Modeling and Simulation for Urban Rail Traffic Problem Based on Cellular Automata

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Commun. Theor. Phys. 58 (2012) 847–855 Vol. 58, No. 6, December 15, 2012

Modeling and Simulation for Urban Rail Traffic Problem Based on Cellular Automata∗

XU Yan (No), CAO Cheng-Xun (ù¤�),† LI Ming-Hua (o²u), and LUO Jin-Long (Û79)

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

(Received July 9, 2012; revised manuscript received October 23, 2012)

Abstract Based on the Nagel-Schreckenberg model, we propose a new cellular automata model to simulate the

urban rail traffic flow under moving block system and present a new minimum instantaneous distance formula under

pure moving block. We also analyze the characteristics of the urban rail traffic flow under the influence of train density,

station dwell times, the length of train, and the train velocity. Train delays can be decreased effectively through flexible

departure intervals according to the preceding train type before its departure. The results demonstrate that a suitable

adjustment of the current train velocity based on the following train velocity can greatly shorten the minimum departure

intervals and then increase the capacity of rail transit.

PACS numbers: 05.40.-a, 05.60.-k, 89.40.BbKey words: cellular automata model, moving block system, minimum departure intervals, train delays

1 Introduction

Urban rail transit system is one of the most effectivemeans to solve traffic congestion in the modern city, be-cause of its high capacity, low energy consumption, smallenvironment effect and other advantages. The populationof city has expanded during the urbanization, and caus-ing higher demands for the capacity of urban rail transit.There are some measures to increase the capacity of urbanrail transit system, one of the measures is to increase thenumber of trailers of a train, and another is to shortenthe train tracking intervals by using the more effectiveblock control means. A few decades ago, moving blocksystem (MBS) was proposed to replace the fixed blocksystem, and it is widely applied in real world train oper-ations recently.[1] Compared with the fixed block system,MBS can decrease the train tracking intervals, and thenincrease the capacity of rail lines and improve the effi-ciency of train operations. It is important to emphasizethat MBS can not only greatly shorten the train trackingintervals and increase the train density, but also bring thegreat flexibility of train schedules.

Cellular automata (CA) model was proposed by Cre-mer and Ludwig in the early 1980s.[2] The well-knownNagel–Schreckenberg (NS) model was proposed by Nageland Schrechenberg in 1992, which was applied to the roadtraffic flows.[3] The NS model can reproduce real worldtraffic phenomena very well, although it is very simple.With considerations of some more complex real worldproblems, many scholars proposed several improved mod-els based on the NS model.[4−7] Li et al.[8] firstly pointedout that the car position could be updated not only de-pending on the position of the successive cars, but also

the relative velocity, then they proposed a new CA modelof road traffic flows. Zhang and Chen[9] proposed a newCA model to simulate the traffic flow when a certain trainbreaks down and be rescued by its following train. Be-cause of the great performance on simulation of road traf-fic flow, the CA theory is also applied to railways. In 2005,Li et al.[10−11] firstly applied CA theory to railway systemand presented a CA model by changing the update rules.In Ref. [12], a new CA model is proposed for the simula-tion of the fixed block system. In Ref. [13], the authorsproposed a CA model of moving-like block system, andanalyzed the influence of departure intervals and initialdelay on train delays propagation. Li et al.[14] analyzedthe influence of the departure intervals, the proportion offreight train, the station dwell times, and the number ofthe train platforms on the train average delay time underthe four fixed blocks system. According to the subwaytrain operation characteristics, Fu et al.[15] established aCA model of the fixed block system, and studied the per-formance of Beijing metro line 2. However, most of theabove works limited to rail transit system under the fixedblocks system, and little attention has been given to themoving block system, which is applied extensively in realworld. In our work, we extend our study to the movingblock system.

