- Email: [email protected]

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 of physicists [1–5]. These models have successfully reproduced many qualitative features observed in real trafﬁc such as trafﬁc jams, etc. More recently, theoretical and numerical results for dangerous situations (DS) in the framework of the CA models have been reported [6–17]. In a DS, there will be no accident if every driver is careful enough. Nevertheless, if the drivers are not so careful (p2 40, see the following text), an accident may occur. In other words, if one denotes the probability of a DS as P, then a car accident will occur with probability P p2 . Boccara et al. [6] were the ﬁrst to propose conditions for DS in the deterministic Nagel–Schreckenberg model. They assume that drivers will probably not respect the safe distance if the speed of the car ahead vnþ1 ðtÞ is positive at time t, because drivers expect the speed of the car ahead vnþ1 ðt þ 1Þ to remain positive at time t þ 1. Thus, the drivers increase their velocity by one unit with probability p2 . It is clear that the careless driving will probably result in an accident if the speed of the car ahead vnþ1 ðt þ 1Þ becomes zero at time t þ 1. Based on this assumption, Bocarra et al. argue that if the following three conditions are satisﬁed, then the car is in a DS: (i) 0pd n ðtÞpvmax ; (ii) vnþ1 ðtÞ40; and (iii) vnþ1 ðt þ 1Þ ¼ 0. Here d n ðtÞ denotes the gap to the front Corresponding author.

E-mail address: [email protected] (R. Jiang). 0378-4371/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2006.11.073

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

634

car and vmax is maximum velocity. Later Huang and coworkers [7,8] as well as Yang and Ma [9] have investigated 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) d n ðt þ 1Þ ¼ 0; (ii) vnþ1 ðtÞ40; and (iii) vn ðt þ 1Þovmax . Jiang et al. [11] also investigated the DS in the velocity-effect model, which is a non-exclusion process: two different DS, i.e., DS caused by stopped cars and DS caused by non-stopped cars, are studied. Moreover, Moussa [12] has investigated the effect of the delayed reaction time on the probability of DS in the Nagel–Schreckenberg model as well as the DS caused by stopped cars and great deceleration. We also note that the DS problem has been studied in the Fukui–Ishibashi model [13,14]. Yang et al. have studied the DS in the open boundary conditions [15] and when disordered effect was considered [16]. Moussa extended the DS investigation to two-lane trafﬁc [17]. We notice that the above mentioned works were carried out in Nagel–Schreckenberg model or velocityeffect 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 quite satisfactorily [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 the conditions of the DS are presented. In Section 3, the simulation results are analyzed and compared with the empirical 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 of the model are as follows: 1. Determination of the randomization parameter pn ðt þ 1Þ: pn ðt þ 1Þ ¼ pðvn ðtÞ; bnþ1 ðtÞ; th;n ; ts;n Þ. 2. Acceleration: if ððbnþ1 ðtÞ ¼ 0 or th;n Xts;n Þ and ðvn ðtÞ40ÞÞ then: vn ðt þ 1Þ ¼ minðvn ðtÞ þ 2; vmax Þ; else if ðvn ðtÞ ¼ 0Þ then: vn ðt þ 1Þ ¼ minðvn ðtÞ þ 1; vmax Þ; else: vn ðt þ 1Þ ¼ vn ðtÞ: 3. Braking rule: vn ðt þ 1Þ ¼ minðd eff n ; vn ðt þ 1ÞÞ. 4. Randomization and braking: if ðrandðÞopn ðt þ 1ÞÞ then : vn ðt þ 1Þ ¼ maxðvn ðt þ 1Þ 1; 0Þ. 5. The determination of bn ðt þ 1Þ: if ðvn ðt þ 1Þovn ðtÞÞ then: bn ðt þ 1Þ ¼ 1; if (vn ðt þ 1Þ4vn ðtÞ) then: bn ðt þ 1Þ ¼ 0; if (vn ðt þ 1Þ ¼ vn ðtÞ) then: bn ðt þ 1Þ ¼ bn ðtÞ. 6. The determination of tst;n : if vn ðt þ 1Þ ¼ 0 then: tst;n ¼ tst;n þ 1; if vn ðt þ 1Þ40 then: tst;n ¼ 0. 7. Car motion: xn ðt þ 1Þ ¼ xn ðtÞ þ vn ðt þ 1Þ.

