A force model for single-line traffic. Abstract

Transcription

1 A force model for single-line traffic Li Ge 1, Qixi Mi 1, Rui Li 1, and J.W. Zhang 1,2 1 School of Physics, Peking University, Beijing 1871, P.R. China 2 Key Laboratory of Quantum Information and Quantum Measurements, Ministry of Education, P.R. China (Dated: October 14, 23) Abstract Based on the assumption that a vehicle always tries to keep a certain speed and a proper distance from the one in front of it, a force model, being a kind of microscopic models which is similar to van der Waal s molecule model, is established. We develop three approaches to study the equilibrium state and the phase transition of traffic: the two-vehicle model in which the traffic is simplified into a leading vehicle and its follower, the numerical simulation in which we study the initially uniformly distributed vehicles, and the corresponding macroscopic model. It is shown that the trajectories of phase transitions in the last two approaches are consistent with each other and all end at the same equilibrium states we anticipate from the results of the first one. At the end of the paper, we point out that some phenomena can only be explained by the microscopic model. PACS numbers: 5.9.+m, i, a Electronic address: james@pku.edu.cn 1

2 I. INTRODUCTION The scientific researches of traffic problems have been carried out for quite a long time. According to our knowledge, since the early 3 s of the last century [1], a considerable volume of articles have been published and several models have been developed to study the relation between traffic flow and vehicle density, as well as the stability of traffic and nonlinear dynamical phenomena, such as the formation of traffic jam [2], stop-and-go traffic [3] and synchronized congested traffic [4, 5]. The main approaches to investigate the traffic problems include the microscopic car-following models, the macroscopic fluid models, the statistic models and the recently developed cellular automata models. Generally speaking, the traffic flow is a kind of many-body systems of strongly interacting vehicles. This characteristic makes the microscopic model a competent candidate to solve traffic problems. The microscopic car-following model has the longest history and has been studied the most among all the approaches we mentioned above. Starting from 195 s [6], the microscopic approach keeps active in the frontier and large numbers of publications appear in a wide range of academic fields. Recently, many new models have been suggested and revised [7 18], which indicate the coming of another heyday of microscopic traffic approaches. In the microscopic car-following model, the basic assumption is that vehicle j is only affected by the vehicle j + 1 ahead of it, which is called the leading vehicle, and all the factors which affect the traffic flow, such as the speed and capability of the vehicle, the relative position, personal driving habits and proficiency, the road and weather condition, can be simplified and condensed into a few cardinal arguments which capture the main characteristics of the traffic dynamics: (1) the headway d j = (x j+1 x j ) or the clearance s j = (d j l j ), where l j stands for the length of the vehicle j; (2) the speed of the front vehicle υ j+1, or alternatively, the optimal speed υ (see the next section); (3) the individual speed fluctuation ξ j. In Sect. II we present our force model integrities. We further analyze the traffic flow in the phase space in Sect. III by simplifying it into the two-vehicle model and investigating the behaviors of initially uniformly distributed vehicles by numerical simulations. At the end of the paper, in Sect. IV we develop the corresponding macroscopic fluid model and compare the outcomes with the results we obtain in Sect. III. 2

