Decentralized Cooperation Strategies in Two-Dimensional Traffic of Cellular Automata

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1 Commun. Theor. Phys. 58 (2012) Vol. 58, No. 6, December 15, 2012 Decentralized Cooperation Strategies in Two-Dimensional Traffic of Cellular Automata FANG Jun (à ), 1,2, QIN Zheng (Æ), 1,2 CHEN Xi-Qun (íí ), 3 LENG Biao ( Â), 4 XU Zhao-Hui (Å Þ), 1 and JIANG Zi-Neng ( Í) 5 1 Department of Computer Science and Technology, Tsinghua University, Beijing , China 2 Key Laboratory for Information System Security, Ministry of Education, Beijing , China 3 Department of Civil Engineering, Tsinghua University, Beijing , China 4 School of Computer Science & Engineering, Beihang University, Beijing , China 5 School of Software, Tsinghua University, Beijing , China (Received March 14, 2012; revised manuscript received August 27, 2012) Abstract We study the two-dimensional traffic of cellular automata using computer simulation. We propose two type of decentralized cooperation strategies, which are called stepping aside (CS-SA) and choosing alternative routes (CS-CAR) respectively. We introduce them into an existing two-dimensional cellular automata (CA) model. CS-SA is designed to prohibit a kind of ping-pong jump when two objects standing together try to move in opposite directions. CS-CAR is designed to change the solution of conflict in parallel update. CS-CAR encourages the objects involved in parallel conflicts choose their alternative routes instead of waiting. We also combine the two cooperation strategies (CS-SA-CAR) to test their combined effects. It is found that the system keeps on a partial jam phase with nonzero velocity and flow until the density reaches one. The ratios of the ping-pong jump and the waiting objects involved in conflict are decreased obviously, especially at the free phase. And the average flow is improved by the three cooperation strategies. Although the average travel time is lengthened a bit by CS-CAR, it is shorten by CS-SA and CS-SA-CAR. In addition, we discuss the advantage and applicability of decentralized cooperation modeling. PACS numbers: Vn, Fh, m, a, n Key words: two-dimensional traffic model, phase transition, decentralized cooperation strategy, cellular automata 1 Introduction The traffic problems have attracted many scholars with different types of backgrounds and many traffic models have been developed in the physics literature. [1 4] The two-dimensional traffic models are of considerable interest and have been used to simulate and analyze many kinds of systems of mobile objects, such as the car [5 20] and the pedestrian. [21 38] As a typical model of two-dimensional cellular automata (CA) for unbar traffic, BIHAM MIDDLETON LEVINE model (BML) has drawn the wide attention since it was introduced in Ref. [5]. After that, a large number of generalizations and extensions of the BML model were reported or published. [6 9] It is a interesting problem to introducing the origin-destination (OD) or routing information of cars into two-dimensional traffic and up to now there are a number of published results. [10 15] Except for cars, the pedestrian and evacuation models are another kind of twodimensional traffic model and have been studied widely in the field of transport hub, fire protection, building safety, ship and aircraft evacuation, etc. [21 24] Unlike other mobile objects, the pedestrian group has some adaptability or intelligence and they are able to cooperate with each other for some goals or under some principles. For two people facing each other or one person facing an obstacle, the turning and sidling effect is observed widely in daily experience and has been considered in some pedestrian models. [25 27] Each pedestrian in Ref. [25] occupies only one site. The pair of pedestrians facing each other exchange their positions simultaneously within only one time step. In Ref. [25], when two pedestrians meet face-to-face, one of them turns himself sidelong to let his partner to move. The sideways movement in Ref. [27] is similar to Ref. [26] except that the object in front of the pedestrian is not another pedestrian but some fixed obstacles. Due to the characteristic of pedestrian flow in reality, the parallel update rule is widely adopted in pedestrian modeling. Considering the excluded-volume effect, the model must solve the conflict in parallel update rule. The most common method is choosing one person among all rivals with equal probability [28] or different probabilities. [29] Supported by the National Natural Science Foundation of China under Grant No and the National High-Tech Research and Development Plan of China (863) under Grant No. 2011AA Corresponding author, fangjun06@mails.tsinghua.edu.cn c 2011 Chinese Physical Society and IOP Publishing Ltd