The main contributions of this paper are as follows.First, we propose a new CA model to simulate the urbanrail traffic flow under the moving block system and presenta new minimum instantaneous distance formula under thepure moving block system (PMS). Second, we analyze thecharacteristics of the urban rail traffic flow under the in-fluence of train density, station dwell times, train length,

∗Supported by the National Basic Research Program of China under Grant No. 2012CB725400, the National Natural Science Foundation

of China under Grant No. 71131001-1 and the Research Foundation of State Key Laboratory of Rail Traffic Control and Safety under Grant

No. RCS2011ZZ003, Beijing Jiaotong University†Corresponding author, E-mail: cxcao@bjtu.edu.cn

c© 2011 Chinese Physical Society and IOP Publishing Ltd

http://www.iop.org/EJ/journal/ctp http://ctp.itp.ac.cn

848 Communications in Theoretical Physics Vol. 58

and the train velocity. Third, we show that the length oftrain has impact on train delays and it can be decreased ef-fectively through flexible departure intervals according tothe preceding train type before its departure. Finally, thesimulation results demonstrate that suitable adjustmentof the train velocity based on following train velocity cangreatly shorten the minimum departure intervals and thenincrease the capacity of rail transit.

The rest of the paper is organized as follows. We in-troduce the basic principles of MBS and propose a newminimum instantaneous distance formula in Sec. 2. InSec. 3, we present a new CA model. The simulation re-sults and analytical discussions are presented in Sec. 4.Finally, the main work of this research is summarized inthe conclusion.

2 Theory of Moving Block System

The railway under the fixed block system is dividedinto several sections of track known as blocks, which arefixed by infrastructure. Every block is separated by sig-nals which control the train movement to prevent trainentering an occupied block and prevent trains from colli-sion. Unlike the traditional fixed block systems, the mov-ing block system does not require fixed-block track circuitsfor determining train position. Instead, it relies on con-tinuous bi-directional communication between each con-trolled train and a wayside control center. In this system,the line is usually divided into areas or regions, and eacharea is under the control of computer. The bi-directionalcommunication between each train and the area computeris continuous so that the computer knows the status oftrains which includes, among others parameters, the ex-act location, speed, travel direction and braking distance.This information allows calculation of the area potentiallyoccupied by the train on the track. It also enables thewayside equipment to define the points on the line thatmust never be passed by the other trains on the sametrack. These points are communicated to make the trainsautomatically and continuously adjust their speed whilemaintaining the safety and comfort requirements. So, thetrains continuously receive information regarding the dis-tance to the preceding train and then are able to adjusttheir safety distance accordingly. The control principle ofthe train movement is shown in Fig. 1. The moving blocksystem allows the reduction of the safety distance betweentwo consecutive trains. This results in a reduced headwaybetween consecutive trains and an increased transport ca-pacity.

Several types of moving-block scheme have been dis-cussed so far.[16] Moving space block (MSB) is the sim-plest scheme, in which the minimum instantaneous dis-tance between successive trains is dn = v2

max/(2b) + SM,where vmax is the maximum velocity, b is the decelera-tion of the train n, SM is the safety distance. Anotherscheme is moving time block (MTB), in which the mini-mum instantaneous distance between successive trains is

dn = vmaxvf/(2b) + SM. The last scheme is pure mov-

ing block (PMB), in which the formula of the minimum

instantaneous distance is dn = v2f/(2b) + SM and vf is

the real-time velocity of the train n. In the existent CA

models for rail traffic, MSB scheme is used by consideringsafety factors. As we know, with the development of mo-

bile communication technology, the real-time informationof train position, velocity and acceleration would be ob-

tained more easily and precisely. Therefore, in our work,

we consider the relative distance of moving-block scheme,namely PMB. In addition, the subway train is consisted

of several carriages, and its length is between 20 to 22.5meters. The number of hanging carriages in a train can

be adjusted according to passenger flow and other factors,

so the train length can not be ignored. Therefore, in ourCA model, we also take the length of the train in con-

sideration. Therefore, minimum instantaneous distancebetween successive trains is given as follows:

dn =v2

f

2b+ SM + Ltrain (n−1) , (1)

where Ltrain (n−1) is the length of the front train (n − 1).