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

635

flow

A

B

ρc0

O density 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].

Here xn and vn are the position and velocity of vehicle n (here vehicle n þ 1 precedes vehicle n), d n is the gap of the vehicle n, bn is the status of the brake light (onðoffÞ ! bn ¼ 1ð0Þ). The two times th;n ¼ d n =vn ðtÞ and ts;n ¼ minðvn ðtÞ; hÞ, where h determines the range of interaction with the brake light, are introduced to compare the time th;n needed to reach the position of the leading vehicle with a velocity dependent interaction horizon ts;n . d eff n ¼ d n þ maxðvanti gapsafety ; 0Þ is the effective distance, where vanti ¼ minðd nþ1 ; vnþ1 Þ is the expected velocity 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 randomization parameter p is deﬁned: 8 p if bnþ1 ¼ 1 and th;n ots;n ; > < b pðvn ðtÞ; bnþ1 ðtÞ; th;n ; ts;n Þ ¼ p0 if vn ¼ 0 and tst;n Xtc ; > : p in all other cases: d

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 t þ 1, the gap to the front car equals to zero: d n ðt þ 1Þ ¼ 0; (ii) at time step t þ 1, its own velocity should be smaller than vmax : vn ðt þ 1Þovmax ; and (iii) at time t, the velocity of the front car should be larger than zero: vnþ1 ðtÞ40. 3. Simulation results In this section, the simulation results are presented. Firstly, we consider the DS when the trafﬁc state is on the branch OAB. In Fig. 2, we have shown the plots of DS probability against density. One can see that the DS probability is zero when the density is small. When the density exceeds a critical density rc1 , the DS begins to

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

probality

636

ρc1

density 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 rc1 Þg with g 3; rc1 0:18.

appear on the road. The DS probability increases with the increase of density. We compare the critical density rc1 with critical density rc0 . One ﬁnds that rc1 4rc0 . This means that in the outﬂow from the jam, the DS does not exist. The analytic analysis of DS is difﬁcult to carry out for Fig. 2. But we expect the DS probability behaves as ðr rc1 Þg because with the increase of density, the probability of appearance of stopped cars increases, which is dominating condition in the three conditions . The tendency is well conﬁrmed by the simulation results with g 3; rc1 0:18 (Fig. 2). Since in the model, the anticipation of the movement of the front car is considered, the model is not an exclusion process. Therefore, the question arises: is there existing the DS caused by nonstopped car as in velocity-effect model [11]? Here, however, our answer is no. This can be proved as follows. (i) If d nþ1 ðtÞovnþ1 ðtÞ, then vanti ¼ d nþ1 ðtÞ. So d eff n ¼ d n ðtÞ þ maxðd nþ1 ðtÞ gapsafety ; 0Þ.

If d nþ1 ðtÞXgapsafety ¼ 7, then d eff n ¼ d n ðtÞ þ d nþ1 ðtÞ 7. So the car n can reach at most xn ðtÞ þ d n ðtÞþ d nþ1 ðtÞ 7. While d eff nþ1 Xd nþ1 ðtÞ, we know that car n þ 1 can at least advance by d nþ1 ðtÞ 1 even if it is randomized, i.e., the car n þ 1 can reach at least xnþ1 ðtÞ þ d nþ1 ðtÞ 1. It is obvious that xn ðtÞ þ d n ðtÞ þ d nþ1 ðtÞ 7oðxnþ1 ðtÞ þ d nþ1 ðtÞ 1Þ 5 because xnþ1 ðtÞ xn ðtÞ 5 ¼ d n ðtÞ. If d nþ1 ðtÞogapsafety ¼ 7, then d eff n ¼ d n ðtÞ. So the car n can reach at most xn ðtÞ þ d n ðtÞ. On the other hand, the existence of the DS caused by non-stopped car implies that vnþ1 ðt þ 1Þ40. So xn ðtÞ þ d n ðtÞoðxnþ1 ðtÞþ vnþ1 ðt þ 1ÞÞ 5. (ii) If d nþ1 ðtÞXvnþ1 ðtÞ, then vanti ¼ vnþ1 ðtÞ. So d eff n ¼ d n ðtÞ þ maxðvnþ1 ðtÞ gapsafety ; 0Þ.