2 884 Communications in Theoretical Physics Vol. 58 As an extension, the friction parameter µ was introduced in Refs. [30 33]. In Ref. [36], the calculation of friction parameter µ is formulated in a gaining function and analyzed with the ST (saint & temptation) reciprocity game theory. Summarizing so far, the solutions listed above solve the conflict through restricting all or most part of pedestrians to move to avoid overlapping, which often causes the bottleneck effect. We try to encourage all waiting pedestrians in conflict to move along other alternative routes. The purpose of this paper is to study the jamming transition of point-to-point traffic through cooperation strategies using computer simulation. We propose two cooperation strategies, called stepping aside (CS-SA) and choosing alternative routes (CS-CAR) respectively, and introduce them into an existing two-dimensional CA traffic model. We hope through the two cooperation strategies the average velocity and average flow of the system could be increased and the travel time could be shortened. The potential applications of our models include modeling and simulation of mobile objects in the system of transportation, mechanism, biology, computer network and sociology, for instance, the movement of a great lot of small robots, movement of ant-like social insects, transportation of data packets on the Internet, pedestrian and evacuation dynamic. The paper is organized as follows. Section 2 describes four traffic models. For model 1, mobile objects travel between randomly chosen origins and destinations without any cooperation strategy. Based upon model 1, the models 2 and 3 incorporate CS-SA and CS-CAR, respectively. The model 4 combines the model 2 with model 3 to test the comprehensive effects of the two cooperation strategies. Section 3 presents and compares the simulation results of the four models. Section 4 gives the summary and discussion of our work. 2 Models 2.1 Model 1: Basic Movement Model 1 is a classical point-to-point traffic model, which is the same as Maniccam s model. [10] The traffic rules of model 1 are described as follows: (i) Initialization. At the beginning, each object is associated with a pair of origin-destination sites (OD). Each pair of OD is chosen randomly on the lattice and the origin and destination must be different. The origins of all objects must be different from each other but their destinations are allowed to overlap. (ii) Calculation of the ND-distances. In Fig. 1(a), the destination of object A and B is denoted by D A and D B, respectively. The digit in the upper left corner represents the distance from each neighbor site to the destination of central object (ND-distance). (iii) Movement selection. According the myopic or greedy approach, the object always chooses an unoccupied site from its neighborhood to move that is the nearest to its destination. If the current location and destination of an object is straight aligned, it prefers the lateral neighbors to back neighbor. If two candidate neighbors are both vacant, they are chosen randomly as the next position. It should be noted that objects would move as long as it has at least one vacant neighbor site, even though it goes far away from its destination next time. When an object arrives at one of lattice boundaries, it can either back step or side step but cannot cross the boundary. (iv) Solution to conflicts in parallel update rule. One of rivals will be chosen randomly as the occupant and the others are just waiting until next time. Each rival has the same probability to win. (v) Re-assignment of OD sites. After arriving at its destination, each object uses the destination as its new origin and randomly chooses another site on the lattice as its new destination. Then they continue to travel from point to point unless the simulation has reached the max time steps. Fig. 1 Two typical spatial distributions between the current location and destination of an object: (a) Straight aligned (horizontally or vertically), (b) Diagonally aligned. 2.2 Model 2: Cooperation Strategy of Stepping Aside (CS-SA) Model 1 will generate the ping-pong jump (PPJ), which slows down the average flow of the system and delays objects arriving at their destinations. The PPJ is defined as a type of loop path that an object moves away from a site at time t 2 and returns to the same site at time step t. The objects stand still for two time steps are not included. Our statistics on PPJ have not included the objects that move in a loop path using more than two time steps yet so far. Figure 2 gives an example about how the PPJ happens. We develop the model 1 into model 2 by introducing CS-SA, which is described as follows.

