwu Li

Copyright copy 2015 Chao Wu et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We proposed a model of the train rear-end collision accidents based on stochastic Petri nets (SPN) theory By isomorphic Markovchain model of the proposed accident model we provide the quantitative analysis of the train rear-end collision accidents Fuzzyrandom method is also applied to analyze the performance of the proposed model In addition according to the data extractedfrom a large amount of historical data of the accident statistics we present a case analysis and discussion It showed that the resultsof the proposed train rear-end accident model based on SPN are reasonable in practical applications and can be used to effectivelyanalyze the accidents and prevent loss and the results may be a reference to the department of railway safety management

1 Introduction

Rail transport is an excellent transport mode in reducingpollution and alleviating the traffic congestion and is alsoa safe and economical way of transportation for passengersand goods Nowadays the high-speed railway and urban railtransit have been quickly developed and the intensive depar-ture interval strategy has been widely adopted by the railwayoperators to cope with the demands for transportation Thedeparture intervals have been as short as possible tomaximizethe transportation capacity However intensive departureintervals also increase the risk of train rear-end accidentsFacing such high-intensive departure intervals how to ensurethe safety of train operations is undoubtedly a challengefor the department of railway safety management RecentlyChinese people have a lot of doubts of the safety of the railtransport especially after theWenzhouhigh-speed train rear-end catastrophic accident in July 23 2011 and the ShanghaiMetro Line 10 rear-end accident in September 27 2011 Inorder to ensure the competitive advantage of rail transportrail transport operators must improve the level of safety toreduce the occurrence of train accidentsThereforemodelingand analysis of the train rear-end collision accidents have vitalpractical significance

In the literatures many models of accidents and safetyanalyses have been proposed such as data mining (DM)Bayesian network (BN) fault tree analysis (FTA) and Petrinets (PN) [1ndash5] Mirabadi and Sharifian [1] analyzed the datafrom past accidents of the Iranian Railway (RAI) by applyingthe CRISP-DM reference model and the association rulesto discover and reveal unknown relationships and patternsamong the data De Ona et al [3] presented BN to describeaccidents that involve many interdependent variables Therelationship and structure of the variables can be studied andtrained from accident data It does not need to know any pre-defined relationships between dependent and independentvariables

By the principles of logical deductive analysis the faulttree model analyzes accidents from a possible top eventand analyzes its causes by layers until we find out all thebackground events that lead to the accident Li [6] carriedout the fault tree analysis in which the train rear-end accidentis seen as the top event and investigated the patterns ofthe accident to analyze various events which can causethe accident He also discussed the security elements andstrategies of complex safety-critical system from a macropoint of view

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 602126 9 pageshttpdxdoiorg1011552015602126

2 Mathematical Problems in Engineering

Although all of themodels presented in the literature havebeen recognized as a powerful tool for accident analysis themajority is only suitable for describing small static systemand cannot analyze the dynamic system and it is difficultto analyze the accident quantitatively [7] As a tool fordiscrete eventsrsquo simulation PN can avoid these deficiencies[8] Dynamic changes in the PN induced by transition firingmake it possible to analyze the dynamic system In additionbeing isomorphic to the state space of the Markov chain(MC) the PN makes analyzing the accident quantitativelyconvenient

The main contributions of this paper are as follows Firstwe proposed a model of train rear-end collision accidentsbased SPN theory instead of conventional method suchas fault tree analysis Second our model can be used forquantitative analysis of the accidents by being isomorphic toMC Third fuzzy random method is applied to analyze theperformance of the model which can improve the reliabilityof the results of the quantitative analysis Finally accordingto the data extracted from a large amount of historical data ofthe accident statistics we have carried out a case analysis anddiscussion which may be useful to the department of railwaysafety management

The remainder of this paper is organized as followsFundamental theory of SPN and theoretical basis of systemmodeling and performance analysis using SPN are given inSection 2 The model of train rear-end collision accidentsbased on SPN theory is proposed in Section 3 A case analysisand discussion are presented in Section 4 The last section isthe conclusion of this paper

2 Methods

Dr Petri [9] proposed Petri nets theory in 1962 when hedeveloped information flow model for computer operationsystem At present PN is extensively applied to modelingperformance analysis and control research for discrete eventdynamic system [10] As a system model PN is dynamicand concurrent In PN standard graphic presentation thesign ldquoOrdquo indicates place and the sign ldquo|rdquo or pane indicatestransition [11]

21 Fundamentals of PN

(1) Definition in general a PN is defined as follows sum =

(119875 119879 119865 119870 119882 1198720) where

(a) 119875 = 1199011 1199012 119901

119899 119899 gt 0 refers to the finite set

of place(b) 119879 = 119905

1 1199052 119905

119898 119898 gt 0 refers to the finite set

of transition(c) 119865 is an ordered pair set consisting of 119875 and 119879

and satisfies 119865 sube 119875 times 119879 cup 119879 times 119875 119875 cap 119879 = and119879 cup 119875 =

(d) 119870 is a capacity function(e) 119882 is a weight function weighting all arc lines

119908(119901 119905) or 119908(119905 119901) is used to denote the weight ofdirected arc going from 119901 to 119905 or 119905 to 119901

(f) 119872 is the marking reflecting token distributionin all places and 119872

0is the initial marking

(2) Enabling and stimulation rules of transition are asfollows

(a) transition 119905 isin 119879 is enabled when and onlywhenin respect of all 119901 isin 119875 119908(119901 119905) = 1 119872(119901) gt 0

(b) after the enabled transition 119905 is stimulated by themarking 119872 a new marking 119872

1015840 is generated inrespect of forall119901 isin 119875 which is

1198721015840(119901) =

119872 (119901) minus 119908 (119901 119905) 119901 isin119900119905

119872 (119901) + 119908 (119901 119905) 119901 isin 119905119900

119872 (119901) + 119908 (119901 119905) minus 119908 (119901 119905) 119901 isin119900119905 and 119901 isin 119905

119900

119872 (119901) other(1)

22 Theoretical Basis of System Modeling and PerformanceAnalysis Using SPN Description of a system by PN is usuallybased on two concepts event and state PN does not onlypresent the graph of a system but also provide themathemat-ical description of the system [12] To describe the dynamicaction of a system execution of the processing task can beindicated by corresponding transition firing In the PN thetransferring of tokens expresses the information processThemarkings of the PN denote the special state of the systemForward markings decide the set of all possible states of thesystem with a given initial state

PN provides a new description tool for system perfor-mance analysis [13] In the continuous-time SPN certaintime delay is needed for a transition from being generableto practical firing that is the period from a transition beinggenerable to its firing is regarded as a continuous randomvariant which is subject to exponential distribution Whenthe firing rate of transitions is exponentially distributedand the markings are countable it has been proved that acontinuous-time SPN is isomorphic to a continuous-timeMC [14] So each marking of SPN is mapping into a state ofMC and the occurrence graph of SPN is isomorphic to thestate space of MC

Therefore referring to the SPNoccurrence graphwhich isisomorphic to a homogeneous MC we can utilize the theoryof random processes to analyze ourmodelThe application ofSPN to system performance analysis is usually as follows

(1) Build the SPN model for the system(2) Define the possible states of SPN and derive its

reachable marking set(3) Obtain the isomorphic MC of SPN(4) Analyze the system performance based on the stable

probability of MC

Based on its stable probability we can further analyzethe system performance indexes and time characteristicsanalyze the busyness andwork efficiency of thewhole systems

Mathematical Problems in Engineering 3

or transitions identify main factors affecting the systemperformance and analyze the resources distribution andoptimization plans for the system under different commandand dispatch methods

3 Modeling and Analysis of Train Rear-EndCollision Accidents Based on SPN

31 Modeling of Train Rear-End Collision Accidents Basedon SPN In this paper we consider the case of one trackonly in the one-way and assume that the collision avoidancesystems such as signal control system train distance controlsystem train state communication and control system andthe danger alarm system are applied in the problemThe traintrack signal failure may be caused by some environmentalfactors such as lightning strike According to Li [6] andour knowledge and understandings of the railway trafficthe major procedure of a train rear-end collision accidentis shown in Figure 1 Based on the major procedure of atrain rear-end collision accident and the theory of SPN weproposed the train rear-end collision accidents model asshown in Figure 2

Places and transitions in Figure 2 are explained as followsFinite set of places 119875 are as follows 119875

1 two successive trains

are driving normally 1198752 velocity of the follow-up train is

greater in two successive trains 1198753 velocity of follow-up

train is still greater 1198754 it refers to risk of train rear-end

collision accident 1198755 it refers to intervention of automatic

train protection system (ATP) 1198756 ATP fails to intervene

1198757 ATP intervenes successfully 119875

8 it refers to intervention

of dispatcher 1198759 dispatcher intervenes successfully 119875

10 it

refers to effectiveness judgment of dispatcherrsquos measures 11987511

dispatcher fails to intervene 11987512 it refers to intervention

of driver 11987513 driver detects the abnormal 119875

14 it refers to

effectiveness judgment of braking 11987515 it refers to train rear-

end collision accident 11987516 trains are safe controlled

Finite set of transitions 119879 are as follows 1199051 it refers

to velocity difference in two successive trains (velocity offollow-up train is greater) 119905

2 interval distance in two

successive trains is much larger than the minimum instan-taneous distance 119905

3 interval distance in two successive

trains is approaching the minimum instantaneous distance1199054 velocity difference is eliminated 119905

5 interval distance in

two successive trains is decreasing 1199056 ATP works auto-

matically 1199057 ATP fails to work 119905

8 ATP works normally

1199059 dispatcher intervenes 119905

10 dispatcher does not detect or

neglect the abnormal 11990511 dispatcher detects the abnormal

11990512 dispatcher takes measures 119905

13 dispatcherrsquos measures fail

to work 11990514 driver intervenes 119905

15 ATP adjusts the trainrsquos

driving state automatically 11990516 dispatcherrsquos measures work

11990517 driver detects the abnormal 119905

18 driver does not detect the

abnormal 11990519 driver operates to brake 119905

20 train fails to stop

before rear-end collision accident happened 11990521 driver stops

the train successfully 11990522 the trainrsquos driving state is adjusted

11990523 it refers to cleaning up for restoring the line operation

32 Effectiveness Analysis of the Model Based on MarkovProcess According to the SPN model as shown in Figure 2

Interval distance in the two successive trains is approaching the minimum

instantaneous distance

ATP failed to work

Two successive trains drive toward thesame direction on the same track

Velocity of follow-up train is greater inthe two successive trains

Train rear-end collision accident

The driver dose not detect abnormal orfailed to brake to stop the train

The dispatcher does not detect abnormal or failed to take effective measures

Figure 1Major procedure of occurrence of a train rear-end collisionaccident

we can obtain its isomorphic MC in Figure 3 where 1198721is

the initial marking containing only one token in the place 1198751

and the Markov state space can be obtained as 1198721to 11987216

as a result of the transferring of the token and the firing ofdifferent sets of transitions Since most process service timeis close to the exponential distribution and the exponentialdistribution simplifies the analysis of the randomnetwork weassume that the firing rate of each transition is exponentiallydistributed in the model Here the average firing rate oftransitions 119905