If vnþ1 ðtÞXgapsafety ¼ 7, then d eff n ¼ d n ðtÞ þ vnþ1 ðtÞ 7. So the car n can reach at most xn ðtÞ þ d n ðtÞþ vnþ1 ðtÞ 7. On the other hand, due to that d nþ1 ðtÞXvnþ1 ðtÞ, we know that car n þ 1 can at least advance by vnþ1 ðtÞ 1 even if it is randomized, i.e., the car n þ 1 can reach at least xnþ1 ðtÞ þ vnþ1 ðtÞ 1. It is obvious that xn ðtÞ þ d n ðtÞ þ vnþ1 ðtÞ 7oðxnþ1 ðtÞ þ vnþ1 ðtÞ 1Þ 5. If vnþ1 ðtÞogapsafety ¼ 7, then d eff n ¼ d n ðtÞ. The situation is the same as case 2 in (i).

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

637

From the proof one can see that the result only depends on the parameter gapsafety . The result is applicable provided gapsafety 41, which is a necessary condition to avoid collision. In real trafﬁc, the accident expense depends on the relative speed of the two vehicles involved in the accident. According to this, next we classify the DS into four sub-types and investigate the probability of the four sub-types of DS: (i) trivial DS if vn ðt þ 1Þ vnþ1 ðt þ 1Þo5; (ii) medium DS if 5pvn ðt þ 1Þ vnþ1 ðt þ 1Þo10; (iii) serious DS if 10pvn ðt þ 1Þ vnþ1 ðt þ 1Þo15; and (iv) fatal DS if vn ðt þ 1Þ vnþ1 ðt þ 1ÞX15. In Fig. 3, we show the probability of the four sub-types of DS against the density. Firstly, one can see that the orders of the probability of the trivial DS, medium DS, serious DS, and fatal DS are gradually decreasing. This is consistent with the real trafﬁc, i.e., the trivial accidents happen frequently 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 the increase of the system density. As for the serious DS, it happens most frequently at density r 0:45, then it decreases with the increase of density. The fatal DS happens most frequently at a smaller density r 0:25, and for 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 when initially the system is a megajam. It can be seen that there are two discontinuities. When the trafﬁc density is smaller than rc0 , there is no stopped car, so the DS probability is zero. When the density is 1, there is no

a

b 0.02

0.06

0.04

probability

probability

medium DS

trivial DS

0.05

0.03

0.01

0.02 0.01 0.00 0.0

0.00 0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.1

0.2

density

c

0.4

0.5

0.6

0.4

0.5

0.6

d 0.0000030

serious DS 0.00015

fatal DS

0.0000025 probability

probability

0.3 density

0.00010

0.0000020 0.0000015 0.0000010

0.00005

0.0000005 0.00000 0.0

0.1

0.2

0.3 density

0.4

0.5

0.6

0.0000000 0.0

0.1

0.2

0.3 density

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.

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

probability

638

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.

moving car, so the DS probability is also zero. When the density is in the range rc0 oro1, there exist both moving 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 well as light synchronized ﬂow if rc0 oro1. 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 is approaching the jam. Based on the fact, we may approximately derive the tendency of the relationship between 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 that of the downstream front. Since p0 ¼ 0:5, the speed of the downstream front is therefore 7:5 m=2 s ¼ 3:75 m=s. So the speed of the upstream front of the jam is also 3.75 m/s. This means that every two seconds there is a car whose 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 the stationary state is arrived if rc0 oro1 (Fig. 5). For the case, the DS probability is proportional to the number

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

a

639

b medium DS

probability

probability

trivial DS

density

density

c

d fatal DS

probability

probability

serious DS

density

density

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.

of the jams. For example, in Fig. 5 there are two jams. The simulation results show that its DS probability is 5:74 104 , approximately double 2:85 104 when starting from megajam. We also investigate the four sub-types of DS when the system starts from a megajam. The simulation results are 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 of the relationship between the four sub-types of DS and the density is similar to each other and obeys ð1=2rÞ R.