As shown in Eq. (1), the minimum instantaneous dis-

tance between the successive trains is closely related tothe trains’ velocities and braking performances. By the

real-time train velocity, we can analyze the velocity effectof a train on the successive trains.

As shown in Fig. 2, the minimum instantaneous dis-tance between the successive trains can obviously be re-

duced when considering the velocity effect of the fronttrain, where sn−1 and sn are the braking distances of train

(n− 1) and train n, respectively. The minimum instanta-

neous distance between the two successive trains is givenas follows:

d′n = vfn

( vfn

2bn

+ τn

)

− vfn−1

( vfn−1

2bn−1+ τn−1

)

+ SM + Ltrain (n−1) , (2)

where d′n is the minimum instantaneous distance between

the two successive trains, vfn−1is the velocity of train

(n − 1), vfnis the velocity of train n, bn−1 is the decel-

eration of train (n − 1), b2 is the deceleration of train n,

and τn−1 and τn are respectively the redundant time oftrain (n−1) and train n, which consist of signal cycle and

driver reaction time of driver.

In train movement, there are two braking modes. As

shown in Fig. 3(a), one is to ensure the stability and com-fort in the process of normal parking, called conventional

braking curve; the other is to ensure the security of trainoperations when the emergency appears, called emergent

braking curve, where sc is the common braking distance,

se is the emergency braking distance. When sn−1 = 0and sn = sc, we call it “hit hard wall” tracking operation

mode, shown in Fig. 3(b); when sn−1 = se and sn = sc,we call it “hit soft wall” tracking operation mode, shown

in Fig. 3(c). In this paper, we adopt the “hit soft wall”

No. 6 Communications in Theoretical Physics 849

tracking operation mode when analyzing the velocity ef-

fect of the front train on the successive trains. Ignoring

the signal cycle and reaction time of driver, we propose a

new formula of minimum instantaneous distance between

two successive trains as follows:

d′n =v2

fn

2bn

−

v2fn−1

2be

+ SM + Ltrain (n−1) , (3)

where be is the emergency deceleration of train (n − 1).

Fig. 1 The principle of two different control system, where (a) is the principle of the moving block system, (b)is the principle of the fixed-block signaling system.

Fig. 2 The train movement under MBS.

Fig. 3 Two different tracking operation under moving block signal system, where (a) is the two different brakingcurves, (b) is the “Hit hard wall” tracking operation mode, (c) is the “Hit soft wall” tracking operation mode.

3 Simulation Model

3.1 CA Model of Urban Rail Transit System

In this work, we use CA model to simulate the rail traf-

fic under the moving block signal system. Our investiga-

tion is based on the NS traffic model.[2] The rail line is di-vided into L cells of equal size numbered by i = 1, 2, . . . , L,and the length of a cell is 1m. The simulation time is dis-crete, and the time step is 1s, the initial time t0 = 0. Eachcell can be either empty or occupied by the train with inte-

850 Communications in Theoretical Physics Vol. 58

ger speed vn = 0, 1, . . . , vmax. A station designated at theposition L/2, and each train should dwell enough time be-fore leaving the station. The open boundary condition forCA model is considered. It is defined as: (i) Each periodof Tint time steps later, a train with speed vmax on thefirst cell is created and immediately moves according tothe update rules if the section from site 1 to site is empty.Tint represents the departure intervals of trains and is thesafety departure distance; (ii) At position i = L, the trainssimply move out of the system.

3.2 Update Rules

As we all know, the difference of individual behaviorof driver, especially suddenly accelerated or deceleratedcaused by driver’s excessive reaction, has been taken intoaccount at the step of randomization in the classical NSmodel. However, the train drivers’ individual random be-haviors are forbidden in train movement, and also lesseffect the weather is on the urban rail transit. Therefore,we do not consider the randomization in our CA model.At each discrete time step t → t + 1, the state of thesystem is updated following a well defined prescription.Before introducing update rules of the train movement,we should explain the two situations as follows:

(i) When the (n − 1)-th train is in front of the n-thtrain at the t-th time step, the velocity of the n-th train isvaried by comparing the headway distance ∆Xn and theminimum instantaneous distance dn to determine whetheraccelerating or decelerating.

