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Physica A 377 (2007) 633–640

www.elsevier.com/locate/physa

Dangerous situations in a synchronized flow model

Rui Jiang�, Qing-Song Wu

School of Engineering Science, University of Science and Technology of China, Hefei 230026, People’s Republic of China

Received 9 May 2006; received in revised form 16 November 2006

Available online 18 December 2006

Abstract

This paper studies the dangerous situation (DS) in a synchronized flow model. The DS on the two branches of the

fundamental diagram are investigated, respectively. It is shown that different relationship between DS probability and the

density exists in the synchronized flow and in the jams. Moreover, we prove that there is no DS caused by non-stopped car

although the model itself is a non-exclusion process. We classify the DS into four sub-types and study the probability of

these four sub-types. The simulation result is consistent with the real traffic.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Dangerous situations; Cellular automaton; Synchronized flow

1. Introduction

In the past decade, cellular automata (CA) traffic flow models have attracted the interest of a community ofphysicists [1–5]. These models have successfully reproduced many qualitative features observed in real trafficsuch as traffic jams, etc. More recently, theoretical and numerical results for dangerous situations (DS) in theframework of the CA models have been reported [6–17]. In a DS, there will be no accident if every driver iscareful enough. Nevertheless, if the drivers are not so careful (p240, see the following text), an accident mayoccur. In other words, if one denotes the probability of a DS as P, then a car accident will occur withprobability P� p2.

Boccara et al. [6] were the first to propose conditions for DS in the deterministic Nagel–Schreckenbergmodel. They assume that drivers will probably not respect the safe distance if the speed of the car aheadvnþ1ðtÞ is positive at time t, because drivers expect the speed of the car ahead vnþ1ðtþ 1Þ to remain positiveat time tþ 1. Thus, the drivers increase their velocity by one unit with probability p2. It is clear that thecareless driving will probably result in an accident if the speed of the car ahead vnþ1ðtþ 1Þ becomes zero attime tþ 1.

Based on this assumption, Bocarra et al. argue that if the following three conditions are satisfied, then thecar is in a DS: (i) 0pdnðtÞpvmax; (ii) vnþ1ðtÞ40; and (iii) vnþ1ðtþ 1Þ ¼ 0. Here dnðtÞ denotes the gap to the front

e front matter r 2006 Elsevier B.V. All rights reserved.

ysa.2006.11.073

ing author.

ess: rjiang@ustc.edu.cn (R. Jiang).

ARTICLE IN PRESSR. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640634

car and vmax is maximum velocity. Later Huang and coworkers [7,8] as well as Yang and Ma [9] haveinvestigated DS for different values of randomization p1 and maximum velocity vmax.

To correctly determine the DS, Jiang et al. [10,11] changed the three conditions of Boccara et al. as follows:(i) dnðtþ 1Þ ¼ 0; (ii) vnþ1ðtÞ40; and (iii) vnðtþ 1Þovmax. Jiang et al. [11] also investigated the DS in thevelocity-effect model, which is a non-exclusion process: two different DS, i.e., DS caused by stopped cars andDS caused by non-stopped cars, are studied.

Moreover, Moussa [12] has investigated the effect of the delayed reaction time on the probability of DS inthe Nagel–Schreckenberg model as well as the DS caused by stopped cars and great deceleration. We also notethat the DS problem has been studied in the Fukui–Ishibashi model [13,14]. Yang et al. have studied the DS inthe open boundary conditions [15] and when disordered effect was considered [16]. Moussa extended the DSinvestigation to two-lane traffic [17].

We notice that the above mentioned works were carried out in Nagel–Schreckenberg model or velocity-effect model or Fukui–Ishibashi model. All the three models cannot describe the synchronized flow. Recently,the authors proposed a new cellular automata model, which can reproduce the synchronized flow quitesatisfactorily [18]. Therefore, it is interesting to analyze the problem of DS in the synchronized flow model.

The paper is organized as follows. In Section 2, the synchronized flow model is briefly reviewed and theconditions of the DS are presented. In Section 3, the simulation results are analyzed and compared with theempirical data. The conclusions are given in Section 4.

2. Model

For the sake of the completeness, we briefly recall the synchronized flow model. The parallel update rules ofthe model are as follows:

1.

Determination of the randomization parameter pnðtþ 1Þ:

pnðtþ 1Þ ¼ pðvnðtÞ; bnþ1ðtÞ; th;n; ts;nÞ.

2.

Acceleration:

if ððbnþ1ðtÞ ¼ 0 or th;nXts;nÞ and ðvnðtÞ40ÞÞ then: vnðtþ 1Þ ¼ minðvnðtÞ þ 2; vmaxÞ;

else if ðvnðtÞ ¼ 0Þ then: vnðtþ 1Þ ¼ minðvnðtÞ þ 1; vmaxÞ;

else: vnðtþ 1Þ ¼ vnðtÞ:

3.