4. Conclusions This paper studies the DS in the synchronized ﬂow model, which can reproduce the synchronized ﬂow and the spatial–temporal patterns of real trafﬁc quite satisfactorily. The DS on the two branches of the fundamental diagram are investigated, respectively. It is shown that in the synchronized ﬂow, the DS probability is zero when the density is small. When the density exceeds a critical density rc1 , the DS begins to appear on the road. The DS probability increases with the increase of density. Moreover, we prove that there is no DS caused by non-stopped car although the model

ARTICLE IN PRESS 640

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

itself is not an 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. As for the trafﬁc on the lower branch, it is shown that there are two discontinuities. When the trafﬁc density is smaller than rc0 , there is no stopped car, so the DS probability is zero. When the density is 1, there is no moving car, so the DS probability is also zero. When the density is in the range rc0 oro1, there exist both moving cars and stopped cars, so the DS may occur. The tendency of the relationship between the DS probability and density is derived. The four sub-types of DS are also investigated. It is shown that the result is quite different from that in synchronized ﬂow. A comparison between the DS in synchronized ﬂow and in jam shows that in synchronized ﬂow, the cars move 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 National Natural 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 Intelligent Technologies and Systems of Trafﬁc and Transportation, Ministry of Education, Beijing Jiaotong University. References [1] M. Schreckenberg, D.E. Wolf (Eds.), Trafﬁc and Granular Flow ’97, Springer, Singapore, 1998; D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Trafﬁc and Granular Flow ’99, Springer, Berlin, 2000; M. Fukui, Y. Sugiyama, M. Schreckenberg, D.E. Wolf (Eds.), Trafﬁc and Granular Flow ’01, Springer, Heidelberg, 2003; S.P. Hoogendoorn, P.H.L. Bovy, M. Schreckenberg, D.E. Wolf (Eds.), Trafﬁc and Granular Flow ’03, Springer, Heidelberg, 2005. [2] D. Chowdhury, L. Santen, A. Schadschneider, Phys. Rep. 329 (2000) 199. [3] D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. [4] K. Nagel, et al., Oper. Res. 51 (2003) 681. [5] S. Maerivoet, B. De Moor, Phys. Rep. 419 (2005) 1. [6] N. Boccara, et al., J. Phys. A 30 (1997) 3329. [7] D.W. Huang, Y.P. Wu, Phys. Rev. E 63 (2001) 022301. [8] D.W. Huang, W.C. Tseng, Phys. Rev. E 64 (2001) 057106. [9] X.Q. Yang, Y.Q. Ma, Mod. Phys. Lett. B 16 (2002) 333. [10] R. Jiang, et al., J. Phys. A 36 (2003) 4763. [11] R. Jiang, et al., J. Phys. A 37 (2004) 5777. [12] N. Moussa, Phys. Rev. E 68 (2003) 036127. [13] D.W. Huang, J. Phys. A 31 (1998) 6167. [14] X.Q. Yang, Y.Q. Ma, J. Phys. A 35 (2002) 10539. [15] X.Q. Yang, et al., J. Phys. A 37 (2004) 4743. [16] X.Q. Yang, et al., Phys. Rev. E 73 (2006) 016126. [17] N. Moussa, Int. J. Mod. Phys. C 16 (2005) 1133. [18] R. Jiang, Q.S. Wu, J. Phys. A 36 (2003) 381. [19] R. Jiang, Q.S. Wu, J. Phys. A 37 (2004) 8197. [20] P. Bak, How Nature Works: The Science of Self-Organized Criticality, Springer, New York, 1996; P. Bak, Self-Organized Criticality in the Bak–Tang–Wiesenfeld Model, available at hwww.physics.nau.edu/hart/classes/265_spring_2006/projects/report.pdfi (accessed on 16 November 2006). [21] The self-organized criticality is observed in many physical phenomena including sandpile, earthquake, forest ﬁre and trafﬁc accidents, in which a power-law behavior is found, i.e., the stronger the event, the less frequently it happens [20].