3 II. MODEL To keep the problem analytical and facilitate our later comparison with the corresponding macroscopic model, we set forth two more assumptions before our discussion: First, all the vehicles are identical. Secondly, the traffic flow is of one dimension, i.e., the road has only one entrance and one exit and all the vehicles are moving in the same direction. Behaviors such as overtaking, backing and lane-shifting are neglected. Based on the above assumptions, the behavior of a vehicle can be explained by the following two factors: (1) To keep speed When the headway is quite large, the hurries in our lives urge a driver to drive faster, but at the same time the lawful or psychological conditions limit the speed within a range. These two factors work together and cause the car to reach a limited high speed υ, which we take reasonably as 15 m s 1 in the following discussion. In this way, we have the following equation: a(υ) = a (1 υ υ ), (1) where a and υ stand for the acceleration and the speed, respectively, and a is the acceleration when the vehicle starts moving, which we take as 2 m s 2. If a is only determined by υ, however, the model is bound to be unsound, because it fails to explain the empirically observed density waves [19]. Therefore, we have to take other factors into consideration. (2) To keep distance Despite a driver s specific driving habit and proficiency, most drivers try to keep a optimal distance d opt from the vehicle in front. Here we define d opt = d + k υ 2 in which d and k are two parameters which we take as 9 m and.1 m 1 s 2. The conception of keeping a certain distance between two individuals is similar to that in the molecule dynamics, thus we construct a potential function by simulating Lennard-Jones potential with different exponentials: U(d) 1 4 (d opt d )4 1 2 (d opt d )2. (2) The difference here is that Newton s third law does not necessarily apply any more here, for the force is passed from the front vehicle to its follower but not vice versa. 3

4 The equivalent expression for acceleration is a(d) = k d [ (d opt d )4 + ( d opt d )2 ], (3) in which d is the actual distance between vehicle j and j + 1. Here we take the coefficient k as 5 m 2 s 2. By combining the two factors above and replacing d with the effective density ρ, which is the reciprocal of d, we obtain the complete form of force model as follows a(ρ, υ) = kρ [ (ρ d opt ) 4 + (ρ d opt ) 2 ] + a (1 υ υ ). (4) Here we would like to point out in advance that in a special state, Eq. (4) is not applicable and another rule takes it place. We will explain the reason in the next section. III. ANALYSIS OF TRAFFIC FLOW IN THE PHASE SPACE The two-dimension space with ρ as the abscissa and υ as the ordinate are called the phase space, for these two arguments are the most important elements in our discussion. According to the sign of a(ρ, v), we divided the phase space into two regions: the deceleration region which lies above the equilibrium line a(ρ, v) = and the acceleration region below it. Fig.1 is the contour map for a = 8, 4, 2,, 1, 1.5, 2, 2.5 m s 2, respectively. From it we can see that the max acceleration lies in the line υ = and computation shows that a max = 3.33 m s 2 with ρ =.86 m 1, which is consistent with the common capability of a vehicle on a city street. As for the deceleration, we take its limits as 8 m s 2. In an equilibrium state we have dv dt = a(ρ, v) =, (5) and dρ dt = ρ 2 (v v pre ) =, (6) where υ pre is the speed of the vehicle in front. From Eq. (6) we know that the equilibrium state can only be reached when ρ = or υ = υ pre. In the first case we have υ = υ which reflects the meaning of υ, and we call it the trivial equilibrium solution. In the second case, by substituting υ pre for υ in Eq. (4), we have ω 5 ω 3 c =. (7) 4

5 ρ* / m 1 Fig. 1: The contour map for a = 8, 4, 2,, 1, 1.5, 2, 2.5 m s 2 in the phase space. Here ω and c are defined as follows, respectively, ω = ρ d opt, (8) and c = a k (1 v pre v )(d + k v 2 pre). (9) A. Analysis of the motion equation and the two-vehicle model To reveal the characters lies behind Eq. (7), we introduce the two-vehicle model. In this model, the traffic is simplified into two vehicles, the first one travels at speed υ pre and the following one behaviors according to Eq. (4). υ pre is a very complex variable which has great effect on the following vehicle s trajectory in the phase space. For the sake of keeping the problem analytical, we take υ pre as a constant parameter. In the following subsection, we will present the result of simulation in a case in which v pre is not constant. If we let f(ω) = ω 5 and g(ω) = ω 3 + c, it is not difficult to see that g(ω) ascends more rapidly than f(ω) as ω becomes larger in the range [, 1], while the case is vice versa when ω is larger than 1. In order to study the versatile dynamics of the model, we suppose that υ pre has a wider value range and can go beyond υ in some cases. Also keep in mind that a, k, d, and k are all non-negative defined. (1) υ pre = υ 5