Braking rule:

vnðtþ 1Þ ¼ minðdeffn ; vnðtþ 1ÞÞ.

4.

Randomization and braking:

if ðrandðÞopnðtþ 1ÞÞ then : vnðtþ 1Þ ¼ maxðvnðtþ 1Þ � 1; 0Þ.

5.

The determination of bnðtþ 1Þ:if ðvnðtþ 1ÞovnðtÞÞ then: bnðtþ 1Þ ¼ 1;if (vnðtþ 1Þ4vnðtÞ) then: bnðtþ 1Þ ¼ 0;if (vnðtþ 1Þ ¼ vnðtÞ) then: bnðtþ 1Þ ¼ bnðtÞ.

6.

The determination of tst;n:if vnðtþ 1Þ ¼ 0 then: tst;n ¼ tst;n þ 1;if vnðtþ 1Þ40 then: tst;n ¼ 0.

7.

Car motion:

xnðtþ 1Þ ¼ xnðtÞ þ vnðtþ 1Þ.

ARTICLE IN PRESS

flow

density

B

ρc0

O

A

Fig. 1. The fundamental diagram of the synchronized flow model. The peak is due to a finite size effect. Here the branch AB starts from a

homogeneous initial condition while the branch AC starts from megajam. The dashed line means the synchronized flow can be maintained

for certain times, then the spontaneous jam occurs in the synchronized flow. The parameters are chosen as shown in Section 2, see also

Ref. [18].

R. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640 635

Here xn and vn are the position and velocity of vehicle n (here vehicle nþ 1 precedes vehicle n), dn is the gap ofthe vehicle n, bn is the status of the brake light (onðoffÞ ! bn ¼ 1ð0Þ). The two times th;n ¼ dn=vnðtÞ andts;n ¼ minðvnðtÞ; hÞ, where h determines the range of interaction with the brake light, are introduced to comparethe time th;n needed to reach the position of the leading vehicle with a velocity dependent interaction horizonts;n. deff

n ¼ dn þmaxðvanti � gapsafety; 0Þ is the effective distance, where vanti ¼ minðdnþ1; vnþ1Þ is the expectedvelocity of the preceding vehicle in the next time step and gapsafety controls the effectiveness of the anticipation.randðÞ is a random number between 0 and 1, tst;n denotes the time that the car n stops. The randomizationparameter p is defined:

pðvnðtÞ; bnþ1ðtÞ; th;n; ts;nÞ ¼

pb if bnþ1 ¼ 1 and th;nots;n;

p0 if vn ¼ 0 and tst;nXtc;

pd in all other cases:

8><>:

Here tc is a parameter.As shown in Fig. 1 and in Ref. [19], the model can reproduce the synchronized flow and the empirical

features at the on-ramp. The improvement of the model mainly lies in that the brake light rule is changed.Moreover, the acceleration rule also contributes to the improvement.

In the next section, the simulations are carried out. In the simulations, the parameter values are tc ¼ 10,vmax ¼ 20, pd ¼ 0:1, pb ¼ 0:94, p0 ¼ 0:5, h ¼ 6, gapsafety ¼ 7. The system size is L ¼ 10 000.

The DS in the model is defined as follows: (i) at time step tþ 1, the gap to the front car equals to zero:dnðtþ 1Þ ¼ 0; (ii) at time step tþ 1, its own velocity should be smaller than vmax: vnðtþ 1Þovmax; and (iii) attime t, the velocity of the front car should be larger than zero: vnþ1ðtÞ40.

3. Simulation results

In this section, the simulation results are presented. Firstly, we consider the DS when the traffic state is onthe branch OAB. In Fig. 2, we have shown the plots of DS probability against density. One can see that the DSprobability is zero when the density is small. When the density exceeds a critical density rc1, the DS begins to

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pro

ba

lity

density

ρc1

Fig. 2. The probability of DS against density when the traffic starts from homogeneous initial condition. For the density corresponding to

the dashed line in Fig. 1, the results are obtained before the jam occurs. The solid line is expected tendency ðr� rc1Þg with

g � 3;rc1 � 0:18.

R. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640636

appear on the road. The DS probability increases with the increase of density. We compare the critical densityrc1 with critical density rc0. One finds that rc14rc0. This means that in the outflow from the jam, the DS doesnot exist.

The analytic analysis of DS is difficult to carry out for Fig. 2. But we expect the DS probability behaves asðr� rc1Þ

g because with the increase of density, the probability of appearance of stopped cars increases, whichis dominating condition in the three conditions . The tendency is well confirmed by the simulation results withg � 3;rc1 � 0:18 (Fig. 2).