3 No. 6 Communications in Theoretical Physics 885 (i) Cooperation condition. (a) The pair of objects stand face-to-face horizontally or vertically and try to move in opposite direction; (b) There is at least one of them whose the current location and destination are in a straight line, e.g. the object B in Fig. 4; (c) There are enough vacant neighbor sites around to move. (ii) Cooperation process. When the current locations and destinations of both objects are straight aligned just like Fig. 3, the one with more vacant lateral sites is chosen to move sideways. If both objects have the same number of vacant lateral sites, one of them is chosen randomly. On the other hand, if the current location and destination of an object are diagonally aligned, e.g. the object A in Fig. 4, it is chosen to step aside. If its lateral site facing its destination is not vacant, e.g. the blue site in Fig. 4(a), it and its partner cannot take part in this cooperation. While the chosen object moves sideways, the other stands still and gets ready to move straight ahead next time. (iii) Cooperation priority. The cooperators in step (ii) have some priority in occupying the lateral vacant positions. The objects that take in the cooperations have higher priority than the others that do not do it. But two objects that both take in cooperations have the same priority. This setting is to increase the probability of success for the cooperation. Fig. 2 (Color online) Illustration of ping-pong jump when the current location and destination of one object is diagonally aligned. The choices of object A and B are denoted by blue and pink boxes, respectively. Fig. 3 (Color online) Illustration of CS-SA. Fig. 4 (Color online) Illustration of CS-SA corresponding to Fig Model 3: Cooperation Strategy of Choosing Alternative Routes (CS-CAR) The conventional solution to the conflict in parallel update wastes some alternative routes for waiting objects if they have two choices. This solution decreases the average velocity of the system, and may slow down the average flow and lengthen the travel time. In order to make as many objects as possible moving instead of waiting, the rivals surrounding a conflict site need to cooperate with each other. Therefore, we develop model 1 into model 3 by introducing CS-CAR, which is described as follows: (i) Cooperation process. The rivals surrounding each conflict site are divided into two groups according to the number of choice: the objects with only one choice (C-1) and the other with two (C-2). The members of C-1 have precedence over C-2 in occupying the conflict site. If C-1

4 886 Communications in Theoretical Physics Vol. 58 has more than one member, one of them is chosen randomly to occupy that site and the others are waiting until next time. If C-1 has no member, one of members of C-2 is chosen randomly to occupy the conflict site. Then the members of C-2 choose their alternative routes. (ii) Cooperation priority. The objects that take in the cooperations have higher priority than the others that do not do it. But two objects that both take in cooperations have the same priority. This setting is to increase the probability of success for the cooperation. Fig. 5 (Color online) Illustration of CS-CAR process. The objects A, B, C, and D are competing for the same site colored with orange. The object A and B each has only one direction to move in, while C and D each has an alternative direction. (a) The solid arrow of each object indicates its initial choice and the dash arrows of C and D indicate their alternative directions. (b) Firstly A and B compete the central site. Finally A occupies that site while B stands still. As the central point is occupied, C and D try to move in their alternative direction. (c) C and D choose another routes simultaneously. 2.4 Model 4: the Combined Cooperation Strategy (CS-SA-CAR) Model 4 combines model 2 with model 3 to test their combined effect. The traffic rule of model 4 is described as follows: (i) The objects satisfying the cooperative conditions of CS-SA move according to the rules of CS-SA. (ii) The objects that cannot take part in CS-SA in step (i) move normally according to the traffic rules of model 1. (iii) The Solution to conflict. All rivals surrounding the conflict sites move according to the rules of CS-CAR. However, due to the process sequence, the priority of CS- SA is higher than CS-CAR. If an object has taken part in CS-SA in step (i), it has the priority over other rivals in occupying the conflict site. 3 Simulation and Results Firstly, we investigate the velocity-density diagram of four models. The average velocity v(t) of the system is defined as the ratio of moving objects to all objects on the lattice at time step t. At each density, ten independent simulations are repeated with different randomly initialization and each simulation runs for time steps. The ensemble average velocity v is defined as the average of v(t) during the steady stage 8001 t In model 2, one of the two objects participating CS-SA is restricted from moving. So the velocity of model 2 hardly increases compared with model 1 (0.03% for L = 50 and 0.28% for L = 100). However, due to CS-CAR the velocity of model 3 and model 4 is improved. Averaging the value of velocity at full range of density, the velocity of model 3 and 4 is 11.21% and 9.98% higher than model 1 when L = 50, respectively. And the velocity of model 3 and 4 is 11.89% and 10.19% higher than model 1 when L = 100, respectively. Figure 6 focues on the phase transition in the velocity-density diagram of four