6 2 18 Equilibrium line ρ* / m 1 Fig. 2: Trajectories from different initial states with υ pre = υ. All the initial states will eventually reach the non-trivial equilibrium state except those lie in the narrow area along υ axis and below the equilibrium line. The dashed line υ = υ is intended to guide the eyes. In this case c = and f(ω) has two intersections with g(ω) at ω =, 1, which indicate ρ = and ρ = ρ opt, respectively. Here ρ opt is the reciprocal of d opt. Fig. 2 shows the trajectories of phase transitions from different initial states. If we combine them with the direction field analysis, we can see that all the initial states will eventually reach the nontrivial equilibrium state except for those lie in the narrow area along υ axis and below the equilibrium line. In fact, it is too narrow to be distinguished from the υ axis. Our simulations have shown that if the initial state of the following vehicle is 1 m s 1, ρ must be smaller than.6 m 1 to bring the trajectory to the trivial equilibrium state. (2) υ pre > υ Computation shows that when c equals c cri = 2( 3) 3 2, the curves of f(ω) and g(ω) are 5 5 tangential to each other at ω = ( 3) 1 2. If we express it by ρ, then we have ρ = ( 3) 1 2 ρ 5 5 opt, which corresponds to the summit of the curve a = in the phase space. From Fig. 3 we can see that this equilibrium state can only be reached when the initial state is in the shadowed area and all the other trajectories converge at the trivial equilibrium state. If c cri < c <, two equilibrium states exit and we have ρ < ρ opt in both states. In Fig. 4 we take υ pre = m s 1 and we find that the equilibrium state (a) is metastable since only the states on the thickened line can reach it and even a small disturbance can lead the vehicle to the trivial equilibrium state or the equilibrium state (b). 6

7 2 18 Equilibrium line ρ* / m 1 Fig. 3: Trajectories from different initial states with c = c cri. All the trajectories converged at the trivial equilibrium state except those lies in the shadowed area. The dashed line υ = υ max is intended to guide the eyes Equilibrium line (a) (b) ρ* / m 1 Fig. 4: Trajectories from different initial states with υ pre = m s 1 at which c cri < c <. The equilibrium state (a) is metastable since only the states on the thickened line can reach it and even a small disturbance will lead the vehicle to the trivial equilibrium state or the equilibrium state (b).the dashed line υ = m s 1 is intended to guide the eyes. There is no equilibrium solution for the two-vehicle model when c < c cri except for the trivial one. (3) υ pre < υ In the above two cases, we either have ρ < ρ opt or ρ = ρ opt, which means the traffic 7

8 5 Equilibrium line ρ* / m 1 Fig. 5: Trajectory from ρ =.9 and υ = with υ pre = 1. m s 1. After the initial acceleratingdecelerating process, it reaches ρ axis and moves along it until it reenters the acceleration region and cycles inward to approach the equilibrium state.the dashed line υ = 1. m s 1 is intended to guide the eyes. is operating in a safe state. But when υ pre < υ, we have c > and there is only one equilibrium solution at which ρ > ρ opt. In this case, if the disturbance goes beyond a certain limit, a jam state or even collision is prone to happen. Note that this conclusion is determined by the form of Eq. (4) and independent from the values of a, k, d, and k as long as they are non-negative. But their values do affect the safety level. If we divide the phase space into more regions according to the sign of dρ /dt, we can see that both dρ /dt and dυ/dt are negative in the lowest area on the right. That is to say the trajectory in this region has the possibility to reach the ρ axis, and this is true for all the cases we mentioned above. But the trajectory will not terminate at the ρ axis except in the case that υ pre =, which will definitely cause a dead jam. As we mentioned in the preceding section, Eq. (4) is not applicable here and it is logic to suppose that the following driver will wait until the distance between the vehicle and the one in front of it becomes larger enough that it reenters the accelerate region to resume the driving. In Fig. 6 we show a typical trajectory with υ pre = 1 m s 1 which moves along the ρ axis until it reaches the equilibrium line and cycles inward to approach the equilibrium state. 8