1 1199052 119905

23are 1205821 1205822 120582

23 respectively

From the MC of the SPN model the following conclu-sions can be obtained

(1) No obstruction occurred in the entire process and thetransform of token is smooth The process does notwait indefinitely for a state indicating that all tran-sitions in the process of the train rear-end collisionaccidents are likely to be implemented in a certainperiod of timeThe successful implementation of eachtransition is the basis for the next transition to beimplemented successfully

(2) A state 119872119894does not exist which cannot be reached

namely no state can never occur and a deadlock statedoes not exist in the model

4 Case Analysis and Discussion

41 Performance Analysis of SPN Model Assuming 119899 stateslocated in the MC we can immediately derive the transferrate matrix119876 = [119902

119894119895] 1 le 119894 119895 le 119899The element of the transfer

matrix is obtained intuitively as follows

4 Mathematical Problems in Engineering

P1 P2

P3

P4 P5

P6

P7

P8

P9

P10

P11P12

P13

P14

P15

P16

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12 t13 t14

t15 t16

t17

t18

t19t20

t21

t22

t23

Figure 2 Model of a train rear-end collision accident based on stochastic Petri nets

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16

12058211205822

1205823

1205824 1205825

1205826

1205827

1205828

120582912058210

12058211 12058212

12058213 12058214

12058215

1205821612058217

12058218

12058219

12058220

12058221

12058222

12058223

Figure 3 Markov chain of the stochastic Petri nets model

(1) For the element 119902119894119895which is located in nondiagonal

when there is an arc connecting state 119872119894with state

119872119895 then the firing rate 120582

119894marked on the arc is the

value of 119902119894119895 if there is not an arc connecting state 119872

119894

with state 119872119895 then 119902

119894119895= 0

(2) For the element 119902119894119895which is located in diagonal the

negative number of the sum of the rate 120582119894marked on

the arcs which are outputting from the state 119872119894is the

value of 119902119894119895[12]

Assuming that the stable probability of the 119899 states inthe MC is a row vector 119883 = (119909

1 1199092 119909

119899) we can get the

following linear equations according to the Markov process

119883119876 = 0

sum119894

119909119894= 1 1 le 119894 le 119899

(2)

Since 120582119894is extracted and estimated on the basis of the

statistics data which is collected from the previous train acci-dents investigations there may be incompletion inaccuracyand uncertainty Therefore using different fuzzy degrees (2)can be transferred into the fuzzy stable probability equationsand solving these fuzzy equations we can calculate the steadystate probability Because the value 120582

119894is usually around a

certain value120582 we can use the trianglemembership functions

to represent fuzzy numbers Mathematically the trianglemembership function 119906

120582119894(119909) can be expressed as follows [15]

119906120582119894

(119909) =

0 119909 le 1198861198941

(119909 minus 1198861198941

)

(1198861198942

minus 1198861198941

) 1198861198941

le 119909 le 1198861198942

(1198861198943

minus 119909)

(1198861198943

minus 1198861198942

) 1198861198942

le 119909 le 1198861198943

0 119909 ge 1198861198943

(3)

The fuzzy variables 120582119894can be represented by a triple

(1198861 1198862 1198863) and the parameter 119886

1198942defines the maximum

degree of 120582119894 namely 119906

120582119894(1198861198942

) = 1 parameters 1198861198941and 119886

1198943

define the minimum limit and the maximum limit of thefuzzy number In addition the triangle membership function(1198861 1198862 1198863) defines a 120572-cut of the fuzzy number In fact the

120572-cut 119860(120572)

= [119886(120572)

1 119886(120572)

3] defines a confidence interval of the

triangular fuzzy number expressed as 119860(120572)

= [1198861

+ (1198862

minus

1198861)120572 1198863

minus (1198863

minus 1198862)120572] (see Figure 4)

Because a concrete source which can provide the valuesof firing rate 120582

119894does not exist we first obtained the data of

the frequency of related events of the responding transitionfrom the statistics data of the previous train accidentsinvestigations and according to it we get the values of thefiring rate 120582

119894after our proper estimation and modification

which are listed in Table 1 with a unit of bout per hour Sincethe uncertainty of the data has been considered by applyingthe fuzzy random method the data are representative to beused for the further analysis

Mathematical Problems in Engineering 5

Table 1 Value of the firing rate 120582119894

1205821

1205822

1205823

1205824

1205825

1205826

1205827

1205828

1205829

12058210

12058211

12058212

10 2 2 4 4 10 1 4 10 1 4 1012058213

12058214

12058215

12058216

12058217

12058218

12058219

12058220

12058221

12058222

12058223

1 1 10 4 4 1 10 2 2 10 1

Table 2 Crisp number and triangular number of the firing rate 120582119894

120582119894

Crisp number Spread Triangular fuzzy number1205821 1205826 1205829 12058212

12058215

12058219

12058222

10 plusmn20 (8 10 12)1205824 1205825 1205828 12058211

12058216

12058217

4 plusmn15 (34 4 46)1205822 1205823 12058220

12058221

2 plusmn10 (18 2 22)1205827 12058210

12058213

12058214

12058218

12058223

1 plusmn5 (095 1 105)

1

120583A

(x)

120572

a1 a1205721 a2 a1205723a3

Figure 4 The 120572-cut of a triangular fuzzy number

In order to account for the uncertainties in the data theobtained crisp data are converted into the fuzzy numbersMore specifically crisp numbers in the extracted data areconverted into fuzzy numbers with a known spread Thus wecan get the triangular fuzzy numbers of 120582

119894as listed in Table 2

Taking (09 10 11) as the triangular fuzzy numbers ofthe sum of stable probability we calculate the fuzzy stableprobability in the condition of triangular fuzzy number 120582

119894

Defuzzification is necessary to convert the fuzzy output to acrisp value as most of the actions or decisions implementedby human or machines are binary or crisp Out of theexistence of the various defuzzification techniques in theliterature center of gravity (COG) method is selected due toits property that it is equivalent to meaning of data [16] Ifthe membership function 119906

120582119894(119909) of the output fuzzy set 119860 is

described on the interval [1199091 1199092] then COG defuzzification

value 119909 can be defined as

119909 =int1199092

1199091

119909 sdot 119906120582119894

(119909) 119889119909

int1199092

1199091

119906120582119894

(119909) 119889119909 (4)

Based on the COG method the crisp values of the stableprobability are calculated since the placersquos busy probability

refers to the probability of events or states in the train rear-end accident process So we get the results as follows

119875 [119872 (1199011) = 1] = 119909

1= 0104

119875 [119872 (1199012) = 1] = 119909

2= 0260

119875 [119872 (1199013) = 1] = 119909

3= 0065

119875 [119872 (1199014) = 1] = 119909

4= 0078

119875 [119872 (1199015) = 1] = 119909

5= 0156

119875 [119872 (1199016) = 1] = 119909

6= 0016

119875 [119872 (1199017) = 1] = 119909

7= 0062

119875 [119872 (1199018) = 1] = 119909

8= 0031

119875 [119872 (1199019) = 1] = 119909

9= 0013

119875 [119872 (11990110

) = 1] = 11990910

= 0025

119875 [119872 (11990111

) = 1] = 11990911

= 0056

119875 [119872 (11990112

) = 1] = 11990912

= 0011

119875 [119872 (11990113

) = 1] = 11990913

= 0004

119875 [119872 (11990114

) = 1] = 11990914

= 0011

119875 [119872 (11990115

) = 1] = 11990915

= 0033

119875 [119872 (11990116

) = 1] = 11990916

= 0075

(5)

A comparison of the placersquos busy probability aboutwhether to introduce the fuzzy random method is carriedout In Figure 5 1 2 4 5 and 6 represent the placersquos busyprobability value in the condition that the firing rate of eachtransition takes its 119886

1 119886(05)

1 1198862 119886(05)

3 and 119886

3 respectively

as showed in the triangle membership function (1198861 1198862 1198863)

3 represents each placersquos busy probabilityrsquos defuzzificationvalue We observe that the defuzzification values of placersquosbusy probability are more stable compared to the direct useof crisp firing rate of the transitions which demonstrate thatfuzzy mathematical method accounted for the uncertainties

6 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8000

005

010

015

020

025

030

Prob

abili

ty

Cut level of firing rate of each transition

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 5 Stable probability of varying 120572-cut level of firing rate ofeach transition

of the firing rate of the transitions Thus the result is moreaccurate and reliable

In addition from the above data it can be seen that thebusy probabilities of places 119875

2and 119875

5are larger relatively

Place 1198752refers to the state that velocity of the follow-up

train is greater in two successive trains There are manyreasons leading to this state such as the driver violating thedriving instruction usually represented in speeding or slowlydriving the automatic control system being abnormal or thedispatcherrsquos misjudgment On the one hand the dispatcherrsquosmisjudgment is reflected by the dispatcherrsquos belief that thefront train cannot be caught up by the following train inthe driving sections to conduct the risk instruction whichsuggests that the dispatcher put too much faith in the driverand neglected risk on the other hand it is reflected by thefact that the dispatcher may believe in the collision avoidancesystem which can prevent the accident which suggests thatthe dispatcher put too much faith in the technology

In all of the factors leading to the state 1198752 some cannot

be avoided because it is the need of dispatching operationmanagement In this case we must be careful and shouldnot put too much faith in the driver or technology toneglect the unexpected adventure and to conduct any riskinstruction Other risk factors can be reduced or eliminatedby strengthening the management of drivers and other safetymeasures

The large busy probability of place 1198752inevitably leads to

a relatively large busy probability of place 1198755(intervention

of ATP) Because only a very small number of cases of thisvelocity difference are eliminated naturally a train generallyhas to rely on ATP to adjust automatically the train drivingstatus to ensure the safe driving of trains In case theATP worked normally in the long-term train drivers ordispatchers can easily depend on the vital role of the ATPIt is the reason why the accident tends to happen once ATPis not working properly Therefore it is more crucial for the

1 2 3 4 5 6 7 8 9000005010015020025030035040045050

Prob

abili

ty

Transition rate of transition number 3p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 6 Stable probabilities of varying transition rate 1205823

day-to-day maintenance of railway safeguard equipment andfacilities to ensure the safe driving of trains In additionthe vigilance of the train driver and dispatcher should bestrengthened in case of the ATP abnormalities

Furthermore the obtained result is consistent with theactual one that the busy probability of place 119875

15(the train

rear-end collision accident) is not very large However nomatter how low the probability is it does not mean that theaccident does not happen Even if we can accurately estimatethe accident we cannot accurately predict the moment of theaccident Because it involves significant consequences every-one from the top management to the low level employeesshould always be vigilant and keep a high level of securityalert

42 Discussion If we change the value of the firing rate 1205823

and values of 1205821 1205822 1205824 120582

23remain unchanged we can

get Figure 6 where 1 7 represent the stable probabilityvalue (namely the placersquos busy probability) in the conditionof the firing rate of transition 119905