Since in the model, the anticipation of the movement of the front car is considered, the modelis not an exclusion process. Therefore, the question arises: is there existing the DS caused by non-stopped car as in velocity-effect model [11]? Here, however, our answer is no. This can be proved asfollows.

(i) If dnþ1ðtÞovnþ1ðtÞ, then vanti ¼ dnþ1ðtÞ. So deffn ¼ dnðtÞ þmaxðdnþ1ðtÞ � gapsafety; 0Þ.

�

If dnþ1ðtÞXgapsafety ¼ 7, then deffn ¼ dnðtÞ þ dnþ1ðtÞ � 7. So the car n can reach at most xnðtÞ þ dnðtÞþ

dnþ1ðtÞ � 7. While deffnþ1Xdnþ1ðtÞ, we know that car nþ 1 can at least advance by dnþ1ðtÞ � 1 even if it is

randomized, i.e., the car nþ 1 can reach at least xnþ1ðtÞ þ dnþ1ðtÞ � 1. It is obvious that xnðtÞ þ dnðtÞ þ

dnþ1ðtÞ � 7oðxnþ1ðtÞ þ dnþ1ðtÞ � 1Þ � 5 because xnþ1ðtÞ � xnðtÞ � 5 ¼ dnðtÞ.

� If dnþ1ðtÞogapsafety ¼ 7, then deff

n ¼ dnðtÞ. So the car n can reach at most xnðtÞ þ dnðtÞ. On the other hand,the existence of the DS caused by non-stopped car implies that vnþ1ðtþ 1Þ40. So xnðtÞ þ dnðtÞoðxnþ1ðtÞþ

vnþ1ðtþ 1ÞÞ � 5.

(ii) If dnþ1ðtÞXvnþ1ðtÞ, then vanti ¼ vnþ1ðtÞ. So deffn ¼ dnðtÞ þmaxðvnþ1ðtÞ � gapsafety; 0Þ.

�

If vnþ1ðtÞXgapsafety ¼ 7, then deffn ¼ dnðtÞ þ vnþ1ðtÞ � 7. So the car n can reach at most xnðtÞ þ dnðtÞþ

vnþ1ðtÞ � 7. On the other hand, due to that dnþ1ðtÞXvnþ1ðtÞ, we know that car nþ 1 can at least advance byvnþ1ðtÞ � 1 even if it is randomized, i.e., the car nþ 1 can reach at least xnþ1ðtÞ þ vnþ1ðtÞ � 1. It is obviousthat xnðtÞ þ dnðtÞ þ vnþ1ðtÞ � 7oðxnþ1ðtÞ þ vnþ1ðtÞ � 1Þ � 5.

� If vnþ1ðtÞogapsafety ¼ 7, then deff

n ¼ dnðtÞ. The situation is the same as case 2 in (i).

ARTICLE IN PRESSR. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640 637

provided gapsafety41, which is a necessary condition to avoid collision.In real traffic, the accident expense depends on the relative speed of the two vehicles involved

From the proof one can see that the result only depends on the parameter gapsafety. The result is applicable

in the accident. According to this, next we classify the DS into four sub-types and investigate theprobability of the four sub-types of DS: (i) trivial DS if vnðtþ 1Þ � vnþ1ðtþ 1Þo5; (ii) medium DS if5pvnðtþ 1Þ � vnþ1ðtþ 1Þo10; (iii) serious DS if 10pvnðtþ 1Þ � vnþ1ðtþ 1Þo15; and (iv) fatal DS ifvnðtþ 1Þ � vnþ1ðtþ 1ÞX15. In Fig. 3, we show the probability of the four sub-types of DS against thedensity. Firstly, one can see that the orders of the probability of the trivial DS, medium DS, serious DS, andfatal DS are gradually decreasing. This is consistent with the real traffic, i.e., the trivial accidents happenfrequently while the fatal accidents happen rarely [20,21].

Moreover, one can also see that for the trivial DS and medium DS, the probability increases with theincrease of the system density. As for the serious DS, it happens most frequently at density r � 0:45, then itdecreases with the increase of density. The fatal DS happens most frequently at a smaller density r � 0:25, andfor large density, it will not occur.

Next, we consider the DS when the traffic state is on the branch OAC. In Fig. 4, we show the results wheninitially the system is a megajam. It can be seen that there are two discontinuities. When the traffic density issmaller than rc0, there is no stopped car, so the DS probability is zero. When the density is 1, there is no

0.06

0.05

0.04

0.03

0.02

0.01

0.00

0.0 0.1 0.2 0.3 0.4 0.5 0.6

density

0.0 0.1 0.2 0.3 0.4 0.5 0.6

density

0.0 0.1 0.2 0.3 0.4 0.5 0.6

density

0.0 0.1 0.2 0.3 0.4 0.5 0.6

density

pro

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rob

ab

ility

trivial DS medium DS

fatal DSserious DS

0.00015

0.00010

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0.00000

0.02

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0.00

pro

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typ

rob

ab

ility

0.0000030

0.0000025

0.0000020

0.0000015

0.0000010

0.0000005

0.0000000

a b

c d

Fig. 3. The probability of four types of DS against density when the traffic starts from homogeneous initial condition. For the density

corresponding to the dashed line in Fig. 1, the results are obtained before the jam occurs.

