- Email: [email protected]
Traffic states and fundamental diagram in cellular automaton model of vehicular traffic controlled by signals
Physica A 388 (2009) 1673–1681
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Traffic states and fundamental diagram in cellular automaton model of vehicular traffic controlled by signals Takashi Nagatani Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan
article
info
Article history: Received 30 July 2008 Received in revised form 18 September 2008 Available online 4 January 2009 PACS: 89.40.+k 05.45.-a 82.40.Bj Keywords: Traffic dynamics Cellular automaton Complex system Signal control Nonlinear map
a b s t r a c t We present a cellular automaton (CA) model for vehicular traffic controlled by traffic lights. The CA model is not described by a set of rules, but is given by a simple difference equation. The vehicular motion varies highly with both signals’ characteristics and vehicular density. The dependence of tour time on both cycle time and vehicular density is clarified. In the dilute limit of vehicles, the vehicular motion is compared with that by the nonlinearmap model. The fundamental diagrams are derived numerically. It is shown that the fundamental diagram depends highly on the signals’ characteristics. The traffic states are shown for various values of cycle time in the fundamental diagram. We also study the effect of a slow vehicle on the traffic flow. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Mobility is nowadays one of the most significant ingredients of a modern society. In urban traffic, vehicles are controlled by traffic lights to give priority for a road because the city traffic networks often exceed the capacity. Recently, transportation problems have attracted much attention in the fields of physics [1–5]. The traffic flow, pedestrian flow, and bus-route problem have been studied from a point of view of statistical mechanics and nonlinear dynamics [1–30]. Interesting dynamical behaviors have been found in the transportation system. The jams, chaos, and pattern formation are typical signatures of the complex behavior of transportation [24,25]. Brockfeld et al. have studied optimizing traffic lights for city traffic by using a CA traffic model [31]. They have clarified the effect of signal control strategy on vehicular traffic. Also, they have shown that the city traffic controlled by traffic lights can be reduced to a simpler problem of a single-lane highway. Sasaki and Nagatani have investigated the traffic flow controlled by traffic lights on a single-lane roadway by using the optimal velocity model [32]. They have derived the relationship between the road capacity and jamming transition. Until now, one has studied the periodic traffic controlled by a few traffic lights. It has been concluded that the periodic traffic does not depend on the number of traffic lights [31,32]. Very recently, Lammer and Helbing have studied the vehicular flow by the self- control of traffic signals in urban road networks [33]. In real traffic, the vehicular traffic depends highly on the configuration of traffic lights and the priority of roadways. In the dilute limit of vehicular density, a few works have been carried out for the traffic of vehicles moving through an infinite series of traffic lights with the same interval. The effect of cycle time on vehicular traffic has been clarified by using the
E-mail address: [email protected]. 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.12.046
1674
T. Nagatani / Physica A 388 (2009) 1673–1681
nonlinear-map models [34–38]. Also, it has been shown that the inhomogeneity of a signal’s interval and irregular split have important effects on vehicular traffic [39]. The nonlinear-map model is an effective tool for the vehicular traffic in the dilute limit in which there are no interactions between vehicles. However, the city traffic is affected by both signals’ characteristics and vehicular density. The interaction between vehicles changes with both signal and vehicular density. Therefore, the vehicular traffic controlled by traffic lights changes with its density. There is little known about how the dependence of tour time on the cycle time changes with vehicular density. The vehicular traffic controlled by signals will show a complex behavior. It will be necessary and important to study the vehicular traffic controlled by signals, using a simple dynamic model. In real traffic, the fundamental diagram is very important. It is meaningful to know the relationship between the fundamental diagram and cycle time. Furthermore, when the slow vehicle is introduced in city traffic, it is important to know how the slow vehicle affects the traffic flow. In this paper, we study the vehicular traffic through a series of traffic lights for various values of vehicular density. We present a deterministic cellular automaton model for the vehicular traffic controlled by signals. The CA model is described by the simple difference equation. We investigate the dynamical behavior of vehicular traffic by using the difference equation. We clarify the dynamical behavior of the vehicular traffic through a sequence of signals, by varying both cycle time and vehicular density. We show how the dependence of the tour time on the cycle time changes with the vehicular density. We compare the vehicular traffic with that obtained by the nonlinear-map model. Also, we study the effect of a slow vehicle on the traffic flow. 2. CA model and difference equation We consider the flow of vehicles going through the series of traffic lights on a one-dimensional lattice. Each vehicle does not pass over other vehicles. The traffic lights are positioned homogeneously on a roadway. The interval between signals has a constant value and is given by l. We consider the synchronized strategy for the signal control. In the synchronized strategy, all the traffic lights change simultaneously from red (green) to green (red) with a fixed time period (1 − sp )ts (sp ts ). The period of green is sp ts and the period of red is (1 − sp )ts . Time ts is called the cycle time and fraction sp represents the split which indicates the ratio of green time to cycle time. We set split as sp = 0.5. We extend the deterministic CA model proposed by Fukui and Ishibashi [1,28,40] to take into account traffic lights. We define the position of vehicle i at time t as xi (t ) where x, i, and t are an integer. The CA model of Fukui and Ishibashi is given by xi (t + 1) = min[xi (t ) + vmax , xi+1 (t ) − 1],
(1)
where vmax is the maximum velocity and an integer. Here, min[A, B] is a minimum function and takes the minimum value within A and B. The velocity takes the integer value ranging from 0 to vmax . The velocity depends on the headway. If headway ∆xi (t ) (=xi+1 (t )− xi (t )) is larger than the maximum velocity, the vehicle moves with the maximum velocity. If the headway is less than the maximum velocity, the vehicle moves with velocity ∆xi (t ) − 1. When a vehicle arrives at a traffic light and the traffic light is red, the vehicle stops at the position of the traffic light. Then, when the traffic light changes from red to green, the vehicle goes ahead. On the other hand, when a vehicle arrives at a traffic light and the traffic light is green, the vehicle does not stop and goes ahead without changing speed. The position of the closest signal before vehicle i at time t is given by xi,s (t ) =
int
xi ( t ) l
+ 1 l.
(2)
Then, the position of vehicle i at time t + 1 is given by xi (t + 1) = min[xi (t ) + vmax , xi+1 (t ) − 1]{1 − ϑ(sin(2π t /ts ))}
+ min[xi (t ) + vmax , xi+1 (t ) − 1, xi,s (t ) − 1]ϑ(sin(2π t /ts )).