3taking 0 05120582

3 1205823 151205823

21205823 251205823 31205823 respectively If we change the value of the

firing rate 1205827 1205828 12058210

12058211

12058213

12058216

12058218

12058220

12058221 we can also

get Figures 7ndash15 similarlyIn Figure 6 if 120582

3increases that is the frequency of inter-

val distance in two successive trains is approaching to theminimum instantaneous distance increases the placersquos busyprobability of119875

41198755 and119875

15increases remarkablyThis shows

that the intensive departure interval strategy will significantlyincrease safety risk and the probability of accidents Thus weshould make a tradeoff between the risk of accidents andintensive departure interval strategy which aim to improvethe transport capacity

From Figures 7 and 8 we find out that ATP plays avery important role as the first barrier for protection of trainrunning security Once the frequency of ATP fails to workincreasingly or the frequency of ATPwork normally declines

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

Although all of themodels presented in the literature havebeen recognized as a powerful tool for accident analysis themajority is only suitable for describing small static systemand cannot analyze the dynamic system and it is difficultto analyze the accident quantitatively [7] As a tool fordiscrete eventsrsquo simulation PN can avoid these deficiencies[8] Dynamic changes in the PN induced by transition firingmake it possible to analyze the dynamic system In additionbeing isomorphic to the state space of the Markov chain(MC) the PN makes analyzing the accident quantitativelyconvenient

The main contributions of this paper are as follows Firstwe proposed a model of train rear-end collision accidentsbased SPN theory instead of conventional method suchas fault tree analysis Second our model can be used forquantitative analysis of the accidents by being isomorphic toMC Third fuzzy random method is applied to analyze theperformance of the model which can improve the reliabilityof the results of the quantitative analysis Finally accordingto the data extracted from a large amount of historical data ofthe accident statistics we have carried out a case analysis anddiscussion which may be useful to the department of railwaysafety management

The remainder of this paper is organized as followsFundamental theory of SPN and theoretical basis of systemmodeling and performance analysis using SPN are given inSection 2 The model of train rear-end collision accidentsbased on SPN theory is proposed in Section 3 A case analysisand discussion are presented in Section 4 The last section isthe conclusion of this paper

2 Methods

Dr Petri [9] proposed Petri nets theory in 1962 when hedeveloped information flow model for computer operationsystem At present PN is extensively applied to modelingperformance analysis and control research for discrete eventdynamic system [10] As a system model PN is dynamicand concurrent In PN standard graphic presentation thesign ldquoOrdquo indicates place and the sign ldquo|rdquo or pane indicatestransition [11]

21 Fundamentals of PN

(1) Definition in general a PN is defined as follows sum =

(119875 119879 119865 119870 119882 1198720) where

(a) 119875 = 1199011 1199012 119901

119899 119899 gt 0 refers to the finite set

of place(b) 119879 = 119905

1 1199052 119905

119898 119898 gt 0 refers to the finite set

of transition(c) 119865 is an ordered pair set consisting of 119875 and 119879

and satisfies 119865 sube 119875 times 119879 cup 119879 times 119875 119875 cap 119879 = and119879 cup 119875 =

(d) 119870 is a capacity function(e) 119882 is a weight function weighting all arc lines

119908(119901 119905) or 119908(119905 119901) is used to denote the weight ofdirected arc going from 119901 to 119905 or 119905 to 119901

(f) 119872 is the marking reflecting token distributionin all places and 119872

0is the initial marking

(2) Enabling and stimulation rules of transition are asfollows

(a) transition 119905 isin 119879 is enabled when and onlywhenin respect of all 119901 isin 119875 119908(119901 119905) = 1 119872(119901) gt 0

(b) after the enabled transition 119905 is stimulated by themarking 119872 a new marking 119872

1015840 is generated inrespect of forall119901 isin 119875 which is

1198721015840(119901) =

119872 (119901) minus 119908 (119901 119905) 119901 isin119900119905

119872 (119901) + 119908 (119901 119905) 119901 isin 119905119900

119872 (119901) + 119908 (119901 119905) minus 119908 (119901 119905) 119901 isin119900119905 and 119901 isin 119905

119900

119872 (119901) other(1)

22 Theoretical Basis of System Modeling and PerformanceAnalysis Using SPN Description of a system by PN is usuallybased on two concepts event and state PN does not onlypresent the graph of a system but also provide themathemat-ical description of the system [12] To describe the dynamicaction of a system execution of the processing task can beindicated by corresponding transition firing In the PN thetransferring of tokens expresses the information processThemarkings of the PN denote the special state of the systemForward markings decide the set of all possible states of thesystem with a given initial state

PN provides a new description tool for system perfor-mance analysis [13] In the continuous-time SPN certaintime delay is needed for a transition from being generableto practical firing that is the period from a transition beinggenerable to its firing is regarded as a continuous randomvariant which is subject to exponential distribution Whenthe firing rate of transitions is exponentially distributedand the markings are countable it has been proved that acontinuous-time SPN is isomorphic to a continuous-timeMC [14] So each marking of SPN is mapping into a state ofMC and the occurrence graph of SPN is isomorphic to thestate space of MC

Therefore referring to the SPNoccurrence graphwhich isisomorphic to a homogeneous MC we can utilize the theoryof random processes to analyze ourmodelThe application ofSPN to system performance analysis is usually as follows

(1) Build the SPN model for the system(2) Define the possible states of SPN and derive its

reachable marking set(3) Obtain the isomorphic MC of SPN(4) Analyze the system performance based on the stable

probability of MC

Based on its stable probability we can further analyzethe system performance indexes and time characteristicsanalyze the busyness andwork efficiency of thewhole systems

Mathematical Problems in Engineering 3

or transitions identify main factors affecting the systemperformance and analyze the resources distribution andoptimization plans for the system under different commandand dispatch methods

3 Modeling and Analysis of Train Rear-EndCollision Accidents Based on SPN

31 Modeling of Train Rear-End Collision Accidents Basedon SPN In this paper we consider the case of one trackonly in the one-way and assume that the collision avoidancesystems such as signal control system train distance controlsystem train state communication and control system andthe danger alarm system are applied in the problemThe traintrack signal failure may be caused by some environmentalfactors such as lightning strike According to Li [6] andour knowledge and understandings of the railway trafficthe major procedure of a train rear-end collision accidentis shown in Figure 1 Based on the major procedure of atrain rear-end collision accident and the theory of SPN weproposed the train rear-end collision accidents model asshown in Figure 2

Places and transitions in Figure 2 are explained as followsFinite set of places 119875 are as follows 119875

1 two successive trains

are driving normally 1198752 velocity of the follow-up train is

greater in two successive trains 1198753 velocity of follow-up

train is still greater 1198754 it refers to risk of train rear-end

collision accident 1198755 it refers to intervention of automatic

train protection system (ATP) 1198756 ATP fails to intervene

1198757 ATP intervenes successfully 119875

8 it refers to intervention

of dispatcher 1198759 dispatcher intervenes successfully 119875

10 it

refers to effectiveness judgment of dispatcherrsquos measures 11987511

dispatcher fails to intervene 11987512 it refers to intervention

of driver 11987513 driver detects the abnormal 119875

14 it refers to

effectiveness judgment of braking 11987515 it refers to train rear-

end collision accident 11987516 trains are safe controlled

Finite set of transitions 119879 are as follows 1199051 it refers

to velocity difference in two successive trains (velocity offollow-up train is greater) 119905

2 interval distance in two

successive trains is much larger than the minimum instan-taneous distance 119905

3 interval distance in two successive

trains is approaching the minimum instantaneous distance1199054 velocity difference is eliminated 119905

5 interval distance in

two successive trains is decreasing 1199056 ATP works auto-

matically 1199057 ATP fails to work 119905

8 ATP works normally

1199059 dispatcher intervenes 119905

10 dispatcher does not detect or

neglect the abnormal 11990511 dispatcher detects the abnormal

11990512 dispatcher takes measures 119905

13 dispatcherrsquos measures fail

to work 11990514 driver intervenes 119905

15 ATP adjusts the trainrsquos

driving state automatically 11990516 dispatcherrsquos measures work

11990517 driver detects the abnormal 119905

18 driver does not detect the

abnormal 11990519 driver operates to brake 119905

20 train fails to stop

before rear-end collision accident happened 11990521 driver stops

the train successfully 11990522 the trainrsquos driving state is adjusted

11990523 it refers to cleaning up for restoring the line operation

32 Effectiveness Analysis of the Model Based on MarkovProcess According to the SPN model as shown in Figure 2

Interval distance in the two successive trains is approaching the minimum

instantaneous distance

ATP failed to work

Two successive trains drive toward thesame direction on the same track

Velocity of follow-up train is greater inthe two successive trains

Train rear-end collision accident

The driver dose not detect abnormal orfailed to brake to stop the train

The dispatcher does not detect abnormal or failed to take effective measures

Figure 1Major procedure of occurrence of a train rear-end collisionaccident

we can obtain its isomorphic MC in Figure 3 where 1198721is

the initial marking containing only one token in the place 1198751

and the Markov state space can be obtained as 1198721to 11987216

as a result of the transferring of the token and the firing ofdifferent sets of transitions Since most process service timeis close to the exponential distribution and the exponentialdistribution simplifies the analysis of the randomnetwork weassume that the firing rate of each transition is exponentiallydistributed in the model Here the average firing rate oftransitions 119905

1 1199052 119905

23are 1205821 1205822 120582

23 respectively

From the MC of the SPN model the following conclu-sions can be obtained

(1) No obstruction occurred in the entire process and thetransform of token is smooth The process does notwait indefinitely for a state indicating that all tran-sitions in the process of the train rear-end collisionaccidents are likely to be implemented in a certainperiod of timeThe successful implementation of eachtransition is the basis for the next transition to beimplemented successfully

(2) A state 119872119894does not exist which cannot be reached

namely no state can never occur and a deadlock statedoes not exist in the model

4 Case Analysis and Discussion

41 Performance Analysis of SPN Model Assuming 119899 stateslocated in the MC we can immediately derive the transferrate matrix119876 = [119902

119894119895] 1 le 119894 119895 le 119899The element of the transfer

matrix is obtained intuitively as follows

4 Mathematical Problems in Engineering

P1 P2

P3

P4 P5

P6

P7

P8

P9

P10

P11P12

P13

P14

P15

P16

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12 t13 t14

t15 t16

t17

t18

t19t20

t21

t22

t23

Figure 2 Model of a train rear-end collision accident based on stochastic Petri nets

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16

12058211205822

1205823

1205824 1205825

1205826

1205827

1205828

120582912058210

12058211 12058212

12058213 12058214

12058215

1205821612058217

12058218

12058219

12058220

12058221

12058222

12058223

Figure 3 Markov chain of the stochastic Petri nets model

(1) For the element 119902119894119895which is located in nondiagonal

when there is an arc connecting state 119872119894with state

119872119895 then the firing rate 120582

119894marked on the arc is the

value of 119902119894119895 if there is not an arc connecting state 119872

119894

with state 119872119895 then 119902

119894119895= 0

(2) For the element 119902119894119895which is located in diagonal the

negative number of the sum of the rate 120582119894marked on

the arcs which are outputting from the state 119872119894is the

value of 119902119894119895[12]