(ii) When the empty station is in front of the n-thtrain, the n-th train must pull in and dwell enough timebefore leaving. The variation of velocity must ensure then-th train can stop at the station. According to the kine-matics equations, we know that v2

t − v20 = 2b · gap (n),

where vt is the velocity of the n-th train in the brakingprocess, v0 is the target velocity, usually v0 = 0 when thetrain stops at the station, b is the deceleration of the n-thtrain, gap (n) is the distance between the station and then-th train. In considering the integer characteristics ofCA model, the velocity of train is integer. Therefore, wegive an integer speed formula as follows:

vt = int (√

2b · gap (n) ) . (4)

Therefore, the update rules for implementing the railtraffic under MBS are as follows:

(i) The n-th train is behind the (n − 1)-th trainStep 1 Acceleration

if ∆Xn(t) ≻ dn, vn = min(vn + a, vmax); else if∆Xn(t) ≺ dn, vn = max(vn − b, 0); else vn = vn end.Step 2 Movement

Xn(t + 1) = Xn(t) + vn.(ii) The n-th train is behind the station(a) If the station is occupied by a train, the update

rules are the same as that used in the case (i); (b) If not(the station is empty), the update rules as follows:Step 1 Acceleration

if gap (n) ≻ Xc, vn = min(vn + a, vmax); else if

gap (n) ≺ Xc, vn = max(vn − b, 0); else vn = vn, end.

Step 2 Deceleration

vn = min(vn, int (√

2b · gap (n))).

Step 3 Movement

Xn(t + 1) = Xn(t) + vn.

(iii) The n-th train is at the station

Step 1 Acceleration

if tdwell = Td and ∆Xn(t) ≻ Ls, vn = min(vn +

a, vmax), tdwell = 0; else if tdwell ≺ Td, vn = 0, tdwell =

tdwell + 1; end.

Step 2 Movement

Xn(t + 1) = Xn(t) + vn, where Xc is the braking dis-

tance of the train with the current velocity vn(t) slowing

down to zero, tdwell is the dwell times that the n-th train

has dwelled at the station, Td is the station dwell times of

the train.

4 Simulations

4.1 Characteristics of Urban Rail Rraffic

In the proposed CA model, the system of L = 4000 is

considered, and the length of evolution time is T = 2000.

The length of train is Ltrain, and it depends on the num-

ber of the carriage that the train has taken. The average

length of carriage is 20 m. In urban rail transit system, the

velocity of train is no more than 100 km/h. The maximum

acceleration passenger can bear is no more than 1.4 m/s2,

and the average acceleration of train is 0.9 m/s2. The av-

erage deceleration is 1.0 m/s2, and the emergency deceler-

ation is 1.3 m/s2. Combined with the actual parameters,

we assume the maximum velocity is vmax = 20, accelera-

tion is a = 1, deceleration b = 1, emergency deceleration

be = 1.5, and safety distance SM = 10.

4.2 Analysis of Train Delays with Different

Station Dwell Times

The train has different dwell times in different metro

station, generally 50 s in the hub-station and 30 s–40 s in

normal station. In this section, we focus on the character-

istics of the rail traffic flow under the influence of different

station dwell times at the same station. We define that

the station dwell times of train are increased from 30 s to

50 s gradually, and the length of train Ltrain = 120, the

departure intervals Tint = 63. The simulation results of

the proposed CA model are shown in Fig. 4.

As shown in Fig. 4(a), the longer the station dwell

times have, the greater delays the follow-up trains make.

Because if the departure intervals are fixed, then the train

density on the rail lines is not changed too, and the follow-

up trains can only wait outside for pulling in when the

station dwell times are increased. According to update

rules, the delays will propagate back constantly and ac-

cumulate. In this case, only by increasing the departure

No. 6 Communications in Theoretical Physics 851

intervals to reduce the train density, can the delays bereduced ultimately.