models. The critical densities of model 1 4 at L = 50 are 0.08 ±0.01, ± 0.01, ± 0.01 and ± 0.01, respectively. Moreover, the critical densities of model 1 4 at L = 100 are 0.07 ± 0.01, 0.07 ±0.01, ± 0.01 and ± 0.01, respectively. Unlike the velocity-density diagram of the BML model, [5] the jam phase remains partial throughout the full range of density and the velocity is above zero unless the lattice is full of objects. This is due to the special jam pattern shown in Fig. 7. The jam pattern of model 1 is a kind of configuration of central aggregation. Most of objects concentrate in the middle of the lattice forming a jam cluster and the rest scatter outside the cluster. There are some thinly scattered cavities across the jam cluster. Most of moving objects lie along the surface of the cluster and the other stay outside or at the edge of each cavity in the middle of the cluster. The nonzero velocity at jam phase is mainly attributed by the motion of objects lying outside and along the surface of the cluster. Due to the configuration of central aggregation, the objects inside the cluster hardly move out and similarly the objects outside hardly move in. Secondly, we investigate the flow-density diagram of four models. Like the definition of average velocity, the flow f(t) of the system is defined as the number of objects that arrive at their destinations at time step t. The ensemble average flow f is defined as the average of f(t) during the steady stage 8001 t The max flows f max of four models at their respective critical density, from the highest to the lowest in rank, are model 2, model 4, model 1, and model 3. It should be noted that

5 No. 6 Communications in Theoretical Physics 887 the flow of model 4 is the highest of four models at free phase (ρ 0.07 for L = 50 and ρ 0.06 for L = 100 in Fig. 8). Just because the traffic in model 4 gets into the jam phase a bit earlier than in model 2, the max flow of model 2 exceeds that of model 4 at the critical density of model 2. Fig. 6 (Color online) The region of phase transition in the velocity-density diagram of four models at system size (a) L = 50, (b) L = 100. Fig. 7 (Color online) A typical configuration of jam cluster at t = for model 1 at system size L = 50 and density ρ = 0.2. Why the increment of flow are small in relation to the increment of velocity for model 3 and 4? We divide the movement of an object into two classes according to the change of distance from its location to its destination after this movement: advance and detour. Through the advanced movement, an object gets closer to its destination. In contrast, the detour means going far way from its destination, including side steps and back steps. The flow is contributed by the average velocity and the ratio of advanced movement simultaneously, not only the average velocity. The ratios of advanced movement to all types of movements (advance + detour) for the four models are given in Fig. 9. The rank in ratio of advanced movement of four models is the same as the rank in the flow of Fig. 8. For model 3, although the velocity is increased obviously, the ratio of advanced movement does not be increased with the increase of the velocity. The increase in the amount of advance is canceled by the increase in the amount of detour. However, the detour effectively reduces traffic jam and improves the critical density, especially when there are four moving directions and no separated lanes for traffic in opposite directions. In order to verify the effectiveness of detour, we reproduce the experiments of the fourth model in Ref. [10] and show a typical traffic configuration on the lattice in Fig. 10(a). The fourth model in Ref. [10] is similar to the model 1 except that each object only travels in the shortest path without detour. In Fig. 10(a), all objects are trapped into a complete jam after t = 989 even ρ = While for the traffic in Fig. 10(b), the system is still at free phase after t = 10, 000 at ρ = If the objects have to move ahead without detour, the objects coming from four directions will form a gridlock easily. In this gridlock, everyone is waiting for others to make way for itself, and as a result, no one can move again. The average travel times of four models at free phase are compared in Fig. 11. From the shortest to the longest time in rank, they are model 4, model 2, model 1, and model 3 at system size L = 50 and L = 100. As the OD pairs are randomly chosen from the whole lattice, the OD distance assigned to each object is largely different with each other, which ranges from one site to 2 (L 1) sites in the closed boundary. In order to measure the traveling smoothness of each object and calculate the average values of all objects, we introduce the time-distance ratio (TDR), i.e. the travel time of each object divided by its OD distance. When an object arrives at its destination without hindrance, its TDR equals just one; otherwise, it is greater than one. Averaging the value of TDR over the densities at free phase, the TDRs of model 2 and model 4 decrease by 4.35% and 7.01% compared with model 1 at L = 50, respectively. Moreover, the TDRs of model 2 and model 4 decrease by 8.90% and 8.48% at L = 100, respectively. However, the TDR of model 3 is 1.94% and 1.62% higher than that of model 1 when L = 50 and L = 100, respectively. We also calculate the TDRs of four models at jam phase, which are all far more than two and increase linearly with the increase of density. During the period of jam phase, part of objects cannot arrive at their destinations until the simulations end.