Assuming that the stable probability of the 119899 states inthe MC is a row vector 119883 = (119909

1 1199092 119909

119899) we can get the

following linear equations according to the Markov process

119883119876 = 0

sum119894

119909119894= 1 1 le 119894 le 119899

(2)

Since 120582119894is extracted and estimated on the basis of the

statistics data which is collected from the previous train acci-dents investigations there may be incompletion inaccuracyand uncertainty Therefore using different fuzzy degrees (2)can be transferred into the fuzzy stable probability equationsand solving these fuzzy equations we can calculate the steadystate probability Because the value 120582

119894is usually around a

certain value120582 we can use the trianglemembership functions

to represent fuzzy numbers Mathematically the trianglemembership function 119906

120582119894(119909) can be expressed as follows [15]

119906120582119894

(119909) =

0 119909 le 1198861198941

(119909 minus 1198861198941

)

(1198861198942

minus 1198861198941

) 1198861198941

le 119909 le 1198861198942

(1198861198943

minus 119909)

(1198861198943

minus 1198861198942

) 1198861198942

le 119909 le 1198861198943

0 119909 ge 1198861198943

(3)

The fuzzy variables 120582119894can be represented by a triple

(1198861 1198862 1198863) and the parameter 119886

1198942defines the maximum

degree of 120582119894 namely 119906

120582119894(1198861198942

) = 1 parameters 1198861198941and 119886

1198943

define the minimum limit and the maximum limit of thefuzzy number In addition the triangle membership function(1198861 1198862 1198863) defines a 120572-cut of the fuzzy number In fact the

120572-cut 119860(120572)

= [119886(120572)

1 119886(120572)

3] defines a confidence interval of the

triangular fuzzy number expressed as 119860(120572)

= [1198861

+ (1198862

minus

1198861)120572 1198863

minus (1198863

minus 1198862)120572] (see Figure 4)

Because a concrete source which can provide the valuesof firing rate 120582

119894does not exist we first obtained the data of

the frequency of related events of the responding transitionfrom the statistics data of the previous train accidentsinvestigations and according to it we get the values of thefiring rate 120582

119894after our proper estimation and modification

which are listed in Table 1 with a unit of bout per hour Sincethe uncertainty of the data has been considered by applyingthe fuzzy random method the data are representative to beused for the further analysis

Mathematical Problems in Engineering 5

Table 1 Value of the firing rate 120582119894

1205821

1205822

1205823

1205824

1205825

1205826

1205827

1205828

1205829

12058210

12058211

12058212

10 2 2 4 4 10 1 4 10 1 4 1012058213

12058214

12058215

12058216

12058217

12058218

12058219

12058220

12058221

12058222

12058223

1 1 10 4 4 1 10 2 2 10 1

Table 2 Crisp number and triangular number of the firing rate 120582119894

120582119894

Crisp number Spread Triangular fuzzy number1205821 1205826 1205829 12058212

12058215

12058219

12058222

10 plusmn20 (8 10 12)1205824 1205825 1205828 12058211

12058216

12058217

4 plusmn15 (34 4 46)1205822 1205823 12058220

12058221

2 plusmn10 (18 2 22)1205827 12058210

12058213

12058214

12058218

12058223

1 plusmn5 (095 1 105)

1

120583A

(x)

120572

a1 a1205721 a2 a1205723a3

Figure 4 The 120572-cut of a triangular fuzzy number

In order to account for the uncertainties in the data theobtained crisp data are converted into the fuzzy numbersMore specifically crisp numbers in the extracted data areconverted into fuzzy numbers with a known spread Thus wecan get the triangular fuzzy numbers of 120582

119894as listed in Table 2

Taking (09 10 11) as the triangular fuzzy numbers ofthe sum of stable probability we calculate the fuzzy stableprobability in the condition of triangular fuzzy number 120582

119894

Defuzzification is necessary to convert the fuzzy output to acrisp value as most of the actions or decisions implementedby human or machines are binary or crisp Out of theexistence of the various defuzzification techniques in theliterature center of gravity (COG) method is selected due toits property that it is equivalent to meaning of data [16] Ifthe membership function 119906

120582119894(119909) of the output fuzzy set 119860 is

described on the interval [1199091 1199092] then COG defuzzification

value 119909 can be defined as

119909 =int1199092

1199091

119909 sdot 119906120582119894

(119909) 119889119909

int1199092

1199091

119906120582119894

(119909) 119889119909 (4)

Based on the COG method the crisp values of the stableprobability are calculated since the placersquos busy probability

refers to the probability of events or states in the train rear-end accident process So we get the results as follows

119875 [119872 (1199011) = 1] = 119909

1= 0104

119875 [119872 (1199012) = 1] = 119909

2= 0260

119875 [119872 (1199013) = 1] = 119909

3= 0065

119875 [119872 (1199014) = 1] = 119909

4= 0078

119875 [119872 (1199015) = 1] = 119909

5= 0156

119875 [119872 (1199016) = 1] = 119909

6= 0016

119875 [119872 (1199017) = 1] = 119909

7= 0062

119875 [119872 (1199018) = 1] = 119909

8= 0031

119875 [119872 (1199019) = 1] = 119909

9= 0013

119875 [119872 (11990110

) = 1] = 11990910

= 0025

119875 [119872 (11990111

) = 1] = 11990911

= 0056

119875 [119872 (11990112

) = 1] = 11990912

= 0011

119875 [119872 (11990113

) = 1] = 11990913

= 0004

119875 [119872 (11990114

) = 1] = 11990914

= 0011

119875 [119872 (11990115

) = 1] = 11990915

= 0033

119875 [119872 (11990116

) = 1] = 11990916

= 0075

(5)

A comparison of the placersquos busy probability aboutwhether to introduce the fuzzy random method is carriedout In Figure 5 1 2 4 5 and 6 represent the placersquos busyprobability value in the condition that the firing rate of eachtransition takes its 119886

1 119886(05)

1 1198862 119886(05)

3 and 119886

3 respectively

as showed in the triangle membership function (1198861 1198862 1198863)

3 represents each placersquos busy probabilityrsquos defuzzificationvalue We observe that the defuzzification values of placersquosbusy probability are more stable compared to the direct useof crisp firing rate of the transitions which demonstrate thatfuzzy mathematical method accounted for the uncertainties

6 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8000

005

010

015

020

025

030

Prob

abili

ty

Cut level of firing rate of each transition

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 5 Stable probability of varying 120572-cut level of firing rate ofeach transition

of the firing rate of the transitions Thus the result is moreaccurate and reliable

In addition from the above data it can be seen that thebusy probabilities of places 119875

2and 119875

5are larger relatively

Place 1198752refers to the state that velocity of the follow-up

train is greater in two successive trains There are manyreasons leading to this state such as the driver violating thedriving instruction usually represented in speeding or slowlydriving the automatic control system being abnormal or thedispatcherrsquos misjudgment On the one hand the dispatcherrsquosmisjudgment is reflected by the dispatcherrsquos belief that thefront train cannot be caught up by the following train inthe driving sections to conduct the risk instruction whichsuggests that the dispatcher put too much faith in the driverand neglected risk on the other hand it is reflected by thefact that the dispatcher may believe in the collision avoidancesystem which can prevent the accident which suggests thatthe dispatcher put too much faith in the technology

In all of the factors leading to the state 1198752 some cannot

be avoided because it is the need of dispatching operationmanagement In this case we must be careful and shouldnot put too much faith in the driver or technology toneglect the unexpected adventure and to conduct any riskinstruction Other risk factors can be reduced or eliminatedby strengthening the management of drivers and other safetymeasures

The large busy probability of place 1198752inevitably leads to

a relatively large busy probability of place 1198755(intervention

of ATP) Because only a very small number of cases of thisvelocity difference are eliminated naturally a train generallyhas to rely on ATP to adjust automatically the train drivingstatus to ensure the safe driving of trains In case theATP worked normally in the long-term train drivers ordispatchers can easily depend on the vital role of the ATPIt is the reason why the accident tends to happen once ATPis not working properly Therefore it is more crucial for the

1 2 3 4 5 6 7 8 9000005010015020025030035040045050

Prob

abili

ty

Transition rate of transition number 3p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 6 Stable probabilities of varying transition rate 1205823

day-to-day maintenance of railway safeguard equipment andfacilities to ensure the safe driving of trains In additionthe vigilance of the train driver and dispatcher should bestrengthened in case of the ATP abnormalities

Furthermore the obtained result is consistent with theactual one that the busy probability of place 119875

15(the train

rear-end collision accident) is not very large However nomatter how low the probability is it does not mean that theaccident does not happen Even if we can accurately estimatethe accident we cannot accurately predict the moment of theaccident Because it involves significant consequences every-one from the top management to the low level employeesshould always be vigilant and keep a high level of securityalert

42 Discussion If we change the value of the firing rate 1205823

and values of 1205821 1205822 1205824 120582

23remain unchanged we can

get Figure 6 where 1 7 represent the stable probabilityvalue (namely the placersquos busy probability) in the conditionof the firing rate of transition 119905

3taking 0 05120582

3 1205823 151205823

21205823 251205823 31205823 respectively If we change the value of the

firing rate 1205827 1205828 12058210

12058211

12058213

12058216

12058218

12058220

12058221 we can also

get Figures 7ndash15 similarlyIn Figure 6 if 120582

3increases that is the frequency of inter-

val distance in two successive trains is approaching to theminimum instantaneous distance increases the placersquos busyprobability of119875

41198755 and119875

15increases remarkablyThis shows

that the intensive departure interval strategy will significantlyincrease safety risk and the probability of accidents Thus weshould make a tradeoff between the risk of accidents andintensive departure interval strategy which aim to improvethe transport capacity

From Figures 7 and 8 we find out that ATP plays avery important role as the first barrier for protection of trainrunning security Once the frequency of ATP fails to workincreasingly or the frequency of ATPwork normally declines

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

or transitions identify main factors affecting the systemperformance and analyze the resources distribution andoptimization plans for the system under different commandand dispatch methods

3 Modeling and Analysis of Train Rear-EndCollision Accidents Based on SPN

31 Modeling of Train Rear-End Collision Accidents Basedon SPN In this paper we consider the case of one trackonly in the one-way and assume that the collision avoidancesystems such as signal control system train distance controlsystem train state communication and control system andthe danger alarm system are applied in the problemThe traintrack signal failure may be caused by some environmentalfactors such as lightning strike According to Li [6] andour knowledge and understandings of the railway trafficthe major procedure of a train rear-end collision accidentis shown in Figure 1 Based on the major procedure of atrain rear-end collision accident and the theory of SPN weproposed the train rear-end collision accidents model asshown in Figure 2

Places and transitions in Figure 2 are explained as followsFinite set of places 119875 are as follows 119875