Fig. 4 The analysis of delays with different dwell time:L = 1200, SM = 10, vmax = 20, Tint = 63: (a) Totaldelays of trains with different station dwell times; (b)Delays of each train with different station dwell times.

4.3 Analysis of Delays with Different Length of

Train

The property of train mainly includes length, accelera-tion or deceleration, and maximum velocity, etc. Combin-ing with actual condition, the maximum acceleration oftrain is not more than 1.4 m/s2, and the real-time veloc-ity of train can be accessed in our CA model. Therefore,when analyzing the influence of train performance on therail traffic flow, we only take the length of train into ac-count.

Firstly, we assume that the properties of the two typesof train are the same, and L1 = L2 = 120, SM = 10,a = b = 1, Td = 40, vmax 1 = vmax 2 = 20, where L1

and L2 are respectively the length of two different typeof train. According to Ref. [1], we calculate the theoret-ical minimum departure intervals. Based on the theoret-ical minimum departure intervals, we use the proposedCA model to simulate the minimum departure intervalsTmin = 71, which can ensure that the trains can be op-erated completely independently. We assume Tint = Tmin

and the trains can move freely. The “Velocity-Time” di-agram and “Time-Space” diagram of 15-th and 16-th areshown in the Fig. 5. Similarly, the results of the assump-tions of Ltrain = 140, Tmin = 72 and the assumptions of

Ltrain = 160, Tmin = 74 are shown in Fig. 6.

Fig. 5 The operation characters of the 15-th and 16-thtrain in free traffic flow of the assumptions of Ltrain =120, Tmin = 71. (a) “Velocity-time” diagram; (b) “Time-space” diagram.

As shown in Fig. 6(a), we can know that trains withdifferent length have different critical value of departureintervals when the traffic flow changes from crowded tofree. The longer train has greater critical value. WhenLtrain = 120, the critical value is 64 s, when Ltrain = 140,the critical value is 66 s, and when Ltrain = 160, the crit-ical value is 68 s. As the minimum departure intervalapproaches to the critical value, the total delays can berepresented by a linear trend. On the contrary, there arealmost no delays when the departure interval is greaterthan the critical value. When the departure interval isfixed, the train density on the rail line would not changetoo. So, we can easily see that the smaller the departureinterval is, the greater the train density is, and the greatertrain delays. In Fig. 6(b), we can know that the longertrains trigger greater delays with the same departure in-tervals and the delays usually propagate backward andaccumulate constantly. We can conclude that the lengthof train has a direct impact on the minimum tracking op-eration intervals.

Then, we discuss the characteristics of the rail traf-fic flow when trains mixed with different length. Bythe preceding analysis, we use the CA model to simulatethree cases respectively as follows: L1 = 120, L2 = 140;L1 = 120, L2 = 160; L1 = 140, L2 = 160, L1, and L2 arerespectively the length of two different type train. Theproportions of the shorter train are set to 30%, 50%, 70%.

852 Communications in Theoretical Physics Vol. 58

Fig. 6 The analysis of the minimum departure intervals of train with different length, where (a) is the totaldelays with different departure intervals, (b) is the delay of each train with different length at the same departureintervals: vmax = 20, Td = 40, SM = 10.

Fig. 7 The delays of mixed train flow with different length: vmax = 20, Td = 40, SM = 10, Tint = 59.

Figures 7 and 8 display the simulation results. As

shown in Fig. 7, the proportion of longer train is greater,

and the cumulative delays are longer. While the length of

train is longer than 120, delays are longer too, and the to-tal delays can be represented by a linear trend. As shown

in Fig. 8(a), there is almost no congestion when the ra-

tio of trains with length L1 = 120 to trains with length

L2 = 140 is 1:1. But there is an obvious congestion when

the ratio of trains with length L1 = 140 to trains withlength L2 = 160 is 1:1, shown in Fig. 8(c). The results

shown in Fig. 8 demonstrate that the follow-up trains are

forced to decelerate or stop when the front trains are de-

celerating in front of the station to pull in. And the delays

appear and propagate backward, accumulate constantly.