(3)
Here, ϑ(t ) is the step function. It takes one if t > 0 and zero if t ≤ 0. When there are no signals, Eq. (3) reduces to Eq. (1). If the signal just before vehicle i is green, ϑ(sin(2π t /ts )) = 0 and xi (t +1) = min[xi (t )+vmax , xi+1 (t )−1]. Eq. (3) reduces to Eq. (1). Otherwise, if the signal just before vehicle i is red, ϑ(sin(2π t /ts )) = 1 and xi (t +1) = min[xi (t )+vmax , xi+1 (t )−1, xi,s (t )−1]. Then, if the headway is larger than vmax , vehicle i stops at site xi,s (t ) − 1 just before the signal. Also, if xi+1 (t ) is higher than xi,s (t ), vehicle i stops at site xi,s (t ) − 1 just before the signal. Thus, Eq. (3) presents the CA model for the vehicular traffic through a series of traffic lights. Eq. (3) is a single difference equation. Until now, the CA model for the signal traffic has been described by a set of CA rules. However, model (3) is of great advantage to simulate because the difference equation is simple. It will be expected that the vehicular traffic exhibits a complex behavior. We study how the vehicular traffic changes by varying both cycle time and vehicular density. 3. Nonlinear-map model In the dilute limit of vehicular density, the interaction between vehicles is negligible. Vehicles do not interfere with each other. Then, it is sufficient only to consider a motion of a single vehicle through a series of traffic lights. Here, we consider the
T. Nagatani / Physica A 388 (2009) 1673–1681
1675
Fig. 1. (a) Plots of mean current against density for ts = 15, 25, 45 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. For comparison, the fundamental diagram is shown for the vehicular traffic with no signals. (b) Plots of mean velocity against density.
motion of a single vehicle through the series of signals. The vehicular motion has been described in terms of the nonlinearmap model [36–39]. For later convenience, we summarize the formulation. We define the arrival time of the vehicle at traffic light n as t (n). The arrival time at traffic light n + 1 is given by t (n + 1) = t (n) + l/v + (r (n) − t (n)) H (t (n) − (int((t (n))/ts )ts ) − ts /2)
(4)
with r (n) = (int ((t (n))/ts ) + 1) · ts , where H (t ) is the Heaviside function: H (t ) = 1 for t ≥ 0 and H (t ) = 0 for t < 0. H (t ) = 1 if the traffic light is red, while H (t ) = 0 if the traffic light is green. l/v is the time it takes for the vehicle to move between lights n and n + 1. r (n) is such time that the traffic light just changed from red to green. The third term on the right hand side of Eq. (4) represents such time that the vehicle stops if traffic light n is red. The number n of iteration increases one by one when the vehicle moves through the traffic light. The iteration corresponds to the going ahead on the highway. We will compare the nonlinear-map model (4) with the CA model (3) at the dilute limit of vehicular density in the simulation. 4. Simulation result We investigate the fundamental diagram and traffic states by using Eq. (3). We calculate the current (flow) and velocity for various values of cycle time ts . Fig. 1(a) shows the plots of mean current against density for ts = 15, 25, 45 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. For comparison, the fundamental diagram is shown for the vehicular traffic with no signals. Fig. 1(b) shows the plots of mean velocity against density. For the traffic flow with no signals, the mean velocity keeps the maximal value until ρ = 1/(vmax + 1) = 0.2 and then decreases with increasing density. The mean current increases linearly with density, reaches the maximal current at ρ = 0.2, and then decreases with increasing density. For the traffic flow controlled by signals, the current and velocity curves change greatly from those of no signals. For the traffic flow at cycle time ts = 15, the current increases with increasing density, saturates at point b1 , keeps a constant value until point c1 , and then decreases with increasing density. The velocity decreases slowly with increasing density, then decreases abruptly from point b1 , and there exists the cusp at point c1 . Thus, the two transitions occur at points b1 and c1 . For the traffic flow at cycle time ts = 25, the current increases with increasing density, saturates at point b2 , keeps a constant value until point c2 , and then decreases with increasing density. However, the transition point a2 appears. The velocity keeps a constant value until point a2 , then decreases abruptly with increasing density until point b2 , furthermore decreases until point c2 , and there exist the two cusps at points b2 and c2 . Thus, the three transitions occur at points a2 , b2 and c2 . For the traffic flow at cycle time ts = 45, the current and velocity display the similar behavior to that at ts = 25 and the three transitions occur at points a3 , b3 and c3 . Later, we study how the traffic state changes at the transition points. We study the dependence of mean velocity on the cycle time for various values of density. Fig. 2 shows the plot of mean velocity hvi against cycle time ts at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Open circles indicate the simulation result. The mean velocity displays the saw-tooth shape. The mean velocity takes minimum value 2 at cycle times ts = 20, 40, 60, 80. The solid line indicates the mean velocity obtained by nonlinear-map model (4) in the traffic of a single vehicle. At low density, the mean velocity of CA model (3) is consistent with that of nonlinear-map model (4) except for the neighborhoods of ts = 20, 40, 60, 80. Fig. 3 shows the plots of mean velocity hvi against cycle time ts at low densities ρ = 0.01, 0.05, 0.25, 0.5 At low densities ρ = 0.01, 0.05, the mean velocity displays the saw-tooth shape. The saw-toothed shape decays with increasing density. The curve of the mean velocity becomes smoother and smoother with increasing density. At an intermediate density, the dependence of mean velocity on the cycle time weakens. The mean velocity takes almost a constant value over the cycle time.
1676
T. Nagatani / Physica A 388 (2009) 1673–1681
Fig. 2. Plot of mean velocity hvi against cycle time ts at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Open circles indicate the simulation result. The solid line indicates the mean velocity obtained by nonlinear-map model (4).
Fig. 3. Plots of mean velocity hvi against cycle time ts at low densities ρ = 0.01, 0.05, 0.25, 0.5.