1 two successive trains

are driving normally 1198752 velocity of the follow-up train is

greater in two successive trains 1198753 velocity of follow-up

train is still greater 1198754 it refers to risk of train rear-end

collision accident 1198755 it refers to intervention of automatic

train protection system (ATP) 1198756 ATP fails to intervene

1198757 ATP intervenes successfully 119875

8 it refers to intervention

of dispatcher 1198759 dispatcher intervenes successfully 119875

10 it

refers to effectiveness judgment of dispatcherrsquos measures 11987511

dispatcher fails to intervene 11987512 it refers to intervention

of driver 11987513 driver detects the abnormal 119875

14 it refers to

effectiveness judgment of braking 11987515 it refers to train rear-

end collision accident 11987516 trains are safe controlled

Finite set of transitions 119879 are as follows 1199051 it refers

to velocity difference in two successive trains (velocity offollow-up train is greater) 119905

2 interval distance in two

successive trains is much larger than the minimum instan-taneous distance 119905

3 interval distance in two successive

trains is approaching the minimum instantaneous distance1199054 velocity difference is eliminated 119905

5 interval distance in

two successive trains is decreasing 1199056 ATP works auto-

matically 1199057 ATP fails to work 119905

8 ATP works normally

1199059 dispatcher intervenes 119905

10 dispatcher does not detect or

neglect the abnormal 11990511 dispatcher detects the abnormal

11990512 dispatcher takes measures 119905

13 dispatcherrsquos measures fail

to work 11990514 driver intervenes 119905

15 ATP adjusts the trainrsquos

driving state automatically 11990516 dispatcherrsquos measures work

11990517 driver detects the abnormal 119905

18 driver does not detect the

abnormal 11990519 driver operates to brake 119905

20 train fails to stop

before rear-end collision accident happened 11990521 driver stops

the train successfully 11990522 the trainrsquos driving state is adjusted

11990523 it refers to cleaning up for restoring the line operation

32 Effectiveness Analysis of the Model Based on MarkovProcess According to the SPN model as shown in Figure 2

Interval distance in the two successive trains is approaching the minimum

instantaneous distance

ATP failed to work

Two successive trains drive toward thesame direction on the same track

Velocity of follow-up train is greater inthe two successive trains

Train rear-end collision accident

The driver dose not detect abnormal orfailed to brake to stop the train

The dispatcher does not detect abnormal or failed to take effective measures

Figure 1Major procedure of occurrence of a train rear-end collisionaccident

we can obtain its isomorphic MC in Figure 3 where 1198721is

the initial marking containing only one token in the place 1198751

and the Markov state space can be obtained as 1198721to 11987216

as a result of the transferring of the token and the firing ofdifferent sets of transitions Since most process service timeis close to the exponential distribution and the exponentialdistribution simplifies the analysis of the randomnetwork weassume that the firing rate of each transition is exponentiallydistributed in the model Here the average firing rate oftransitions 119905

1 1199052 119905

23are 1205821 1205822 120582

23 respectively

From the MC of the SPN model the following conclu-sions can be obtained

(1) No obstruction occurred in the entire process and thetransform of token is smooth The process does notwait indefinitely for a state indicating that all tran-sitions in the process of the train rear-end collisionaccidents are likely to be implemented in a certainperiod of timeThe successful implementation of eachtransition is the basis for the next transition to beimplemented successfully

(2) A state 119872119894does not exist which cannot be reached

namely no state can never occur and a deadlock statedoes not exist in the model

4 Case Analysis and Discussion

41 Performance Analysis of SPN Model Assuming 119899 stateslocated in the MC we can immediately derive the transferrate matrix119876 = [119902

119894119895] 1 le 119894 119895 le 119899The element of the transfer

matrix is obtained intuitively as follows

4 Mathematical Problems in Engineering

P1 P2

P3

P4 P5

P6

P7

P8

P9

P10

P11P12

P13

P14

P15

P16

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12 t13 t14

t15 t16

t17

t18

t19t20

t21

t22

t23

Figure 2 Model of a train rear-end collision accident based on stochastic Petri nets

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16

12058211205822

1205823

1205824 1205825

1205826

1205827

1205828

120582912058210

12058211 12058212

12058213 12058214

12058215

1205821612058217

12058218

12058219

12058220

12058221

12058222

12058223

Figure 3 Markov chain of the stochastic Petri nets model

(1) For the element 119902119894119895which is located in nondiagonal

when there is an arc connecting state 119872119894with state

119872119895 then the firing rate 120582

119894marked on the arc is the

value of 119902119894119895 if there is not an arc connecting state 119872

119894

with state 119872119895 then 119902

119894119895= 0

(2) For the element 119902119894119895which is located in diagonal the

negative number of the sum of the rate 120582119894marked on

the arcs which are outputting from the state 119872119894is the

value of 119902119894119895[12]

Assuming that the stable probability of the 119899 states inthe MC is a row vector 119883 = (119909

1 1199092 119909

119899) we can get the

following linear equations according to the Markov process

119883119876 = 0

sum119894

119909119894= 1 1 le 119894 le 119899

(2)

Since 120582119894is extracted and estimated on the basis of the

statistics data which is collected from the previous train acci-dents investigations there may be incompletion inaccuracyand uncertainty Therefore using different fuzzy degrees (2)can be transferred into the fuzzy stable probability equationsand solving these fuzzy equations we can calculate the steadystate probability Because the value 120582

119894is usually around a

certain value120582 we can use the trianglemembership functions

to represent fuzzy numbers Mathematically the trianglemembership function 119906

120582119894(119909) can be expressed as follows [15]

119906120582119894

(119909) =

0 119909 le 1198861198941

(119909 minus 1198861198941

)

(1198861198942

minus 1198861198941

) 1198861198941

le 119909 le 1198861198942

(1198861198943

minus 119909)

(1198861198943

minus 1198861198942

) 1198861198942

le 119909 le 1198861198943

0 119909 ge 1198861198943

(3)

The fuzzy variables 120582119894can be represented by a triple

(1198861 1198862 1198863) and the parameter 119886

1198942defines the maximum

degree of 120582119894 namely 119906

120582119894(1198861198942

) = 1 parameters 1198861198941and 119886

1198943

define the minimum limit and the maximum limit of thefuzzy number In addition the triangle membership function(1198861 1198862 1198863) defines a 120572-cut of the fuzzy number In fact the

120572-cut 119860(120572)

= [119886(120572)

1 119886(120572)

3] defines a confidence interval of the

triangular fuzzy number expressed as 119860(120572)

= [1198861

+ (1198862

minus

1198861)120572 1198863

minus (1198863

minus 1198862)120572] (see Figure 4)

Because a concrete source which can provide the valuesof firing rate 120582

119894does not exist we first obtained the data of

the frequency of related events of the responding transitionfrom the statistics data of the previous train accidentsinvestigations and according to it we get the values of thefiring rate 120582

119894after our proper estimation and modification

which are listed in Table 1 with a unit of bout per hour Sincethe uncertainty of the data has been considered by applyingthe fuzzy random method the data are representative to beused for the further analysis

Mathematical Problems in Engineering 5

Table 1 Value of the firing rate 120582119894

1205821

1205822

1205823

1205824

1205825

1205826

1205827

1205828

1205829

12058210

12058211

12058212

10 2 2 4 4 10 1 4 10 1 4 1012058213

12058214

12058215

12058216

12058217

12058218

12058219

12058220

12058221

12058222

12058223

1 1 10 4 4 1 10 2 2 10 1

Table 2 Crisp number and triangular number of the firing rate 120582119894

120582119894

Crisp number Spread Triangular fuzzy number1205821 1205826 1205829 12058212

12058215

12058219

12058222

10 plusmn20 (8 10 12)1205824 1205825 1205828 12058211

12058216

12058217

4 plusmn15 (34 4 46)1205822 1205823 12058220

12058221

2 plusmn10 (18 2 22)1205827 12058210

12058213

12058214

12058218

12058223

1 plusmn5 (095 1 105)

1

120583A

(x)

120572

a1 a1205721 a2 a1205723a3

Figure 4 The 120572-cut of a triangular fuzzy number

In order to account for the uncertainties in the data theobtained crisp data are converted into the fuzzy numbersMore specifically crisp numbers in the extracted data areconverted into fuzzy numbers with a known spread Thus wecan get the triangular fuzzy numbers of 120582

119894as listed in Table 2

Taking (09 10 11) as the triangular fuzzy numbers ofthe sum of stable probability we calculate the fuzzy stableprobability in the condition of triangular fuzzy number 120582

119894

Defuzzification is necessary to convert the fuzzy output to acrisp value as most of the actions or decisions implementedby human or machines are binary or crisp Out of theexistence of the various defuzzification techniques in theliterature center of gravity (COG) method is selected due toits property that it is equivalent to meaning of data [16] Ifthe membership function 119906

120582119894(119909) of the output fuzzy set 119860 is

described on the interval [1199091 1199092] then COG defuzzification

value 119909 can be defined as

119909 =int1199092

1199091

119909 sdot 119906120582119894

(119909) 119889119909

int1199092

1199091

119906120582119894

(119909) 119889119909 (4)

Based on the COG method the crisp values of the stableprobability are calculated since the placersquos busy probability

refers to the probability of events or states in the train rear-end accident process So we get the results as follows

119875 [119872 (1199011) = 1] = 119909

1= 0104

119875 [119872 (1199012) = 1] = 119909

2= 0260

119875 [119872 (1199013) = 1] = 119909

3= 0065

119875 [119872 (1199014) = 1] = 119909

4= 0078

119875 [119872 (1199015) = 1] = 119909

5= 0156

119875 [119872 (1199016) = 1] = 119909

6= 0016

119875 [119872 (1199017) = 1] = 119909

7= 0062

119875 [119872 (1199018) = 1] = 119909

8= 0031

119875 [119872 (1199019) = 1] = 119909

9= 0013

119875 [119872 (11990110

) = 1] = 11990910

= 0025

119875 [119872 (11990111

) = 1] = 11990911

= 0056

119875 [119872 (11990112

) = 1] = 11990912

= 0011

119875 [119872 (11990113

) = 1] = 11990913

= 0004

119875 [119872 (11990114

) = 1] = 11990914

= 0011

119875 [119872 (11990115

) = 1] = 11990915

= 0033

119875 [119872 (11990116

) = 1] = 11990916

= 0075

(5)

A comparison of the placersquos busy probability aboutwhether to introduce the fuzzy random method is carriedout In Figure 5 1 2 4 5 and 6 represent the placersquos busyprobability value in the condition that the firing rate of eachtransition takes its 119886

1 119886(05)

1 1198862 119886(05)

3 and 119886

3 respectively

as showed in the triangle membership function (1198861 1198862 1198863)

3 represents each placersquos busy probabilityrsquos defuzzificationvalue We observe that the defuzzification values of placersquosbusy probability are more stable compared to the direct useof crisp firing rate of the transitions which demonstrate thatfuzzy mathematical method accounted for the uncertainties

6 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8000

005

010

015

020

025

030

Prob

abili

ty

Cut level of firing rate of each transition

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 5 Stable probability of varying 120572-cut level of firing rate ofeach transition

of the firing rate of the transitions Thus the result is moreaccurate and reliable