6 888 Communications in Theoretical Physics Vol. 58 Fig. 8 (Color online) The region of phase transition in the flow-density diagram of four models at system size (a) L = 50, (b) L = 100. Fig. 9 (Color online) The ratio (r a) of advanced movement to all types of movements (advance + detour) against density (ρ) for four models at system size (a) L = 50, (b) L = 100. Fig. 10 (Color online) The typical traffic configurations of two models at L = 50: (a) t = 989, the model without detour (the fourth model in Ref. [10] at ρ = 0.01, (b) t = 10000, the model with detour (model 1) at ρ = In order to illustrate the results of Fig. 11, in Fig. 12 we calculate the ratio composition of three types of motion states at free phase: the pause, detour and advance. For the three motion states, we count the sum of each state from all objects during t = 8001 to t = , respectively. Then we divide the sum of each state by the product of the total number of objects and 2000 time steps, respectively. The travel time cost of the three states, from the least to the most in rank, is the advance, pause and detour. Comparing the ratio compositions of four models in Fig. 12, the ratio of detour in model 3 is the highest, which leads to the longest travel time. The ratio of advanced motion in model 3 is the lowest of four models at the free phase shown in Fig. 9. Compared with model 1, the model 2 decreases the ratio of detour and the model 4 decreases the ratio of detour and pause simultaneously. Therefore, their travel times are lower than that of model 1. The model 3 does not shorten the travel time. However, combining the CS-CAR with CS-SA, the travel time is shorten in model 4.

7 No. 6 Communications in Theoretical Physics 889 Fig. 11 The travel time-distance ratio (TDR) of four models at free phase: (a) L = 50, (b) L = 100. Fig. 12 The stacked bar graph containing ratios of three types of motion states at L = 50 at free phase: (a) Model 1, (b) Model 2, (c) Model 3, (d) Model 4. 4 Summary and Discussion We investigate and compare the four models mainly from the fundamental diagrams, travel time-distance ratio, occurrence and elimination of ping-pong jump and parallel conflict. Table 1 sums up the results. For model 2, the average velocity is almost the same as that of model 1 while the flow is increased and travel time is shortened. For model 3, the average velocity and flow are increased but the travel time is lengthened. For model 4, all traffic parameters are improved. In particular, the growth rate of average flow in model 4 is the highest of the three models. It should be noted that the flow of model 4 is the largest of four models at free phase (ρ 0.07 for L = 50 and ρ 0.06 for L = 100), which are shown in Figs. 8(a) and 8(b). It should be noted that the two cooperation strategies are both a type of decentralized cooperation strategy (DCS). In this strategy, several objects form a group temporarily to cooperate with each other. Usually, the number of group members is small and the group lasts for a short period. The group members take part in cooperation according to only their local environment. After this group splits up, the partnership of all members breaks up and they form some new groups with other objects again. Unlike the centralized cooperation strategy (CCS), there is not a global strategy, which can coordinate all objects