We study the dependence of the tour time on the cycle time. The tour time is defined as such a time that it takes for a vehicle to travel between a signal and its next signal. For comparison, we show the plot of tour time Dt against cycle time ts in Fig. 4(a) for the case of the single-vehicle traffic. The plot is obtained from the nonlinear-map model (4). Fig. 4(b) shows the plots of tour time Dt against cycle time ts at density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. The tour-time diagram in Fig. 4(b) agrees nearly with that in Fig. 4(a). Because the velocity takes an integer value in CA model (3), the small discontinuous segments in Fig. 4(b) occur. The dependence of tour time on the cycle time at low density is described by the nonlinear-map model (4). Fig. 4(c) shows the plots of tour time Dt against cycle time ts at density ρ = 0.1. The interaction between vehicles strengthens with increasing density. The dependence of tour time on the cycle time deviates from that of the single-vehicle traffic. Fig. 4(d) shows the plots of tour time Dt against cycle time ts at density ρ = 0.2. The tour time displays nearly the linear dependence on the cycle time. Thus, the dependence of tour time on the cycle time changes highly with increasing density. We study the variation of tour time with both cycle time and density when vehicles go ahead through the series of traffic signals. Fig. 5 shows the plots of tour time Dt against signal number n for various values of cycle time at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for cycle time ts = 15, 25, 45, 65. If all signals are green when a vehicle goes through signals, the tour time is Dt = 10. Otherwise, the tour time is higher than 10. In the case of ts = 15 of diagram (a), the vehicles always stop at all signals because the signals are always red when the vehicles go through the signals. The behavior occurs for 10 < ts < 20 in Fig. 4(b). In the case of ts = 25 of diagram (b), the vehicles stop every two signals because the vehicles meet the red every two signals. The behavior occurs for 20 < ts < 40 in Fig. 4(b). In the case of ts = 45 of diagram (c), the vehicles stop every three signals because the vehicles meet the red every three signals. The behavior occurs for 40 < ts < 60 in Fig. 4(b). In the case of ts = 65 of diagram (d), the vehicles stop every four signals because the vehicles meet the red every four signals. The behavior occurs for 60 < ts < 80 in Fig. 4(b). These behaviors agree with those obtained by the nonlinear-map model (4) because the interaction between vehicles is negligible when the traffic density is very low. Fig. 6 shows the plots of tour time Dt against signal number n at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3. At very low density, the vehicles stop every two signals as the diagram in
T. Nagatani / Physica A 388 (2009) 1673–1681
1677
Fig. 4. (a) Plot of tour time Dt against cycle time ts for the case of the single-vehicle traffic. The plot is obtained from the nonlinear-map model (4). (b)–(d) Plots of tour time Dt against cycle time ts at density ρ = 0.01, 0.1, 02 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4.
Fig. 5. Plots of tour time Dt against signal number n for various values of cycle time at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for cycle time ts = 15, 25, 45, 65.
Fig. 6(a). When the density increases, the interaction between vehicles strengthens. The tour time in Fig. 6(a) changes for the diagrams (b)–(d). The vehicles stop more times at signals with increasing density. In diagram (d) at density ρ = 0.3, the vehicles always stop at all signals. The waiting time at signals increases with density. Thus, the traffic behavior changes highly with both cycle time and density, as seen in Figs. 5 and 6.
1678
T. Nagatani / Physica A 388 (2009) 1673–1681
Fig. 6. Plots of tour time Dt against signal number n at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3.
We study the traffic pattern (vehicular trajectories) for various values of density at cycle time ts = 25. Fig. 7 shows the trajectories of all vehicles at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3. These patterns correspond respectively to those in Fig. 6(a)–(d). In diagram (a), a pair of two vehicles move together and stop every two signals. In diagram (b), a group of four vehicles move together. The last vehicle in the group stops two times every seven signals. It changes the first vehicle in the group every seven signals. In diagram (c), a group of five vehicles move together. The fourth and fifth vehicles in the group stop two times every four signals. In diagram (d), a group of eleven vehicles move together and all vehicles always stop at all signals. The stop- and go-wave is induced by stopping at the signal. The vehicles stop temporarily by the stop- and go-wave before they reach the signal. Thus, the vehicular trajectories change greatly with density. 5. Effect of a slow vehicle on traffic We study the effect of a slow vehicle on the traffic flow through a series of signals. We introduce a slow vehicle with maximal velocity vmax = 2 into the vehicular traffic with maximal velocity vmax = 4. For comparison, we present the fundamental and velocity-density diagrams for the traffic flow with no signals. Fig. 8(a) shows the plots of mean current against density where the road length is L = 400. The bold line indicates the fundamental diagram for the traffic with a slow vehicle. The solid line represents the fundamental diagram for the case with no slow vehicles. Fig. 8(b) shows the plots of mean velocity against density. By introducing a slow vehicle into the vehicular traffic, the velocity of vehicles with vmax = 4 reduces to vmax = 2 because the vehicles with vmax = 4 do not pass over the slow vehicle. The fundamental diagram for the traffic with a slow vehicle is consistent with that for the traffic in which all vehicles have maximal velocity vmax = 2. Fig. 9(a) shows the plots of mean current against density at cycle time ts = 30 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). Fig. 9(b) shows the plots of mean velocity against density at cycle time ts = 30 for the three cases. The bold line indicates the velocity for case (3). In case (3), the current and velocity agree with those for case (2) at low density. When the density is higher than 0.3, the current and velocity agree with those for case (1). Thus, the traffic is restricted by a slow vehicle at a low density, while it is independent on the slow vehicle for an intermediate density higher than 0.3. Fig. 10(a) shows the plots of mean current against density at cycle time ts = 45 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). Fig. 10(b) shows the plots of mean velocity against density at cycle time ts = 45 for the three cases. The bold line indicates the velocity for case (3). When the density is less than 0.35, the current and velocity have the middle value between cases (1) and (2). If the density is higher than 0.35, the current and velocity agree with that for case (1). Thus, the traffic flow changes with the cycle time. By introducing a slow vehicle into the traffic, the traffic flow changes highly.
T. Nagatani / Physica A 388 (2009) 1673–1681
1679
Fig. 7. Trajectories of all vehicles at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3. These patterns correspond respectively to those in Fig. 6(a)–(d).
Fig. 8. (a) Plots of mean current against density for the traffic flow with no signals where the road length is L = 400. The bold line indicates the fundamental diagram. The solid line represents the fundamental diagram for the case with no slow vehicles. (b) Plots of mean velocity against density.