The reasons of forming this phenomenon are as follows:

(i) as previously mentioned, different lengths of the trainshave different minimum departure intervals that ensure all

of the trains moving freely; (ii) known by Eq. (1), differ-

ent length of train has different minimum instantaneous

distance. In order to ensure the safety of train opera-

tions, there must be greater minimum instantaneous dis-tance when the front train is longer. Besides, the longer

trains require more time to leave the station. From anal-

ysis above, we can see that train delays can be reduced

No. 6 Communications in Theoretical Physics 853

by extending the departure intervals. But if the fixed de-parture interval, which sets according to the longer train,is applied, it may be a waste for the shorter trains in thesystem. Therefore, this paper modifies the departure rulesand proposes another flexible departure interval by taking

account of various factors. That is to adjust the depar-

ture intervals according to the front train type before its

departure. The departure rules after modifying are as fol-

lows:

Fig. 8 The “space-time” diagrams of mixed train flow with different length: vmax = 20, Td = 40, SM = 10,ρ = 50%, Tint = 65. (a) L1 = 120, L2 = 140; (b) L1 = 120, L2 = 160; (c) L1 = 140, L2 = 160.

If the safety distance before the origin is not occupied.

If the front train is shorter.

A train with the velocity vmax is created after the sys-tem updating for Tint1 time steps.

Else if the front train is longer. A train with the ve-

locity vmax is created after the system updating for Tint2

time steps, end.

End

where Tint1 and Tint2 are respectively the minimum de-parture intervals of the shorter and longer train. After areasonable adjustment of the departure rules, the simula-

tion results are shown in Fig. 9.

Fig. 9 The delays of different proportions of differ-ent length of train after adjusting the departure rules:vmax = 20, Td = 40, SM = 10.

As shown in Fig. 9, we know that all of the trainsrun freely on the rail line, and the follow-up trains can

be hardly impacted by the front trains; the proportion ofthe two types of train has no effect on delays too. It can

absolutely avoid the unnecessary deceleration or waitingbefore pulling in. Thereby, the improved departure rulescan increase the capacity of rail lines.

4.4 Analysis of Delays When Considering

Velocity Effect of the Front Train on the

Successive Trains

As the preceding analysis, the velocity of the fronttrain has a greater effect on the calculation of the min-imum instantaneous distance under PMB. In this sec-tion, we focus on the velocity effect of the front train onthe successive trains and we call it VE-PMB. In this pa-per, we apply the “hit soft wall” of tracking operationmode, and the minimum instantaneous distance is com-puted through the formula dn = v2

f1/(2b) − v2

f2/(2be) +

SM + Ltrain (n−1). As the previously described, the for-mula of the minimum instantaneous distance under PMBscheme is dn = v2

f/(2b) + SM + Ltrain (n−1), and it’s

dn = v2max/(2b) + SM + Ltrain (n−1) under MSB scheme.

Firstly, we investigate the change of the headway be-tween two successive trains before analyzing the velocityeffect of the front train on the successive trains. We as-sume that L = 120, vmax = 20, a = b = 1, be = 1.5,Tint = 57, SM = 10. Here, a train is tracked at the timet = 800 by accessing its position, velocity, distance head-way and etc. The simulations are shown in Fig. 10.