In addition from the above data it can be seen that thebusy probabilities of places 119875

2and 119875

5are larger relatively

Place 1198752refers to the state that velocity of the follow-up

train is greater in two successive trains There are manyreasons leading to this state such as the driver violating thedriving instruction usually represented in speeding or slowlydriving the automatic control system being abnormal or thedispatcherrsquos misjudgment On the one hand the dispatcherrsquosmisjudgment is reflected by the dispatcherrsquos belief that thefront train cannot be caught up by the following train inthe driving sections to conduct the risk instruction whichsuggests that the dispatcher put too much faith in the driverand neglected risk on the other hand it is reflected by thefact that the dispatcher may believe in the collision avoidancesystem which can prevent the accident which suggests thatthe dispatcher put too much faith in the technology

In all of the factors leading to the state 1198752 some cannot

be avoided because it is the need of dispatching operationmanagement In this case we must be careful and shouldnot put too much faith in the driver or technology toneglect the unexpected adventure and to conduct any riskinstruction Other risk factors can be reduced or eliminatedby strengthening the management of drivers and other safetymeasures

The large busy probability of place 1198752inevitably leads to

a relatively large busy probability of place 1198755(intervention

of ATP) Because only a very small number of cases of thisvelocity difference are eliminated naturally a train generallyhas to rely on ATP to adjust automatically the train drivingstatus to ensure the safe driving of trains In case theATP worked normally in the long-term train drivers ordispatchers can easily depend on the vital role of the ATPIt is the reason why the accident tends to happen once ATPis not working properly Therefore it is more crucial for the

1 2 3 4 5 6 7 8 9000005010015020025030035040045050

Prob

abili

ty

Transition rate of transition number 3p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 6 Stable probabilities of varying transition rate 1205823

day-to-day maintenance of railway safeguard equipment andfacilities to ensure the safe driving of trains In additionthe vigilance of the train driver and dispatcher should bestrengthened in case of the ATP abnormalities

Furthermore the obtained result is consistent with theactual one that the busy probability of place 119875

15(the train

rear-end collision accident) is not very large However nomatter how low the probability is it does not mean that theaccident does not happen Even if we can accurately estimatethe accident we cannot accurately predict the moment of theaccident Because it involves significant consequences every-one from the top management to the low level employeesshould always be vigilant and keep a high level of securityalert

42 Discussion If we change the value of the firing rate 1205823

and values of 1205821 1205822 1205824 120582

23remain unchanged we can

get Figure 6 where 1 7 represent the stable probabilityvalue (namely the placersquos busy probability) in the conditionof the firing rate of transition 119905

3taking 0 05120582

3 1205823 151205823

21205823 251205823 31205823 respectively If we change the value of the

firing rate 1205827 1205828 12058210

12058211

12058213

12058216

12058218

12058220

12058221 we can also

get Figures 7ndash15 similarlyIn Figure 6 if 120582

3increases that is the frequency of inter-

val distance in two successive trains is approaching to theminimum instantaneous distance increases the placersquos busyprobability of119875

41198755 and119875

15increases remarkablyThis shows

that the intensive departure interval strategy will significantlyincrease safety risk and the probability of accidents Thus weshould make a tradeoff between the risk of accidents andintensive departure interval strategy which aim to improvethe transport capacity

From Figures 7 and 8 we find out that ATP plays avery important role as the first barrier for protection of trainrunning security Once the frequency of ATP fails to workincreasingly or the frequency of ATPwork normally declines

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

P1 P2

P3

P4 P5

P6

P7

P8

P9

P10

P11P12

P13

P14

P15

P16

t1

t2

t3

t4

t5

t6

t7

t8

t9

t10

t11

t12 t13 t14

t15 t16

t17

t18

t19t20

t21

t22

t23

Figure 2 Model of a train rear-end collision accident based on stochastic Petri nets

M1 M2 M3 M4 M5 M6 M7 M8 M9 M10 M11 M12 M13 M14 M15 M16

12058211205822

1205823

1205824 1205825

1205826

1205827

1205828

120582912058210

12058211 12058212

12058213 12058214

12058215

1205821612058217

12058218

12058219

12058220

12058221

12058222

12058223

Figure 3 Markov chain of the stochastic Petri nets model

(1) For the element 119902119894119895which is located in nondiagonal

when there is an arc connecting state 119872119894with state

119872119895 then the firing rate 120582

119894marked on the arc is the

value of 119902119894119895 if there is not an arc connecting state 119872

119894

with state 119872119895 then 119902

119894119895= 0

(2) For the element 119902119894119895which is located in diagonal the

negative number of the sum of the rate 120582119894marked on

the arcs which are outputting from the state 119872119894is the

value of 119902119894119895[12]

Assuming that the stable probability of the 119899 states inthe MC is a row vector 119883 = (119909

1 1199092 119909

119899) we can get the

following linear equations according to the Markov process

119883119876 = 0

sum119894

119909119894= 1 1 le 119894 le 119899

(2)

Since 120582119894is extracted and estimated on the basis of the

statistics data which is collected from the previous train acci-dents investigations there may be incompletion inaccuracyand uncertainty Therefore using different fuzzy degrees (2)can be transferred into the fuzzy stable probability equationsand solving these fuzzy equations we can calculate the steadystate probability Because the value 120582

119894is usually around a

certain value120582 we can use the trianglemembership functions

to represent fuzzy numbers Mathematically the trianglemembership function 119906

120582119894(119909) can be expressed as follows [15]

119906120582119894

(119909) =

0 119909 le 1198861198941

(119909 minus 1198861198941

)

(1198861198942

minus 1198861198941

) 1198861198941

le 119909 le 1198861198942

(1198861198943

minus 119909)

(1198861198943

minus 1198861198942

) 1198861198942

le 119909 le 1198861198943

0 119909 ge 1198861198943

(3)

The fuzzy variables 120582119894can be represented by a triple

(1198861 1198862 1198863) and the parameter 119886

1198942defines the maximum

degree of 120582119894 namely 119906

120582119894(1198861198942

) = 1 parameters 1198861198941and 119886

1198943

define the minimum limit and the maximum limit of thefuzzy number In addition the triangle membership function(1198861 1198862 1198863) defines a 120572-cut of the fuzzy number In fact the

120572-cut 119860(120572)

= [119886(120572)

1 119886(120572)

3] defines a confidence interval of the

triangular fuzzy number expressed as 119860(120572)

= [1198861

+ (1198862

minus

1198861)120572 1198863

minus (1198863

minus 1198862)120572] (see Figure 4)

Because a concrete source which can provide the valuesof firing rate 120582

119894does not exist we first obtained the data of

the frequency of related events of the responding transitionfrom the statistics data of the previous train accidentsinvestigations and according to it we get the values of thefiring rate 120582

119894after our proper estimation and modification

which are listed in Table 1 with a unit of bout per hour Sincethe uncertainty of the data has been considered by applyingthe fuzzy random method the data are representative to beused for the further analysis

Mathematical Problems in Engineering 5

Table 1 Value of the firing rate 120582119894

1205821

1205822

1205823

1205824

1205825

1205826

1205827

1205828

1205829

12058210

12058211

12058212

10 2 2 4 4 10 1 4 10 1 4 1012058213

12058214

12058215

12058216

12058217

12058218

12058219

12058220

12058221

12058222

12058223

1 1 10 4 4 1 10 2 2 10 1

Table 2 Crisp number and triangular number of the firing rate 120582119894

120582119894

Crisp number Spread Triangular fuzzy number1205821 1205826 1205829 12058212

12058215

12058219

12058222

10 plusmn20 (8 10 12)1205824 1205825 1205828 12058211

12058216

12058217

4 plusmn15 (34 4 46)1205822 1205823 12058220

12058221

2 plusmn10 (18 2 22)1205827 12058210

12058213

12058214

12058218

12058223

1 plusmn5 (095 1 105)

1

120583A

(x)

120572

a1 a1205721 a2 a1205723a3

Figure 4 The 120572-cut of a triangular fuzzy number

In order to account for the uncertainties in the data theobtained crisp data are converted into the fuzzy numbersMore specifically crisp numbers in the extracted data areconverted into fuzzy numbers with a known spread Thus wecan get the triangular fuzzy numbers of 120582

119894as listed in Table 2

Taking (09 10 11) as the triangular fuzzy numbers ofthe sum of stable probability we calculate the fuzzy stableprobability in the condition of triangular fuzzy number 120582

119894

Defuzzification is necessary to convert the fuzzy output to acrisp value as most of the actions or decisions implementedby human or machines are binary or crisp Out of theexistence of the various defuzzification techniques in theliterature center of gravity (COG) method is selected due toits property that it is equivalent to meaning of data [16] Ifthe membership function 119906

120582119894(119909) of the output fuzzy set 119860 is

described on the interval [1199091 1199092] then COG defuzzification

value 119909 can be defined as

119909 =int1199092

1199091

119909 sdot 119906120582119894

(119909) 119889119909

int1199092

1199091

119906120582119894

(119909) 119889119909 (4)

Based on the COG method the crisp values of the stableprobability are calculated since the placersquos busy probability

refers to the probability of events or states in the train rear-end accident process So we get the results as follows

119875 [119872 (1199011) = 1] = 119909

1= 0104

119875 [119872 (1199012) = 1] = 119909

2= 0260

119875 [119872 (1199013) = 1] = 119909

3= 0065

119875 [119872 (1199014) = 1] = 119909

4= 0078

119875 [119872 (1199015) = 1] = 119909

5= 0156

119875 [119872 (1199016) = 1] = 119909

6= 0016

119875 [119872 (1199017) = 1] = 119909

7= 0062

119875 [119872 (1199018) = 1] = 119909

8= 0031

119875 [119872 (1199019) = 1] = 119909

9= 0013

119875 [119872 (11990110

) = 1] = 11990910

= 0025

119875 [119872 (11990111

) = 1] = 11990911

= 0056

119875 [119872 (11990112

) = 1] = 11990912

= 0011

119875 [119872 (11990113

) = 1] = 11990913

= 0004

119875 [119872 (11990114

) = 1] = 11990914

= 0011

119875 [119872 (11990115

) = 1] = 11990915

= 0033

119875 [119872 (11990116

) = 1] = 11990916

= 0075

(5)

A comparison of the placersquos busy probability aboutwhether to introduce the fuzzy random method is carriedout In Figure 5 1 2 4 5 and 6 represent the placersquos busyprobability value in the condition that the firing rate of eachtransition takes its 119886

1 119886(05)

1 1198862 119886(05)

3 and 119886

3 respectively

as showed in the triangle membership function (1198861 1198862 1198863)

3 represents each placersquos busy probabilityrsquos defuzzificationvalue We observe that the defuzzification values of placersquosbusy probability are more stable compared to the direct useof crisp firing rate of the transitions which demonstrate thatfuzzy mathematical method accounted for the uncertainties

6 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8000

005

010

015

020

025

030

Prob

abili

ty

Cut level of firing rate of each transition

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 5 Stable probability of varying 120572-cut level of firing rate ofeach transition

of the firing rate of the transitions Thus the result is moreaccurate and reliable