We study the dependence of the velocity on the cycle time in case (3) that a slow vehicle exists in the group of fast vehicles. Fig. 11(a) shows the plots of mean velocity hvi against cycle time ts at density ρ = 0.1. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2). Except for two regions of 40 < ts < 50 and 80 < ts < 90, the mean velocity agrees with that for case (2). For two regions of 40 < ts < 50 and 80 < ts < 90, the mean velocity is higher than that for case (2). Over all regions of cycle time, the mean velocity is less than that for case (1).
1680
T. Nagatani / Physica A 388 (2009) 1673–1681
Fig. 9. (a) Plots of mean current against density at cycle time ts = 30 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). (b) Plots of mean velocity against density at cycle time ts = 30 for the three cases.
Fig. 10. (a) Plots of mean current against density at cycle time ts = 45 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). (b) Plots of mean velocity against density at cycle time ts = 45 for the three cases.
Fig. 11. (a) Plots of mean velocity hvi against cycle time ts at density ρ = 0.1. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2). (b) Plots of mean velocity hvi against cycle time ts at density ρ = 0.25. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2).
T. Nagatani / Physica A 388 (2009) 1673–1681
1681
Fig. 11(b) shows the plots of mean velocity hvi against cycle time ts at density ρ = 0.25. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2). For two regions of 27 < ts < 40 and 67 < ts < 80, the mean velocity agrees with that for case (2). Except for the two regions, the mean velocity is higher than that for case (2). Over all regions of cycle time, the mean velocity is less than that for case (1). Thus, the dependence of mean velocity on the cycle time changes highly with both cycle time and density by introducing the slow vehicle into the traffic flow of fast vehicles. 6. Summary We have presented the cellular automaton model for vehicular traffic through a series of traffic signals. The CA model has been described by the simple difference equation. We have studied the effect of both vehicular density and signals’ characteristics on dynamical behavior of vehicles by using the CA model. We have derived the fundamental diagram for the vehicular traffic. We have clarified the effect of the cycle time on the fundamental diagram. We have shown that the current-density diagram depends highly on the signals’ characteristics. We have clarified the vehicular motion in the traffic flow controlled by signals. We have shown how the vehicular motion changes with both signal characteristics and density. Also, we have shown the traffic patterns at various values of both density and cycle time. We have compared the simulation result obtained from the CA model with that of the nonlinear-map model in the dilute limit. We have shown that the traffic behavior of the CA model agrees with that of the nonlinear-map model at very low density. We have also studied the effect of a slow vehicle on the traffic through the series of signals. We have shown that the vehicular traffic changes highly by introducing a slow vehicle into the group of vehicles. The CA model proposed by this paper will be useful for the traffic flow controlled by signals, because it is described by the simple difference equation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
T. Nagatani, Rep. Progr. Phys. 65 (2002) 1331. D. Helbing, Rev. Modern. Phys. 73 (2001) 1067. D. Chowdhury, L. Santen, A. Schadscheider, Phys. Rep. 329 (2000) 199. B.S. Kerner, The Physics of Traffic, Springer, Heidelberg, 2004. D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow ’99, Springer, Heidelberg, 2000. K. Nagel, M. Schreckenberg, J. Phys. I France 2 (1992) 2221. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. E. Tomer, L. Safonov, S. Havlin, Phys. Rev. Lett. 84 (2000) 382. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.K. Lee, H.-W Lee, D. Kim, Phys. Rev. E 64 (2001) 056126. I. Lubashevsky, S. Kalenkov, R. Mahnke, Phys. Rev. E 65 (2002) 036140. I. Lubashevsky, R. Mahnke, P. Wagner, S. Kalenkov, Phys. Rev. E 66 (2002) 016117. A. Kirchner, A. Schadschneider, Physica A 312 (2002) 260. K. Nishinari, D. Chowdhury, A. Schadschneider, Phys. Rev. E 67 (2002) 036120. K. Yamamoto, S. Kokubo, K. Nishinar, Physica A 379 (2007) 654. S. Maniccam, Physica A 321 (2003) 653. O.J. O’loan, M.R. Evans, M.E. Cates, Phys. Rev. E 58 (1998) 1404. D. Chowdhury, R.C. Desai, Eur. Phys. J. B 15 (2000) 375. T. Nagatani, Phys. Rev. E 63 (2001) 036116. H.J.C. Huijberts, Physica A 308 (2002) 489. S.A. Hill, cond-mat/0206008 2002. T. Nagatani, Physica A 297 (2001) 260. T. Nagatani, Phys. Rev. E 66 (2002) 046103. T. Nagatani, Phys. Rev. E 60 (1999) 1535. L.A. Safonov, E. Tomer, V.V. Strygin, Y. Ashkenazy, S. Havlin, Chaos 12 (2002) 1006. R. Jiang, Q. Wu, Z. Zhu, Phys. Rev. E 64 (2002) 017101. H.X. Ge, S.Q. Dai, Y. Xue, L.Y. Dong, Phys. Rev. E 71 (2005) 066119. K. Nishinari, M. Fukui, A. Schadschneider, J. Phys. A 37 (2004) 3101. S. Lammer, R. Donner, D. Helbing, Eur. Phys. J. B 63 (2008) 341. W. Zhang, W. Zhang, X. Yang, Physica A 387 (2008) 4657. E. Brockfeld, R. Barlovic, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 64 (2001) 056132. M. Sasaki, T. Nagatani, Physica A 325 (2003) 531. S. Lammer, D. Helbing, J. Stat. Mech: Theory & Exp. (2008) P04019. B.A. Toledo, V. Munoz, J. Rogan, C. Tenreiro, Phys. Rev. E 70 (2004) 016107. B.A. Toledo, E. Cerda, J. Rogan, V. Munoz, C. Tenreiro, R. Zarama, J.A. Valdivia, Phys. Rev. E 75 (2007) 189701. T. Nagatani, Physica A 347 (2005) 673. T. Nagatani, Physica A 368 (2006) 560. T. Nagatani, Physica A 377 (2007) 651. T. Nagatani, Physica A 387 (2008) 1637. M. Fukui, Y. Ishibashi, J. Phys. Soc. Japan 65 (1996) 1868.