9 B. Simulations of phase transition in different conditions In the above subsection, we analyze the characters of the motion equation and present the equilibrium solutions in all conditions. But all of these are based on the two-vehicle model, whose validity and whether it can reveal all the dynamics in the force model are still questionable. In this subsection, we will display the outcomes of several simulations in different initial conditions and compare it with the results we have got in the previous subsection. Our first object is the behaviors of the vehicles at the traffic light after it turns green. Here we suppose that ρ =.2 m 1 and the queue of vehicles extends 1 m along the road. We call the last vehicle in the queue vehicle No.1 and the vehicles in front of it vehicle No. 2, No. 3,..., and so on. The first vehicle is assigned a steady acceleration of 2 m s 2 until it reaches υ at which it keeps from then on. The following vehicles start moving one by one when d j = 6 m and behavior according to Eq. (4). Simulation shows that it takes the last vehicle nearly 35 seconds to get started and another 12 seconds to cross the intersection of 3 meters long. Allowing 1 seconds for the free traffic hehind the waiting queue to cross the intersection, the appropriate time for green light is about 6 seconds. Undoubtedly, this depends on the virtual length of the queue, the extent of the crossroad and some other factors, but averagely speaking, the conclusion from the simulation is consistent with the real condition. Another item we concern in this problem is how to determine the spatial interval between two consecutive traffic lights. Based on the above assumptions, we can recur to Fig. 6 for the solution. If one doesn t inspect closely, he may think that the traffic flow has reached the equilibrium state when t is large than 1 s, but the truth is that the following vehicles are a little faster than the first one. The small variance cause a vibration which begins approximately five minutes after the traffic starts moving and lasts for about 8 s. Furthermore, the aptitude and duration of the vibration keep growing if there are more vehicles in the traffic flow. In order to avoid this disadvantage, we can let the first vehicle stops before this vibration begin and a reasonable way to achieve this is setting another traffic light about 4.5 km from the first one, which is the distance the first vehicle have travelled before then. Simulations show that this will not cause extra vibration in the course of deceleration. If we take on-ramps into consideration, further research is needed to 9

10 t / s Fig. 6: The velocity-time plot of the vehicles at the traffic light after it turns green. A vibration appears approximately at t = 3 s and lasts for about 8 s. For the reason of clarity, we only plot the velocity lines of odd ordinal number vehicles. solve the problem in a specific situation, but the conclusion that 4.5 km is the upper limit for the interval between two traffic lights is tenable in most cases. We use the mean values of ρ and υ to represent the phase and show the phase transition in Fig. 11. The trajectory is similar to the one in Fig. 2 and terminates at the equilibrium state we expected, but here the area above the equilibrium line no longer strictly represents the decelerate region due to the average effect. Next, we focus on a section of traffic flow which extends 5 m. We suppose that the vehicles are initially uniformly distributed with non-zero equal speed υ ini. (1) υ ini < υ In this case, we let ρ =.5m 1 and υ ini = 1 m s 1. Surely this is not a equilibrium state and from Fig. 7 we can see that a congest area appears immediately after the simulation begins and it evolves into a stop-and-go wave which lasts about 2 s. After that, a density wave forms and propagates approximately at the speed of 18 kph, which is within the observed range of 15 ± 5 kph [19]. Fig. 12 shows the phase transition and we can see that the trajectory nearly overlaps the equilibrium line for some time, then it moves down and circles to approach the equilibrium point we expected in the previous subsection. Fig. 8 shows the spatial-temporal plot in which v pre vibrates with an aptitude of 2 m s 1 and a period of 2 s. We can see that from the second period on, the formations of the tracks in each period are very similar to that in Fig. 7 and the period of the stop-and-go waves are 1