In addition from the above data it can be seen that thebusy probabilities of places 119875

2and 119875

5are larger relatively

Place 1198752refers to the state that velocity of the follow-up

train is greater in two successive trains There are manyreasons leading to this state such as the driver violating thedriving instruction usually represented in speeding or slowlydriving the automatic control system being abnormal or thedispatcherrsquos misjudgment On the one hand the dispatcherrsquosmisjudgment is reflected by the dispatcherrsquos belief that thefront train cannot be caught up by the following train inthe driving sections to conduct the risk instruction whichsuggests that the dispatcher put too much faith in the driverand neglected risk on the other hand it is reflected by thefact that the dispatcher may believe in the collision avoidancesystem which can prevent the accident which suggests thatthe dispatcher put too much faith in the technology

In all of the factors leading to the state 1198752 some cannot

be avoided because it is the need of dispatching operationmanagement In this case we must be careful and shouldnot put too much faith in the driver or technology toneglect the unexpected adventure and to conduct any riskinstruction Other risk factors can be reduced or eliminatedby strengthening the management of drivers and other safetymeasures

The large busy probability of place 1198752inevitably leads to

a relatively large busy probability of place 1198755(intervention

of ATP) Because only a very small number of cases of thisvelocity difference are eliminated naturally a train generallyhas to rely on ATP to adjust automatically the train drivingstatus to ensure the safe driving of trains In case theATP worked normally in the long-term train drivers ordispatchers can easily depend on the vital role of the ATPIt is the reason why the accident tends to happen once ATPis not working properly Therefore it is more crucial for the

1 2 3 4 5 6 7 8 9000005010015020025030035040045050

Prob

abili

ty

Transition rate of transition number 3p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 6 Stable probabilities of varying transition rate 1205823

day-to-day maintenance of railway safeguard equipment andfacilities to ensure the safe driving of trains In additionthe vigilance of the train driver and dispatcher should bestrengthened in case of the ATP abnormalities

Furthermore the obtained result is consistent with theactual one that the busy probability of place 119875

15(the train

rear-end collision accident) is not very large However nomatter how low the probability is it does not mean that theaccident does not happen Even if we can accurately estimatethe accident we cannot accurately predict the moment of theaccident Because it involves significant consequences every-one from the top management to the low level employeesshould always be vigilant and keep a high level of securityalert

42 Discussion If we change the value of the firing rate 1205823

and values of 1205821 1205822 1205824 120582

23remain unchanged we can

get Figure 6 where 1 7 represent the stable probabilityvalue (namely the placersquos busy probability) in the conditionof the firing rate of transition 119905

3taking 0 05120582

3 1205823 151205823

21205823 251205823 31205823 respectively If we change the value of the

firing rate 1205827 1205828 12058210

12058211

12058213

12058216

12058218

12058220

12058221 we can also

get Figures 7ndash15 similarlyIn Figure 6 if 120582

3increases that is the frequency of inter-

val distance in two successive trains is approaching to theminimum instantaneous distance increases the placersquos busyprobability of119875

41198755 and119875

15increases remarkablyThis shows

that the intensive departure interval strategy will significantlyincrease safety risk and the probability of accidents Thus weshould make a tradeoff between the risk of accidents andintensive departure interval strategy which aim to improvethe transport capacity

From Figures 7 and 8 we find out that ATP plays avery important role as the first barrier for protection of trainrunning security Once the frequency of ATP fails to workincreasingly or the frequency of ATPwork normally declines

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Table 1 Value of the firing rate 120582119894

1205821

1205822

1205823

1205824

1205825

1205826

1205827

1205828

1205829

12058210

12058211

12058212

10 2 2 4 4 10 1 4 10 1 4 1012058213

12058214

12058215

12058216

12058217

12058218

12058219

12058220

12058221

12058222

12058223

1 1 10 4 4 1 10 2 2 10 1

Table 2 Crisp number and triangular number of the firing rate 120582119894

120582119894

Crisp number Spread Triangular fuzzy number1205821 1205826 1205829 12058212

12058215

12058219

12058222

10 plusmn20 (8 10 12)1205824 1205825 1205828 12058211

12058216

12058217

4 plusmn15 (34 4 46)1205822 1205823 12058220

12058221

2 plusmn10 (18 2 22)1205827 12058210

12058213

12058214

12058218

12058223

1 plusmn5 (095 1 105)

1

120583A

(x)

120572

a1 a1205721 a2 a1205723a3

Figure 4 The 120572-cut of a triangular fuzzy number

In order to account for the uncertainties in the data theobtained crisp data are converted into the fuzzy numbersMore specifically crisp numbers in the extracted data areconverted into fuzzy numbers with a known spread Thus wecan get the triangular fuzzy numbers of 120582

119894as listed in Table 2

Taking (09 10 11) as the triangular fuzzy numbers ofthe sum of stable probability we calculate the fuzzy stableprobability in the condition of triangular fuzzy number 120582

119894

Defuzzification is necessary to convert the fuzzy output to acrisp value as most of the actions or decisions implementedby human or machines are binary or crisp Out of theexistence of the various defuzzification techniques in theliterature center of gravity (COG) method is selected due toits property that it is equivalent to meaning of data [16] Ifthe membership function 119906

120582119894(119909) of the output fuzzy set 119860 is

described on the interval [1199091 1199092] then COG defuzzification

value 119909 can be defined as

119909 =int1199092

1199091

119909 sdot 119906120582119894

(119909) 119889119909

int1199092

1199091

119906120582119894

(119909) 119889119909 (4)

Based on the COG method the crisp values of the stableprobability are calculated since the placersquos busy probability

refers to the probability of events or states in the train rear-end accident process So we get the results as follows

119875 [119872 (1199011) = 1] = 119909

1= 0104

119875 [119872 (1199012) = 1] = 119909

2= 0260

119875 [119872 (1199013) = 1] = 119909

3= 0065

119875 [119872 (1199014) = 1] = 119909

4= 0078

119875 [119872 (1199015) = 1] = 119909

5= 0156

119875 [119872 (1199016) = 1] = 119909

6= 0016

119875 [119872 (1199017) = 1] = 119909

7= 0062

119875 [119872 (1199018) = 1] = 119909

8= 0031

119875 [119872 (1199019) = 1] = 119909

9= 0013

119875 [119872 (11990110

) = 1] = 11990910

= 0025

119875 [119872 (11990111

) = 1] = 11990911

= 0056

119875 [119872 (11990112

) = 1] = 11990912

= 0011

119875 [119872 (11990113

) = 1] = 11990913

= 0004

119875 [119872 (11990114

) = 1] = 11990914

= 0011

119875 [119872 (11990115

) = 1] = 11990915

= 0033

119875 [119872 (11990116

) = 1] = 11990916

= 0075

(5)

A comparison of the placersquos busy probability aboutwhether to introduce the fuzzy random method is carriedout In Figure 5 1 2 4 5 and 6 represent the placersquos busyprobability value in the condition that the firing rate of eachtransition takes its 119886

1 119886(05)

1 1198862 119886(05)

3 and 119886

3 respectively

as showed in the triangle membership function (1198861 1198862 1198863)

3 represents each placersquos busy probabilityrsquos defuzzificationvalue We observe that the defuzzification values of placersquosbusy probability are more stable compared to the direct useof crisp firing rate of the transitions which demonstrate thatfuzzy mathematical method accounted for the uncertainties

6 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8000

005

010

015

020

025

030

Prob

abili

ty

Cut level of firing rate of each transition

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 5 Stable probability of varying 120572-cut level of firing rate ofeach transition

of the firing rate of the transitions Thus the result is moreaccurate and reliable

In addition from the above data it can be seen that thebusy probabilities of places 119875

2and 119875

5are larger relatively

Place 1198752refers to the state that velocity of the follow-up

train is greater in two successive trains There are manyreasons leading to this state such as the driver violating thedriving instruction usually represented in speeding or slowlydriving the automatic control system being abnormal or thedispatcherrsquos misjudgment On the one hand the dispatcherrsquosmisjudgment is reflected by the dispatcherrsquos belief that thefront train cannot be caught up by the following train inthe driving sections to conduct the risk instruction whichsuggests that the dispatcher put too much faith in the driverand neglected risk on the other hand it is reflected by thefact that the dispatcher may believe in the collision avoidancesystem which can prevent the accident which suggests thatthe dispatcher put too much faith in the technology

In all of the factors leading to the state 1198752 some cannot

be avoided because it is the need of dispatching operationmanagement In this case we must be careful and shouldnot put too much faith in the driver or technology toneglect the unexpected adventure and to conduct any riskinstruction Other risk factors can be reduced or eliminatedby strengthening the management of drivers and other safetymeasures

The large busy probability of place 1198752inevitably leads to

a relatively large busy probability of place 1198755(intervention

of ATP) Because only a very small number of cases of thisvelocity difference are eliminated naturally a train generallyhas to rely on ATP to adjust automatically the train drivingstatus to ensure the safe driving of trains In case theATP worked normally in the long-term train drivers ordispatchers can easily depend on the vital role of the ATPIt is the reason why the accident tends to happen once ATPis not working properly Therefore it is more crucial for the

1 2 3 4 5 6 7 8 9000005010015020025030035040045050

Prob

abili

ty

Transition rate of transition number 3p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 6 Stable probabilities of varying transition rate 1205823

day-to-day maintenance of railway safeguard equipment andfacilities to ensure the safe driving of trains In additionthe vigilance of the train driver and dispatcher should bestrengthened in case of the ATP abnormalities

Furthermore the obtained result is consistent with theactual one that the busy probability of place 119875

15(the train

rear-end collision accident) is not very large However nomatter how low the probability is it does not mean that theaccident does not happen Even if we can accurately estimatethe accident we cannot accurately predict the moment of theaccident Because it involves significant consequences every-one from the top management to the low level employeesshould always be vigilant and keep a high level of securityalert

42 Discussion If we change the value of the firing rate 1205823

and values of 1205821 1205822 1205824 120582

23remain unchanged we can

get Figure 6 where 1 7 represent the stable probabilityvalue (namely the placersquos busy probability) in the conditionof the firing rate of transition 119905

3taking 0 05120582

3 1205823 151205823

21205823 251205823 31205823 respectively If we change the value of the

firing rate 1205827 1205828 12058210

12058211

12058213

12058216

12058218

12058220

12058221 we can also

get Figures 7ndash15 similarlyIn Figure 6 if 120582

3increases that is the frequency of inter-

val distance in two successive trains is approaching to theminimum instantaneous distance increases the placersquos busyprobability of119875

41198755 and119875

15increases remarkablyThis shows

that the intensive departure interval strategy will significantlyincrease safety risk and the probability of accidents Thus weshould make a tradeoff between the risk of accidents andintensive departure interval strategy which aim to improvethe transport capacity

From Figures 7 and 8 we find out that ATP plays avery important role as the first barrier for protection of trainrunning security Once the frequency of ATP fails to workincreasingly or the frequency of ATPwork normally declines