Contents lists available at ScienceDirect
Physica A journal homepage: www.elsevier.com/locate/physa
Traffic states and fundamental diagram in cellular automaton model of vehicular traffic controlled by signals Takashi Nagatani Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan
article
info
Article history: Received 30 July 2008 Received in revised form 18 September 2008 Available online 4 January 2009 PACS: 89.40.+k 05.45.-a 82.40.Bj Keywords: Traffic dynamics Cellular automaton Complex system Signal control Nonlinear map
a b s t r a c t We present a cellular automaton (CA) model for vehicular traffic controlled by traffic lights. The CA model is not described by a set of rules, but is given by a simple difference equation. The vehicular motion varies highly with both signals’ characteristics and vehicular density. The dependence of tour time on both cycle time and vehicular density is clarified. In the dilute limit of vehicles, the vehicular motion is compared with that by the nonlinearmap model. The fundamental diagrams are derived numerically. It is shown that the fundamental diagram depends highly on the signals’ characteristics. The traffic states are shown for various values of cycle time in the fundamental diagram. We also study the effect of a slow vehicle on the traffic flow. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Mobility is nowadays one of the most significant ingredients of a modern society. In urban traffic, vehicles are controlled by traffic lights to give priority for a road because the city traffic networks often exceed the capacity. Recently, transportation problems have attracted much attention in the fields of physics [1–5]. The traffic flow, pedestrian flow, and bus-route problem have been studied from a point of view of statistical mechanics and nonlinear dynamics [1–30]. Interesting dynamical behaviors have been found in the transportation system. The jams, chaos, and pattern formation are typical signatures of the complex behavior of transportation [24,25]. Brockfeld et al. have studied optimizing traffic lights for city traffic by using a CA traffic model [31]. They have clarified the effect of signal control strategy on vehicular traffic. Also, they have shown that the city traffic controlled by traffic lights can be reduced to a simpler problem of a single-lane highway. Sasaki and Nagatani have investigated the traffic flow controlled by traffic lights on a single-lane roadway by using the optimal velocity model [32]. They have derived the relationship between the road capacity and jamming transition. Until now, one has studied the periodic traffic controlled by a few traffic lights. It has been concluded that the periodic traffic does not depend on the number of traffic lights [31,32]. Very recently, Lammer and Helbing have studied the vehicular flow by the self- control of traffic signals in urban road networks [33]. In real traffic, the vehicular traffic depends highly on the configuration of traffic lights and the priority of roadways. In the dilute limit of vehicular density, a few works have been carried out for the traffic of vehicles moving through an infinite series of traffic lights with the same interval. The effect of cycle time on vehicular traffic has been clarified by using the
E-mail address: [email protected]. 0378-4371/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2008.12.046
1674
T. Nagatani / Physica A 388 (2009) 1673–1681
nonlinear-map models [34–38]. Also, it has been shown that the inhomogeneity of a signal’s interval and irregular split have important effects on vehicular traffic [39]. The nonlinear-map model is an effective tool for the vehicular traffic in the dilute limit in which there are no interactions between vehicles. However, the city traffic is affected by both signals’ characteristics and vehicular density. The interaction between vehicles changes with both signal and vehicular density. Therefore, the vehicular traffic controlled by traffic lights changes with its density. There is little known about how the dependence of tour time on the cycle time changes with vehicular density. The vehicular traffic controlled by signals will show a complex behavior. It will be necessary and important to study the vehicular traffic controlled by signals, using a simple dynamic model. In real traffic, the fundamental diagram is very important. It is meaningful to know the relationship between the fundamental diagram and cycle time. Furthermore, when the slow vehicle is introduced in city traffic, it is important to know how the slow vehicle affects the traffic flow. In this paper, we study the vehicular traffic through a series of traffic lights for various values of vehicular density. We present a deterministic cellular automaton model for the vehicular traffic controlled by signals. The CA model is described by the simple difference equation. We investigate the dynamical behavior of vehicular traffic by using the difference equation. We clarify the dynamical behavior of the vehicular traffic through a sequence of signals, by varying both cycle time and vehicular density. We show how the dependence of the tour time on the cycle time changes with the vehicular density. We compare the vehicular traffic with that obtained by the nonlinear-map model. Also, we study the effect of a slow vehicle on the traffic flow. 2. CA model and difference equation We consider the flow of vehicles going through the series of traffic lights on a one-dimensional lattice. Each vehicle does not pass over other vehicles. The traffic lights are positioned homogeneously on a roadway. The interval between signals has a constant value and is given by l. We consider the synchronized strategy for the signal control. In the synchronized strategy, all the traffic lights change simultaneously from red (green) to green (red) with a fixed time period (1 − sp )ts (sp ts ). The period of green is sp ts and the period of red is (1 − sp )ts . Time ts is called the cycle time and fraction sp represents the split which indicates the ratio of green time to cycle time. We set split as sp = 0.5. We extend the deterministic CA model proposed by Fukui and Ishibashi [1,28,40] to take into account traffic lights. We define the position of vehicle i at time t as xi (t ) where x, i, and t are an integer. The CA model of Fukui and Ishibashi is given by xi (t + 1) = min[xi (t ) + vmax , xi+1 (t ) − 1],
(1)
where vmax is the maximum velocity and an integer. Here, min[A, B] is a minimum function and takes the minimum value within A and B. The velocity takes the integer value ranging from 0 to vmax . The velocity depends on the headway. If headway ∆xi (t ) (=xi+1 (t )− xi (t )) is larger than the maximum velocity, the vehicle moves with the maximum velocity. If the headway is less than the maximum velocity, the vehicle moves with velocity ∆xi (t ) − 1. When a vehicle arrives at a traffic light and the traffic light is red, the vehicle stops at the position of the traffic light. Then, when the traffic light changes from red to green, the vehicle goes ahead. On the other hand, when a vehicle arrives at a traffic light and the traffic light is green, the vehicle does not stop and goes ahead without changing speed. The position of the closest signal before vehicle i at time t is given by xi,s (t ) =
int
xi ( t ) l
+ 1 l.
(2)
Then, the position of vehicle i at time t + 1 is given by xi (t + 1) = min[xi (t ) + vmax , xi+1 (t ) − 1]{1 − ϑ(sin(2π t /ts ))}
+ min[xi (t ) + vmax , xi+1 (t ) − 1, xi,s (t ) − 1]ϑ(sin(2π t /ts )).