As shown in Fig. 10, compared with PMB scheme, thevelocity of train is more flexible under VE-PMB scheme.At the same time, due to the greater departure intervals,the headway of two successive trains at the origin reachesto 1140 m, and it absolutely meets the requirement ofsafety before departing a train. At t = 878, the certaintrain gradually decelerates but not stops because the fronttrain is waiting outside of the station; at t = 915, the cer-tain train stops outside of the station to wait for pulling

854 Communications in Theoretical Physics Vol. 58

in and the distance headway is 124 m which is less thanthe minimum instantaneous distance dn; at t = 955, thecertain train begins to accelerate to pull in after the fronttrain leaving the station. In order to ensure that the cer-tain train finally stops at the station with zero velocity,we control the velocity of the certain train through thegap distance between the station and the real-time posi-tion of the certain train. The distance headway is grad-ually increasing after the front train is gradually leaving;at t = 1015, the train gradually accelerates to pull out ofthe station. As shown in Fig. 10, it begins to form stable,free rail traffic flow after trains leaving the station, andthe distance headway remains at 1260 m. Compared with

VE-PMB scheme, the certain train begins to decelerateat t = 875 and stops at t = 896. At t = 951 the certaintrain stops again. At last, the certain train pulls out ofthe station at t = 1048. It can be seen that the VE-PMBscheme can effectively control train operations state, andreduce the process of acceleration, deceleration and stopin train operations. And it then reduces the delays.

Secondly, we study train delays under the three dif-ferent schemes. As stated above, we assume Tint = 62and adopt the three different formulas to compute theminimum instantaneous distance. Using the proposed CAmodel to simulate, the “Time-Space” diagrams are shownin Fig. 11.

Fig. 10 The operating characteristics of a certain train when considering or un-considering the velocity effectof the front train on the successive trains, where (a) is the “headway-time”, “gap-time”, and “velocity-time” of acertain train when considering the velocity of the front train on the successive trains, and (b) does not consider:vmax = 20, Td = 40, SM = 10, Tint = 57, Ltrain = 120.

Fig. 11 The ‘Time-Space’ under different scheme, where (a) is under VE-PMB scheme and consider the fronttrains’ velocity effect on the successive trains, (b) is under PMB scheme but UN-consider the front trains velocityeffect, and (c) is the delay time under the MSB scheme: vmax = 20, Td = 40, SM = 10, Tint = 62, Ltrain = 120.

As shown in Fig. 11, the simulations of the three dif-

ferent schemes have great difference with the same depar-

ture intervals. There are serious jam phenomenon under

the simplest MSB scheme, slight jam phenomenon under

PMB scheme, and no jam phenomenon under the pro-

posed VE-PMB scheme. The reason is that the VE-PMB

scheme is based on PMB scheme and it can update the

minimum instantaneous distance by accessing the real-

time train velocities constantly, and shorten the tracking

distance of the successive trains. In general, the VE-PMB

scheme is more reasonable in the real word control of train

movement.

At last, we analyze the departure times under different

schemes. As we all know, one of the efficient measures of

No. 6 Communications in Theoretical Physics 855

improving the carrying capacity of urban rail transit sys-tem is to shorten the train tracking interval. Using theproposed CA model, we simulate and obtain the differ-ent minimum departure intervals under different schemes.The results are shown in Fig. 12.

Fig. 12 The different minimum departure intervals un-der different schemes.

From Fig. 12, we can know that simulation results arein accordance with theoretical analysis. Considering thevelocity effect of the front train on successive trains can

efficiently shorten the minimum departure intervals and

increase the train density, capacity of rail lines.

5 Conclusions

In this paper, we propose a new CA model of urban

rail transit under MBS and adopt the PMB scheme to

control train tracking operations. Under PMB scheme,

the minimum instantaneous distance dn is not fixed, and

we update it by obtaining the real-time train velocities

constantly. With the proposed CA model, the presented

simulations demonstrate that: (i) The length of train has

great effect on urban rail traffic flow. Trains with different

lengths have different departure intervals. Mixed traffic

flow with different lengths and different proportions will

cause different degree of train delays. The delays can dis-

appear after revising departure rules; (ii) Train density

and station dwell times also have great effect on urban

rail traffic flow. The greater train density or train dwell

times will cause longer delays. The delays can be reduced

by setting reasonable departure intervals; (iii) Considering

the velocity effect of the front train on successive trains

can efficiently shorten the minimum departure intervals

and this demonstrates that the simulation results agree

with the theoretical analysis.

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