8 890 Communications in Theoretical Physics Vol. 58 or a major part of objects in the system. The main advantages of DCS are its simple rules and requiring little or even no extra parameters. It is an interesting question to incorporate the CCS into our models, or combine the DCS with CCS in the same traffic model. Some remaining questions are worth further research. Firstly, we are unclear whether the critical density remains non-zero in the thermodynamic limit. We have only carried out simulations on the lattice of size L = 100 at most. Using larger regions is very costly in terms of computer simulation time. Secondly, how to increase the ratio of advanced movement in model 3 and model 4, and at the same time not to decrease the critical density or increase traffic jam. Finally, the scope of cooperation is rather small, e.g. two objects in the CS-SA and no more than four objects in the CS-CAR. It is a beneficial trial to expand the scope of cooperation strategy to make more objects working together. Table 1 The growth rates of model 2 4 relative to model 1 for several important parameters. The positive values present the increase and negative values present the decrease for model 2 4 in relation to model 1. Please note that the lower the TDR is, the better the traffic is. L = 50 Model 2 Model 3 Model 4 v 0.03 % % 9.98 % f 3.12 % 5.73 % 9.58 % f max % 2.00 % 9.89 % TDR 4.35 % 1.94 % 7.01 % L = 100 Model 2 Model 3 Model 4 v 0.28 % % % f 3.35 % 3.82 % 6.01 % f max % 0.67 % 7.08 % TDR 8.90 % 1.62 % 8.48 % References [1] D. Chowdhury, L. Santen, and A. Schadschneider, Phys. Rep. 329 (2000) 199. [2] D. Helbing, Rev. Mod. Phys. 73 (2001) [3] T. Nagatani, Rep. Prog. Phys. 65 (2002) [4] S. Maerivoet, B. De Moor, Phys. Rep. 419 (2005) 1. [5] O. Biham, A.A. Middleton, and D. Levine, Phys. Rev. A 46 (1992) [6] O.K. Tonguz, W. Viriyasitavat, and F. Bai, IEEE Commun. Mag. 47 (2009) 142. [7] M. Fukui and Y. Ishibashi, Physica A 389 (2010) [8] Z.J. Ding, R. Jiang, W. Huang, and J. Stat. Mech. Theory Exp. (2011) P [9] Q.H. Sui, Z.J. Ding, and R. Jiang, Comput. Phys. Commun. 183 (2012) 547. [10] S. Maniccam, Physica A 331 (2004) 669. [11] N. Moussa, Int. J. Mod. Phys. C 18 (2007) [12] J. In-nami and H. Toyoki, Physica A 378 (2007) 485. [13] D. Huang and W. Huang, Chin. J. Phys. 45 (2007) 708. [14] D.W. Huang and W.N. Huang, Traffic and Granular Flow 07, ed. Cécile Appert Rolland, Springer-Verlag, Berlin (2009) 333. [15] J. Fang, J. Shi, X.Q. Chen, and Z. Qin, Int. J. Mod. Phys. C 21 (2010) 221. [16] C.Q. Mei and Y.J. Liu, Commun. Theor. Phys. 56 (2011) 945. [17] Y.S. Qian, W.J. Li, and J.W. Zeng, Commun. Theor. Phys. 56 (2011) 785. [18] Y. Peng, H.Y. Shang, and H.P. Lu, Commun. Theor. Phys. 56 (2011) 177. [19] T.Q. Tang, P. Li, and Y.H. Wu, Commun. Theor. Phys. 58 (2012) 300. [20] T.Q. Tang, C.Y. Li, and H.J. Huang, Nonlinear Dyn. 67 (2012) [21] D. Helbing and P. Molnar, Phys. Rev. E 51 (1995) [22] M. Muramatsu, T. Irie, and T. Nagatani, Physica A 267 (1999) 487. [23] A. Schadschneider, W. Klingsch, H. Kluepfel, T. Kretz, C. Rogsch, and A. Seyfried, Encyclopedia of Complexity and System Science, ed. R.A. Meyers, Springer-Verlag, Berlin (2009) [24] J.B. Zeng, B. Leng, Z. Xiong, and Z. Qin, Int. J. Mod. Phys. C 22 (2011) 775. [25] J. Li, L. Yang, and D. Zhao, Physica A 354 (2005) 619. [26] M. Fukamachi and T. Nagatani, Physica A 377 (2007) 269. [27] A. Matsui, T. Mashiko, and T. Nagatani, Physica A 388 (2009) 157. [28] Q.Y. Hao, M.B. Hu, X.Q. Cheng, W.G. Song, R. Jiang, and Q.S. Wu, Phys. Rev. E 82 (2010) [29] C. Burstedde, K. Klauck, A. Schadschneider, and J. Zittartz, Physica A 295 (2001) 507. [30] A. Kirchner, K. Nishinari, and A. Schadschneider, Phys. Rev. E 67 (2003) [31] A. Kirchner, H. Klüpfel, K. Nishinari, A. Schadschneider, and M. Schreckenberg, Physica A 324 (2003) 689. [32] D. Yanagisawa and K. Nishinari, Phys. Rev. E 76 (2007) [33] A. Schadschneider and A. Seyfried, Cybern. Syst. 40 (2009) 367. [34] W. Song, X. Xu, B.H. Wang, and S. Ni, Physica A 363 (2006) 492. [35] S. Marconi and B. Chopard, Proceedings of Cellular Automata, eds. S. Bandini, B. Chopard, M. Tomassini, Springer-Verlag, Berlin (2002) 231. [36] J. Tanimoto, A. Hagishimaa, and Y. Tanaka, Physica A 389 (2010) [37] R.Y. Guo and H.J. Huang, Transport. Res. C-emer 24 (2012) 50. [38] R.Y. Guo, H.J. Huang, and S.C. Wong, Transport. Res. B-meth 46 (2012) 669.