(3)
Here, ϑ(t ) is the step function. It takes one if t > 0 and zero if t ≤ 0. When there are no signals, Eq. (3) reduces to Eq. (1). If the signal just before vehicle i is green, ϑ(sin(2π t /ts )) = 0 and xi (t +1) = min[xi (t )+vmax , xi+1 (t )−1]. Eq. (3) reduces to Eq. (1). Otherwise, if the signal just before vehicle i is red, ϑ(sin(2π t /ts )) = 1 and xi (t +1) = min[xi (t )+vmax , xi+1 (t )−1, xi,s (t )−1]. Then, if the headway is larger than vmax , vehicle i stops at site xi,s (t ) − 1 just before the signal. Also, if xi+1 (t ) is higher than xi,s (t ), vehicle i stops at site xi,s (t ) − 1 just before the signal. Thus, Eq. (3) presents the CA model for the vehicular traffic through a series of traffic lights. Eq. (3) is a single difference equation. Until now, the CA model for the signal traffic has been described by a set of CA rules. However, model (3) is of great advantage to simulate because the difference equation is simple. It will be expected that the vehicular traffic exhibits a complex behavior. We study how the vehicular traffic changes by varying both cycle time and vehicular density. 3. Nonlinear-map model In the dilute limit of vehicular density, the interaction between vehicles is negligible. Vehicles do not interfere with each other. Then, it is sufficient only to consider a motion of a single vehicle through a series of traffic lights. Here, we consider the
T. Nagatani / Physica A 388 (2009) 1673–1681
1675
Fig. 1. (a) Plots of mean current against density for ts = 15, 25, 45 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. For comparison, the fundamental diagram is shown for the vehicular traffic with no signals. (b) Plots of mean velocity against density.
motion of a single vehicle through the series of signals. The vehicular motion has been described in terms of the nonlinearmap model [36–39]. For later convenience, we summarize the formulation. We define the arrival time of the vehicle at traffic light n as t (n). The arrival time at traffic light n + 1 is given by t (n + 1) = t (n) + l/v + (r (n) − t (n)) H (t (n) − (int((t (n))/ts )ts ) − ts /2)
(4)
with r (n) = (int ((t (n))/ts ) + 1) · ts , where H (t ) is the Heaviside function: H (t ) = 1 for t ≥ 0 and H (t ) = 0 for t < 0. H (t ) = 1 if the traffic light is red, while H (t ) = 0 if the traffic light is green. l/v is the time it takes for the vehicle to move between lights n and n + 1. r (n) is such time that the traffic light just changed from red to green. The third term on the right hand side of Eq. (4) represents such time that the vehicle stops if traffic light n is red. The number n of iteration increases one by one when the vehicle moves through the traffic light. The iteration corresponds to the going ahead on the highway. We will compare the nonlinear-map model (4) with the CA model (3) at the dilute limit of vehicular density in the simulation. 4. Simulation result We investigate the fundamental diagram and traffic states by using Eq. (3). We calculate the current (flow) and velocity for various values of cycle time ts . Fig. 1(a) shows the plots of mean current against density for ts = 15, 25, 45 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. For comparison, the fundamental diagram is shown for the vehicular traffic with no signals. Fig. 1(b) shows the plots of mean velocity against density. For the traffic flow with no signals, the mean velocity keeps the maximal value until ρ = 1/(vmax + 1) = 0.2 and then decreases with increasing density. The mean current increases linearly with density, reaches the maximal current at ρ = 0.2, and then decreases with increasing density. For the traffic flow controlled by signals, the current and velocity curves change greatly from those of no signals. For the traffic flow at cycle time ts = 15, the current increases with increasing density, saturates at point b1 , keeps a constant value until point c1 , and then decreases with increasing density. The velocity decreases slowly with increasing density, then decreases abruptly from point b1 , and there exists the cusp at point c1 . Thus, the two transitions occur at points b1 and c1 . For the traffic flow at cycle time ts = 25, the current increases with increasing density, saturates at point b2 , keeps a constant value until point c2 , and then decreases with increasing density. However, the transition point a2 appears. The velocity keeps a constant value until point a2 , then decreases abruptly with increasing density until point b2 , furthermore decreases until point c2 , and there exist the two cusps at points b2 and c2 . Thus, the three transitions occur at points a2 , b2 and c2 . For the traffic flow at cycle time ts = 45, the current and velocity display the similar behavior to that at ts = 25 and the three transitions occur at points a3 , b3 and c3 . Later, we study how the traffic state changes at the transition points. We study the dependence of mean velocity on the cycle time for various values of density. Fig. 2 shows the plot of mean velocity hvi against cycle time ts at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Open circles indicate the simulation result. The mean velocity displays the saw-tooth shape. The mean velocity takes minimum value 2 at cycle times ts = 20, 40, 60, 80. The solid line indicates the mean velocity obtained by nonlinear-map model (4) in the traffic of a single vehicle. At low density, the mean velocity of CA model (3) is consistent with that of nonlinear-map model (4) except for the neighborhoods of ts = 20, 40, 60, 80. Fig. 3 shows the plots of mean velocity hvi against cycle time ts at low densities ρ = 0.01, 0.05, 0.25, 0.5 At low densities ρ = 0.01, 0.05, the mean velocity displays the saw-tooth shape. The saw-toothed shape decays with increasing density. The curve of the mean velocity becomes smoother and smoother with increasing density. At an intermediate density, the dependence of mean velocity on the cycle time weakens. The mean velocity takes almost a constant value over the cycle time.
1676
T. Nagatani / Physica A 388 (2009) 1673–1681
Fig. 2. Plot of mean velocity hvi against cycle time ts at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Open circles indicate the simulation result. The solid line indicates the mean velocity obtained by nonlinear-map model (4).
Fig. 3. Plots of mean velocity hvi against cycle time ts at low densities ρ = 0.01, 0.05, 0.25, 0.5.