11 15 space / m time / s Fig. 7: The space-time plot of uniformly distributed vehicles with the same initially velocity of υini = 1 m s 1 and ρ? =.5 m 1. A congest area appears immediately after the simulation begins and it evolves into a stop-and-go wave which lasts about 2 s. After that, a density wave forms and propagates approximately at the speed of 18 kph, which is within the observed range of 15 ± 5 kph. 25 space / m time / s Fig. 8: The space-time plot of uniformly distributed vehicles with the same initially velocity of υini = 1 m s 1, and ρ? =.5 m 1. Different from Fig. 7, here υpre vibrates around 1 m s 1 with an aptitude of 2 m s 1 and a period of 4 s. From the plot we can see that from the second period on, the formation of the tracks in each period is very similar to that in Fig. 7 and the period of the stop-and-go waves are also 2 s. 11

12 t / s Fig. 9: The time-velocity plot of uniformly distributed vehicles with the same initial velocity of υ ini = m s 1 and ρ =.25 m s 1. The traffic flow evolves into two parts automatically, with the front four vehicles constituting a group which travels at the uniform speed of υ ini = m s 1, and the following nine vehicles another group travelling at υ. also 2 s. (2) υ ini = υ The value of ρ in the equilibrium state is very close to.5 m 1, so we take υ ini =.4 m 1 instead of.5 m 1 to see the phase transition more clearly. From Fig. 13 we can see that the trajectory is more smooth that the previous one and takes fewer time to reach the equilibrium state which is the same in Fig. 2. (3) υ ini > υ max In the two-vehicle model, as we have pointed out in Sect. III.A, there is only one equilibrium state in this case the trivial equilibrium state. But this is not to say that no stable solution exist for the real traffic. In fact, we can expect the distance between the first two vehicles keeps growing and the velocity of the second one approaches υ little by little. At the same time, as the effect the first vehicle exerts on the second one diminishes, the second vehicle becomes to function as the real leading vehicle in the traffic. Thus we can expect the trajectory (the first two vehicles excluded) to terminate at the equilibrium state at which υ = 15 m s 1 and Fig. 14 proves this thought. (4) υ < υ ini < υ max This is an very interesting case. From Fig. 9 we can see that the traffic flow evolves into two parts automatically, with the front four vehicles constituting a group which travels at 12

13 2 Equilibrium line ρ* / m 1 Fig. 1: Phase transition of uniformly distributed vehicles with the same initial velocity of υ ini = m s 1 and ρ =.25 m s 1. Each point on the trajectory is taken every 2 s. The dashed lines υ = 15. m s 1 and υ =. m s 1 are intended to guide the eyes. the uniform speed of υ ini = m s 1, and the following nine vehicles another group travelling at υ. Fig. 1 shows the trajectories of phase transition of the two groups respectively, which ends exactly at the equilibrium states we expected. From the results of the simulations, we can see that the two-vehicle model not only approximately depicts the trajectories of the phase transitions, but also predicts the equilibrium states precisely. As a matter of fact, if we regard the traffic flow of N vehicles as a chain of N-1 two-vehicle couples, we can expect the first couple to reach the equilibrium state predicted by the two-vehicle model, since the second vehicle is only affected by the leading one. In the course of phase transition of the first couple, the second vehicle gradually plays the role of the leading vehicle for the third one and the second two-vehicle couple also behaviors as the two-vehicle model predicted. As the progress is carried downwards, we can expect the whole traffic to reach the equilibrium state we expect by using the two-vehicle model. IV. CORRESPONDING MACROSCOPIC MODEL The macroscopic traffic approach treats traffic as an one-dimensional fluid, thus, in contrast to microscopic models, macroscopic models describe the collective dynamics in terms 13