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

1 2 3 4 5 6 7 8000

005

010

015

020

025

030

Prob

abili

ty

Cut level of firing rate of each transition

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 5 Stable probability of varying 120572-cut level of firing rate ofeach transition

of the firing rate of the transitions Thus the result is moreaccurate and reliable

In addition from the above data it can be seen that thebusy probabilities of places 119875

2and 119875

5are larger relatively

Place 1198752refers to the state that velocity of the follow-up

train is greater in two successive trains There are manyreasons leading to this state such as the driver violating thedriving instruction usually represented in speeding or slowlydriving the automatic control system being abnormal or thedispatcherrsquos misjudgment On the one hand the dispatcherrsquosmisjudgment is reflected by the dispatcherrsquos belief that thefront train cannot be caught up by the following train inthe driving sections to conduct the risk instruction whichsuggests that the dispatcher put too much faith in the driverand neglected risk on the other hand it is reflected by thefact that the dispatcher may believe in the collision avoidancesystem which can prevent the accident which suggests thatthe dispatcher put too much faith in the technology

In all of the factors leading to the state 1198752 some cannot

be avoided because it is the need of dispatching operationmanagement In this case we must be careful and shouldnot put too much faith in the driver or technology toneglect the unexpected adventure and to conduct any riskinstruction Other risk factors can be reduced or eliminatedby strengthening the management of drivers and other safetymeasures

The large busy probability of place 1198752inevitably leads to

a relatively large busy probability of place 1198755(intervention

of ATP) Because only a very small number of cases of thisvelocity difference are eliminated naturally a train generallyhas to rely on ATP to adjust automatically the train drivingstatus to ensure the safe driving of trains In case theATP worked normally in the long-term train drivers ordispatchers can easily depend on the vital role of the ATPIt is the reason why the accident tends to happen once ATPis not working properly Therefore it is more crucial for the

1 2 3 4 5 6 7 8 9000005010015020025030035040045050

Prob

abili

ty

Transition rate of transition number 3p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 6 Stable probabilities of varying transition rate 1205823

day-to-day maintenance of railway safeguard equipment andfacilities to ensure the safe driving of trains In additionthe vigilance of the train driver and dispatcher should bestrengthened in case of the ATP abnormalities

Furthermore the obtained result is consistent with theactual one that the busy probability of place 119875

15(the train

rear-end collision accident) is not very large However nomatter how low the probability is it does not mean that theaccident does not happen Even if we can accurately estimatethe accident we cannot accurately predict the moment of theaccident Because it involves significant consequences every-one from the top management to the low level employeesshould always be vigilant and keep a high level of securityalert

42 Discussion If we change the value of the firing rate 1205823

and values of 1205821 1205822 1205824 120582

23remain unchanged we can

get Figure 6 where 1 7 represent the stable probabilityvalue (namely the placersquos busy probability) in the conditionof the firing rate of transition 119905

3taking 0 05120582

3 1205823 151205823

21205823 251205823 31205823 respectively If we change the value of the

firing rate 1205827 1205828 12058210

12058211

12058213

12058216

12058218

12058220

12058221 we can also

get Figures 7ndash15 similarlyIn Figure 6 if 120582

3increases that is the frequency of inter-

val distance in two successive trains is approaching to theminimum instantaneous distance increases the placersquos busyprobability of119875

41198755 and119875

15increases remarkablyThis shows

that the intensive departure interval strategy will significantlyincrease safety risk and the probability of accidents Thus weshould make a tradeoff between the risk of accidents andintensive departure interval strategy which aim to improvethe transport capacity

From Figures 7 and 8 we find out that ATP plays avery important role as the first barrier for protection of trainrunning security Once the frequency of ATP fails to workincreasingly or the frequency of ATPwork normally declines

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

000

005

010

015

020

025

030

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 7

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 7 Stable probabilities of varying transition rate 1205827

000002004006008010012014016018020022024026028030032034036

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 8

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 8 Stable probabilities of varying transition rate 1205828

the probability of accidents increases evidently The ldquo723rdquoand ldquo927rdquo accidents in China which have been mentionedin the front of this paper are both mainly caused by the ATPmalfunctionThe former is because of striking by lighting andthe latter resulted from loss of power Therefore we must tryour best efforts to guarantee that ATP works normally fromboth designing and maintaining

From Figures 9ndash12 we observe that the dispatcher alsoplays a very important role as the second barrier for pro-tection of train running security Whether dispatcher candetect the abnormal timely (Figures 9 and 10) and whetherdispatcherrsquos measures work effectively after realizing abnor-mal situation (Figures 11 and 12) are critical to the occurrenceof accident and its influence is almost equivalent to ATP

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 10

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 9 Stable probabilities of varying transition rate 12058210

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 11

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 10 Stable probabilities of varying transition rate 12058211

implied by the value of11987515 For instance in the ldquo723rdquo accident

the dispatcher is not such sensitive to the abnormal ldquoredbandrdquo from the signal system and had not taken effectivemeasures in the critical period of 60min before the accidentwhich resulted in the accident finally

Figures 13ndash15 describe that if the frequency of 11990518and 11990520

increases (Figures 13 and 14) or 11990521decreases (Figure 15) the

probability of accident increases showing that whether thedriver can detect the abnormal and take effective measuresis related to the occurrence of the accident However becausedrivers are generally the recipients of information the impactis not as significant as ATP and dispatcher seen from theplacersquos busy probability value of 119875

15 When the driver noticed

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 13

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 11 Stable probabilities of varying transition rate 12058213

000002004006008010012014016018020022024026028

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 16

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 12 Stable probabilities of varying transition rate 12058216

the abnormal there was not enough time and space distancefor the driver to take measures to prevent the accident

5 Conclusion

In this paper we proposed a model of train rear-end collisionaccidents based on the theory of SPN and verified the validityof the model based on the isomorphic MC Meanwhile weprovided quantitative analysis of the train rear-end collisionaccidents by the isomorphic MC of the SPN model Inthe quantitative analysis we accounted for the uncertaintiesof the firing rate 120582

119894of the transitions and introduced the

triangular fuzzy numbers to fuzzy 120582119894 We took different fuzzy

degrees for 120582119894in the steady state probability equations of the

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 18

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 13 Stable probabilities of varying transition rate 12058218

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 20

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 14 Stable probabilities of varying transition rate 12058220

Markov process converted the equations into fuzzy steadystate probability equations and solved the fuzzy equations toget the steady state probability which is more reliable

By analyzing the steady state probability it is found thatthe busy probability of places 119875

2and 119875

5is larger relatively

Since 1198752is the initial state of the train rear-end accident

process it is easy to reduce accidents in this state Howeverit is also a risk state which is most likely to be ignoredTherefore at the same time of reducing the possibility ofgenerating this state we should pay attention to timely adjust-ment of this state after it is generatedThe busy probability ofplace 119875

5is relatively large indicating that the role of ATP is

crucial for protection of train driving security Meanwhile it

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

000002004006008010012014016018020022024026

Prob

abili

ty

1 2 3 4 5 6 7 8 9Transition rate of transition number 21

p1p2p3p4

p5p6p7p8

p9p10p11p12

p13p14p15p16

Figure 15 Stable probabilities of varying transition rate 12058221

is consistent with the actual situation that the busy probabilityof the place 119875

15is not large

In all the case analysis and discussion showed that theresults of the proposed train rear-end accidents model basedon SPN are reasonable in practical applications and can beused to effectively analyze the accidents or prevent loss andthe results may be useful to the department of railway safetymanagement

However there are also some shortcomings in the modelIn order to avoid the deadlocks we made some idealizedprocessing during the model design stage such as not con-sidering the feedback between the driver and dispatcherignoring the repairing and recovering of ATP after the dis-patcher noticed that it is abnormal which should be thesubject of our further research

Conflict of Interests

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

Acknowledgments

This research was supported in part by the National BasicResearch Program of China (Grant no 2012CB725400) theNational Natural Science Foundation of China (Grant nosU1434209 and 71131001-1) and the Research Foundation ofState Key Laboratory of Rail Traffic Control and SafetyBeijing Jiaotong University China (Grant nos RCS2014ZT19and RCS2014ZZ001)

References

[1] A Mirabadi and S Sharifian ldquoApplication of association rulesin Iranian Railways (RAI) accident data analysisrdquo Safety Sciencevol 48 no 10 pp 1427ndash1435 2010

[2] WWang X Jiang S Xia and Q Cao ldquoIncident tree model andincident tree analysis method for quantified risk assessment anin-depth accident study in traffic operationrdquo Safety Science vol48 no 10 pp 1248ndash1262 2010

[3] J De Ona R O Mujalli and F J Calvo ldquoAnalysis of traffic acci-dent injury severity on Spanish rural highways using Bayesiannetworksrdquo Accident Analysis and Prevention vol 43 no 1 pp402ndash411 2011

[4] T Kontogiannis V Leopoulos and N Marmaras ldquoA compar-ison of accident analysis techniques for safety-critical man-machine systemsrdquo International Journal of Industrial Ergo-nomics vol 25 no 4 pp 327ndash347 2000

[5] N G Leveson and J L Stolzy ldquoSafety analysis using Petri netsrdquoIEEE Transactions on Software Engineering vol 133 no 3 pp386ndash397 1987

[6] Z Z Li ldquoFault tree analysis of train rear-end accidents and atalking on the complex system securityrdquo Industrial Engineeringand Management vol 16 no 4 pp 1ndash8 2011 (Chinese)

[7] L Harms-Ringdahl ldquoRelationships between accident investi-gations risk analysis and safety managementrdquo Journal of Haz-ardous Materials vol 111 no 1ndash3 pp 13ndash19 2004

[8] D Vernez D Buchs and G Pierrehumbert ldquoPerspectives inthe use of coloured Petri nets for risk analysis and accidentmodellingrdquo Safety Science vol 41 no 5 pp 445ndash463 2003

[9] C A Petri Kommunikation Mit Automaten [PhD thesis]Shriften des IIMNr 2 Institute fur InstrumentelleMathematikBonn Germany 1962

[10] R David and H Alla ldquoPetri nets for modeling of dynamicsystems a surveyrdquo Automatica vol 30 no 2 pp 175ndash202 1994

[11] T Murata ldquoPetri nets properties analysis and applicationsrdquoProceedings of the IEEE vol 77 no 4 pp 541ndash580 1989

[12] C Lin Stochastic Petri Nets and System Performance EvaluationTsinghua University Press Beijing China 2005 (Chinese)

[13] J Wang Y Deng and C Jin ldquoPerformance analysis of trafficcontrol systems based upon stochastic timed Petri net modelsrdquoInternational Journal of Software Engineering and KnowledgeEngineering vol 10 no 6 pp 735ndash757 2000

[14] M K Molloy On the integration of delay and throughput mea-sures in distributed processing models [PhD thesis] Universityof California Los Angeles Calif USA 1981

[15] L A Zadeh ldquoFuzzy setsrdquo Information and Computation vol 8pp 338ndash353 1965

[16] T J Ross Fuzzy Logic with Engineering Applications WileyChichester UK 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of