We study the dependence of the tour time on the cycle time. The tour time is defined as such a time that it takes for a vehicle to travel between a signal and its next signal. For comparison, we show the plot of tour time Dt against cycle time ts in Fig. 4(a) for the case of the single-vehicle traffic. The plot is obtained from the nonlinear-map model (4). Fig. 4(b) shows the plots of tour time Dt against cycle time ts at density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. The tour-time diagram in Fig. 4(b) agrees nearly with that in Fig. 4(a). Because the velocity takes an integer value in CA model (3), the small discontinuous segments in Fig. 4(b) occur. The dependence of tour time on the cycle time at low density is described by the nonlinear-map model (4). Fig. 4(c) shows the plots of tour time Dt against cycle time ts at density ρ = 0.1. The interaction between vehicles strengthens with increasing density. The dependence of tour time on the cycle time deviates from that of the single-vehicle traffic. Fig. 4(d) shows the plots of tour time Dt against cycle time ts at density ρ = 0.2. The tour time displays nearly the linear dependence on the cycle time. Thus, the dependence of tour time on the cycle time changes highly with increasing density. We study the variation of tour time with both cycle time and density when vehicles go ahead through the series of traffic signals. Fig. 5 shows the plots of tour time Dt against signal number n for various values of cycle time at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for cycle time ts = 15, 25, 45, 65. If all signals are green when a vehicle goes through signals, the tour time is Dt = 10. Otherwise, the tour time is higher than 10. In the case of ts = 15 of diagram (a), the vehicles always stop at all signals because the signals are always red when the vehicles go through the signals. The behavior occurs for 10 < ts < 20 in Fig. 4(b). In the case of ts = 25 of diagram (b), the vehicles stop every two signals because the vehicles meet the red every two signals. The behavior occurs for 20 < ts < 40 in Fig. 4(b). In the case of ts = 45 of diagram (c), the vehicles stop every three signals because the vehicles meet the red every three signals. The behavior occurs for 40 < ts < 60 in Fig. 4(b). In the case of ts = 65 of diagram (d), the vehicles stop every four signals because the vehicles meet the red every four signals. The behavior occurs for 60 < ts < 80 in Fig. 4(b). These behaviors agree with those obtained by the nonlinear-map model (4) because the interaction between vehicles is negligible when the traffic density is very low. Fig. 6 shows the plots of tour time Dt against signal number n at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3. At very low density, the vehicles stop every two signals as the diagram in
T. Nagatani / Physica A 388 (2009) 1673–1681
1677
Fig. 4. (a) Plot of tour time Dt against cycle time ts for the case of the single-vehicle traffic. The plot is obtained from the nonlinear-map model (4). (b)–(d) Plots of tour time Dt against cycle time ts at density ρ = 0.01, 0.1, 02 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4.
Fig. 5. Plots of tour time Dt against signal number n for various values of cycle time at low density ρ = 0.01 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for cycle time ts = 15, 25, 45, 65.
Fig. 6(a). When the density increases, the interaction between vehicles strengthens. The tour time in Fig. 6(a) changes for the diagrams (b)–(d). The vehicles stop more times at signals with increasing density. In diagram (d) at density ρ = 0.3, the vehicles always stop at all signals. The waiting time at signals increases with density. Thus, the traffic behavior changes highly with both cycle time and density, as seen in Figs. 5 and 6.
1678
T. Nagatani / Physica A 388 (2009) 1673–1681
Fig. 6. Plots of tour time Dt against signal number n at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3.
We study the traffic pattern (vehicular trajectories) for various values of density at cycle time ts = 25. Fig. 7 shows the trajectories of all vehicles at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3. These patterns correspond respectively to those in Fig. 6(a)–(d). In diagram (a), a pair of two vehicles move together and stop every two signals. In diagram (b), a group of four vehicles move together. The last vehicle in the group stops two times every seven signals. It changes the first vehicle in the group every seven signals. In diagram (c), a group of five vehicles move together. The fourth and fifth vehicles in the group stop two times every four signals. In diagram (d), a group of eleven vehicles move together and all vehicles always stop at all signals. The stop- and go-wave is induced by stopping at the signal. The vehicles stop temporarily by the stop- and go-wave before they reach the signal. Thus, the vehicular trajectories change greatly with density. 5. Effect of a slow vehicle on traffic We study the effect of a slow vehicle on the traffic flow through a series of signals. We introduce a slow vehicle with maximal velocity vmax = 2 into the vehicular traffic with maximal velocity vmax = 4. For comparison, we present the fundamental and velocity-density diagrams for the traffic flow with no signals. Fig. 8(a) shows the plots of mean current against density where the road length is L = 400. The bold line indicates the fundamental diagram for the traffic with a slow vehicle. The solid line represents the fundamental diagram for the case with no slow vehicles. Fig. 8(b) shows the plots of mean velocity against density. By introducing a slow vehicle into the vehicular traffic, the velocity of vehicles with vmax = 4 reduces to vmax = 2 because the vehicles with vmax = 4 do not pass over the slow vehicle. The fundamental diagram for the traffic with a slow vehicle is consistent with that for the traffic in which all vehicles have maximal velocity vmax = 2. Fig. 9(a) shows the plots of mean current against density at cycle time ts = 30 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). Fig. 9(b) shows the plots of mean velocity against density at cycle time ts = 30 for the three cases. The bold line indicates the velocity for case (3). In case (3), the current and velocity agree with those for case (2) at low density. When the density is higher than 0.3, the current and velocity agree with those for case (1). Thus, the traffic is restricted by a slow vehicle at a low density, while it is independent on the slow vehicle for an intermediate density higher than 0.3. Fig. 10(a) shows the plots of mean current against density at cycle time ts = 45 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). Fig. 10(b) shows the plots of mean velocity against density at cycle time ts = 45 for the three cases. The bold line indicates the velocity for case (3). When the density is less than 0.35, the current and velocity have the middle value between cases (1) and (2). If the density is higher than 0.35, the current and velocity agree with that for case (1). Thus, the traffic flow changes with the cycle time. By introducing a slow vehicle into the traffic, the traffic flow changes highly.
T. Nagatani / Physica A 388 (2009) 1673–1681
1679
Fig. 7. Trajectories of all vehicles at various values of density for cycle time ts = 25 where the road length is L = 400, the interval between signals is l = 40, and the maximal velocity is vmax = 4. Diagrams (a)–(d) are obtained for density ρ = 0.05, 0.1, 0.15, 0.3. These patterns correspond respectively to those in Fig. 6(a)–(d).
Fig. 8. (a) Plots of mean current against density for the traffic flow with no signals where the road length is L = 400. The bold line indicates the fundamental diagram. The solid line represents the fundamental diagram for the case with no slow vehicles. (b) Plots of mean velocity against density.