14 18 14 Equilibrium line Micro simulation Macro solution ρ* / m 1 Fig. 11: Phase transition of initially stopped traffic with ρ =.2 m 1. The trajectory denoted by squares is the result of microscopic simulation and each square is taken every 4s while the one denoted by triangles is the solutions of corresponding macroscopic model and each triangle is taken every 5 s. Both of the trajectories terminate at the equilibrium state we get from the two-vehicle model. The dashed line υ = 15. m s 1 is intended to guide the eyes. of the spatial density ρ and the average velocity V as a function of x and t. Being the consequence of the conservation of the number of vehicles, one fundamental assumption that virtually all macroscopic model have is the continuity equation: ρ(x, t) t + Q(x, t) x =, (1) where Q(x, t) is the traffic flow which is defined as follows Q(x, t) = ρ(x, t)v (x, t). (11) The following work is how to define ρ(x, t) and V (x, t) in terms of macroscopic fields. As to ρ(x, t), we can recur to the following formula to determine its value in [x j (t), x j+1 (t)] ρ(x, t) = 1 d j+1 (t) [x x d j (t) j(t)] + [x j+1 (t) x]. (12) Since d j (j = 1, 2, 3,, N 1) is the central distance between vehicle j and j + 1, it is always larger than l, the identical length of all the vehicles, which prevents ρ from growing towards infinity. 14

15 18 Equilibrium line Micro simulation Macro solution ρ* / m 1 Fig. 12: Phase transition of uniformly distributed traffic with the initially velocity of υ ini = 1 m s 1 and ρ =.5 m 1. The trajectory denoted by squares is the result of microscopic simulation and each square is taken every 2 s while the one denoted by triangles is the solutions of corresponding macroscopic model and each triangle is taken every 5 s. Both of the trajectories terminate at the equilibrium state we get from the two-vehicle model. The dashed line υ = 1. m s 1 is intended to guide the eyes. Next, we adopt the definition of the average velocity suggested by Helbing et al [21], in which V (x, t) is the linear interpolation between the velocities of vehicle j and j + 1: V (x, t) = v j+1(t)[x x j (t)] + v j (t)[x j+1 (t) x] d j (t), (13) where x j+1 x x j. Here By combining its derivatives with respect to x and t, we have the following equation: V t + V V x A(x, t) = a j+1(t)[x x j (t)] + a j (t)[x j+1 (t) x] d j (t) = A(x, t). (14) is the linear interpolation of the single-vehicle acceleration a j and a j+1 which has the form of Eq. (4). In order to express all the single-vehicle arguments on the right side of Eq. (15) in terms of macroscopic fields, we make the approximation that A(x, t) a(ρ(x, t), V (x, t)). Following these definitions and the approximation, we have the corresponding macroscopic model for the microscopic force model and our problem now is how to determine the (15) 15

16 17 Equilibrium line Micro simulation Macro solution ρ* / m 1 Fig. 13: Phase transition of uniformly distributed traffic with the initially velocity of υ ini = 15 m s 1 and ρ =.4 m 1. The trajectory denoted by squares is the result of microscopic simulation and each square is taken every 2 s while the one denoted by triangles is the solutions of corresponding macroscopic model and each triangle is taken every 5 s. Both of the trajectories terminate at the equilibrium state we get from the two-vehicle model. The dashed line υ = 15. m s 1 is intended to guide the eyes. 18 Equilibrium line Micro simulation Macro solution ρ* / m 1 Fig. 14: Phase transition of uniformly distributed traffic with the initially velocity of υ ini = 18 m s 1 and ρ =.4 m 1. The trajectory denoted by squares is the result of microscopic simulation and each square is taken every 2 s while the one denoted by triangles is the solutions of corresponding macroscopic model and each triangle is taken every 3 s. Both of the trajectories terminate at the equilibrium state we get from the two-vehicle model. The dashed line υ = 15. m s 1 is intended to guide the eyes.