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pro

babili

ty

density

Fig. 4. The probability of DS against density when the traffic starts from megajam. The dashed line is the analytical tendency.

Fig. 5. The spacetime plots of the system when starting from the homogeneous initial condition and after sufficiently long time. The

density r ¼ 0:7.

R. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640638

moving car, so the DS probability is also zero. When the density is in the range rc0oro1, there exist bothmoving cars and stopped cars, so the DS may occur.

We notice that when starting from a megajam, the traffic will be the coexistence of jams and free flow as wellas light synchronized flow if rc0oro1. Furthermore, there exists only one jam, and it is compact. In addition,there is no stopped car in the outflow region of the jam. Therefore, the DS occurs only when the car isapproaching the jam. Based on the fact, we may approximately derive the tendency of the relationshipbetween the DS probability and density.

When the system is stationary, the average moving speed of the upstream front of the jam is the same as thatof the downstream front. Since p0 ¼ 0:5, the speed of the downstream front is therefore 7:5m=2 s ¼ 3:75m=s.So the speed of the upstream front of the jam is also 3.75m/s. This means that every two seconds there is a carwhose speed decreases to zero. We suppose that when a car stops, it will induce a DS with a fixed ratio R.Therefore, the DS probability is ð1=2rÞ � R. In Fig. 4, we also show the curve of ð1=2rÞ � R where R ¼ 0:0004.One can see that it is in good agreement with the simulation results.

We also need to point out if not starting from a megajam, there may exist more than one jam when thestationary state is arrived if rc0oro1 (Fig. 5). For the case, the DS probability is proportional to the number

ARTICLE IN PRESSp

rob

ab

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pro

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pro

ba

bili

typ

rob

ab

ility

density density

densitydensity

trivial DSmedium DS

fatal DSserious DS

a b

c d

Fig. 6. The probability of four types of DS against density when the traffic starts from megajam. The dashed line is the analytical

tendency. (a) R ¼ 0:00034; (b) R ¼ 0:00005; (c) R ¼ 0:0000056; and (d) R ¼ 0:000001.

R. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640 639

of the jams. For example, in Fig. 5 there are two jams. The simulation results show that its DS probability is5:74� 10�4, approximately double 2:85� 10�4 when starting from megajam.

We also investigate the four sub-types of DS when the system starts from a megajam. The simulation resultsare plotted in Fig. 6. One can see that similarly, the orders of the probability of the trivial DS, medium DS,serious DS, and fatal DS gradually decrease. However, different from in synchronized flow, the tendency ofthe relationship between the four sub-types of DS and the density is similar to each other and obeysð1=2rÞ � R.

4. Conclusions

This paper studies the DS in the synchronized flow model, which can reproduce the synchronized flow andthe spatial–temporal patterns of real traffic quite satisfactorily. The DS on the two branches of thefundamental diagram are investigated, respectively.

It is shown that in the synchronized flow, the DS probability is zero when the density is small. When thedensity exceeds a critical density rc1, the DS begins to appear on the road. The DS probability increases withthe increase of density. Moreover, we prove that there is no DS caused by non-stopped car although the model

ARTICLE IN PRESSR. Jiang, Q.-S. Wu / Physica A 377 (2007) 633–640640

itself is not an exclusion process. We classify the DS into four sub-types and study the probability of these foursub-types. The simulation result is consistent with the real traffic.

As for the traffic on the lower branch, it is shown that there are two discontinuities. When the traffic densityis smaller than rc0, there is no stopped car, so the DS probability is zero. When the density is 1, there is nomoving car, so the DS probability is also zero. When the density is in the range rc0oro1, there exist bothmoving cars and stopped cars, so the DS may occur. The tendency of the relationship between the DSprobability and density is derived. The four sub-types of DS are also investigated. It is shown that the result isquite different from that in synchronized flow.

A comparison between the DS in synchronized flow and in jam shows that in synchronized flow, the carsmove faster but the DS probability is large. So it is needed to improve the driving safety without loss of speed.

Acknowledgments

We acknowledge the support of National Basic Research Program of China (2006CB705500), the NationalNatural Science Foundation of China (NNSFC) under Key Project no. 10532060 and Project nos. 10404025,10672160, 70601026, the CAS special Foundation, the open project of Key Laboratory of IntelligentTechnologies and Systems of Traffic and Transportation, Ministry of Education, Beijing Jiaotong University.

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