We study the dependence of the velocity on the cycle time in case (3) that a slow vehicle exists in the group of fast vehicles. Fig. 11(a) shows the plots of mean velocity hvi against cycle time ts at density ρ = 0.1. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2). Except for two regions of 40 < ts < 50 and 80 < ts < 90, the mean velocity agrees with that for case (2). For two regions of 40 < ts < 50 and 80 < ts < 90, the mean velocity is higher than that for case (2). Over all regions of cycle time, the mean velocity is less than that for case (1).
1680
T. Nagatani / Physica A 388 (2009) 1673–1681
Fig. 9. (a) Plots of mean current against density at cycle time ts = 30 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). (b) Plots of mean velocity against density at cycle time ts = 30 for the three cases.
Fig. 10. (a) Plots of mean current against density at cycle time ts = 45 for the three cases: (1) all fast vehicles with vmax = 4, (2) all slow vehicles with vmax = 2, and (3) a slow vehicle with vmax = 2 and fast vehicles with vmax = 4 where the road length is L = 400. The bold line indicates the current for case (3). (b) Plots of mean velocity against density at cycle time ts = 45 for the three cases.
Fig. 11. (a) Plots of mean velocity hvi against cycle time ts at density ρ = 0.1. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2). (b) Plots of mean velocity hvi against cycle time ts at density ρ = 0.25. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2).
T. Nagatani / Physica A 388 (2009) 1673–1681
1681
Fig. 11(b) shows the plots of mean velocity hvi against cycle time ts at density ρ = 0.25. The open circles indicate the simulation result. The upper solid line represents the mean velocity for case (1). The lower solid line indicates the mean velocity for case (2). For two regions of 27 < ts < 40 and 67 < ts < 80, the mean velocity agrees with that for case (2). Except for the two regions, the mean velocity is higher than that for case (2). Over all regions of cycle time, the mean velocity is less than that for case (1). Thus, the dependence of mean velocity on the cycle time changes highly with both cycle time and density by introducing the slow vehicle into the traffic flow of fast vehicles. 6. Summary We have presented the cellular automaton model for vehicular traffic through a series of traffic signals. The CA model has been described by the simple difference equation. We have studied the effect of both vehicular density and signals’ characteristics on dynamical behavior of vehicles by using the CA model. We have derived the fundamental diagram for the vehicular traffic. We have clarified the effect of the cycle time on the fundamental diagram. We have shown that the current-density diagram depends highly on the signals’ characteristics. We have clarified the vehicular motion in the traffic flow controlled by signals. We have shown how the vehicular motion changes with both signal characteristics and density. Also, we have shown the traffic patterns at various values of both density and cycle time. We have compared the simulation result obtained from the CA model with that of the nonlinear-map model in the dilute limit. We have shown that the traffic behavior of the CA model agrees with that of the nonlinear-map model at very low density. We have also studied the effect of a slow vehicle on the traffic through the series of signals. We have shown that the vehicular traffic changes highly by introducing a slow vehicle into the group of vehicles. The CA model proposed by this paper will be useful for the traffic flow controlled by signals, because it is described by the simple difference equation. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]
T. Nagatani, Rep. Progr. Phys. 65 (2002) 1331. D. Helbing, Rev. Modern. Phys. 73 (2001) 1067. D. Chowdhury, L. Santen, A. Schadscheider, Phys. Rep. 329 (2000) 199. B.S. Kerner, The Physics of Traffic, Springer, Heidelberg, 2004. D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow ’99, Springer, Heidelberg, 2000. K. Nagel, M. Schreckenberg, J. Phys. I France 2 (1992) 2221. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. E. Tomer, L. Safonov, S. Havlin, Phys. Rev. Lett. 84 (2000) 382. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.K. Lee, H.-W Lee, D. Kim, Phys. Rev. E 64 (2001) 056126. I. Lubashevsky, S. Kalenkov, R. Mahnke, Phys. Rev. E 65 (2002) 036140. I. Lubashevsky, R. Mahnke, P. Wagner, S. Kalenkov, Phys. Rev. E 66 (2002) 016117. A. Kirchner, A. Schadschneider, Physica A 312 (2002) 260. K. Nishinari, D. Chowdhury, A. Schadschneider, Phys. Rev. E 67 (2002) 036120. K. Yamamoto, S. Kokubo, K. Nishinar, Physica A 379 (2007) 654. S. Maniccam, Physica A 321 (2003) 653. O.J. O’loan, M.R. Evans, M.E. Cates, Phys. Rev. E 58 (1998) 1404. D. Chowdhury, R.C. Desai, Eur. Phys. J. B 15 (2000) 375. T. Nagatani, Phys. Rev. E 63 (2001) 036116. H.J.C. Huijberts, Physica A 308 (2002) 489. S.A. Hill, cond-mat/0206008 2002. T. Nagatani, Physica A 297 (2001) 260. T. Nagatani, Phys. Rev. E 66 (2002) 046103. T. Nagatani, Phys. Rev. E 60 (1999) 1535. L.A. Safonov, E. Tomer, V.V. Strygin, Y. Ashkenazy, S. Havlin, Chaos 12 (2002) 1006. R. Jiang, Q. Wu, Z. Zhu, Phys. Rev. E 64 (2002) 017101. H.X. Ge, S.Q. Dai, Y. Xue, L.Y. Dong, Phys. Rev. E 71 (2005) 066119. K. Nishinari, M. Fukui, A. Schadschneider, J. Phys. A 37 (2004) 3101. S. Lammer, R. Donner, D. Helbing, Eur. Phys. J. B 63 (2008) 341. W. Zhang, W. Zhang, X. Yang, Physica A 387 (2008) 4657. E. Brockfeld, R. Barlovic, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 64 (2001) 056132. M. Sasaki, T. Nagatani, Physica A 325 (2003) 531. S. Lammer, D. Helbing, J. Stat. Mech: Theory & Exp. (2008) P04019. B.A. Toledo, V. Munoz, J. Rogan, C. Tenreiro, Phys. Rev. E 70 (2004) 016107. B.A. Toledo, E. Cerda, J. Rogan, V. Munoz, C. Tenreiro, R. Zarama, J.A. Valdivia, Phys. Rev. E 75 (2007) 189701. T. Nagatani, Physica A 347 (2005) 673. T. Nagatani, Physica A 368 (2006) 560. T. Nagatani, Physica A 377 (2007) 651. T. Nagatani, Physica A 387 (2008) 1637. M. Fukui, Y. Ishibashi, J. Phys. Soc. Japan 65 (1996) 1868.