17 boundary conditions. From the definition of d j (x, t), we know that ρ is undefined at the front point of the section we study. To solve this problem, we can treat this point as the end of another section of traffic in front it which is in the equilibrium state. Thus we can assign it a value ρ fro = ρ equ. As for the situation in which there is no non-trivial equilibrium state exist, we take ρ fro =.494 m 1, the value of ρ equ when V = υ, which is proved by the results to be applicable. To compare the results with the trajectories of phase transitions we get in the previous section, we define ρ (t) as ρ (t) = L ρ(x, t)dx L. () Here L is the length of the section of traffic we study. Similarly, we can treat V (x, t) in the same way. The solutions for different initial conditions are shown in Figs and we can see that the trajectories of phase transition is very similar to those we got by micro-simulations. The only difference here is that when the trajectory gets close to the equilibrium line, it simply moves along it until it arrives at the equilibrium state, while in the previous section, the trajectory always miss the equilibrium state when approaching it for the first time and circles one time or several times before reaching it. Although converting a microscopic car-following model into a corresponding macroscopic model is an effective way to treat the traffic problems and examine the results we get via microscopic approach, there is still some characteristics that it fails to explain, for example, all our efforts to reproduce the results we get in Figs. 9 and 1 are proved to be vain. After all, as we point out before, traffic flow is a kind of many-body systems of strongly interacting vehicles in macro scale and some of its characteristics may roots in the innate discontinuity. What s more, in a car-following model, the influence on a vehicle comes only from the front vehicle, but in a macroscopic fluid model, the character is erased due to the introduction of spatial derivative. V. SUMMARY We have established the microscopic force model for traffic and analyzed the characteristics of the model by simplifying the traffic into the two-vehicle model. We have used numerical simulations and solutions of the corresponding macroscopic model to study the 17

18 equilibrium and phase transition in different conditions. The outcomes show that the results of these two means are consistent with each other, which also corroborate our earlier investigation by the two-vehicles model. By discussing some of the results we have got, we have given a few suggestions on some practical problems such as the duration for green light and the space interval between two consecutive traffic lights. We have also pointed out the fundamental difference between the microscopic model and its corresponding macroscopic one, which may explain why there are some phenomena which can only be explained by the microscopic approach. Acknowledgement This project is partially supported by NNSF and 973 Project of China. Ge, Mi and Li are also grateful for the support from CURE Program by Dr. T.D. Lee. [1] B.D. Greenshields, Proc. Highway Rearch Board vol 14, p 448, (Highway Research Board, Washington, D.C., 1935). [2] B.S. Kerner and P. Konhkäuser, Phys. Rev. E 48, 2335 (1993). [3] B.S. Kerner and P. Konhkäuser, Phys. Rev. E 5, 54 (1993). [4] B.S. Kerner and H. Rehborn, Phys. Rev. E 53, R4275 (1996). [5] B.S. Kerner and H. Rehborn, Phys. Rev. Lett. 79, 43 (1997). [6] L.A. Pipes, J. Appl. Phys. 24, 274 (1953). [7] M. Bando, K. Hasebe, A. Nakayama, A. Shibata, and Y. Sugiyama, Phys. Rev. E 51, 135 (1995). [8] S. Krauss, P. Wagner, and C. Gawron, Phys. Rev. E 54, 377 (1996). [9] S. Krauss, P. Wagner, and C. Gawron, Phys. Rev. E 55, 5597 (1997). [1] D. Helbing, Verkehrsdynamik (Springer, Berlin, 1997). [11] D. Helbing and B. Tilch, Phys. Rev. E 58, 133 (1998). [12] T. Nagatani, Phys. Rev. E 58, 4271 (1998). [13] D.E. Wolf, Physica A 263, 438 (1999). [14] M. Treiber, A. Henneke, and D. Helbing, Traffic and Granular Flow 99, p 365, (Springer, 18

19 Berlin, 1999). [15] M. Treiber, A. Henneke, and D. Helbing, Phys. Rev. E 62, 185 (2). [] E. Tomer, L. Sofonov, and S. Havlin, Phys. Rev. Lett. 84, 382 (2). [17] T. Nagatani, Phys. Rev. E 61, 3534 (2). [18] I. Lubashevsky, S. Kalenkov, and R. Mahnke, Phys. Rev. E 65, 3614 (22). [19] D. Helbing, Rev. Mod. Phys., 73, 4 (21). [2] D. Helbing, A. Hennecke, and M. Treiber, Math. Comput. Model A 38, 489 (22). [21] D. Helbing, A. Hennecke, and M. Treiber, Math. Comput. Model A 35, 517 (22). 19

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