CHINESE JOURNAL OF PHYSICS VOL. 41, NO. 6 DECEMBER 2003 Numerical Simulations of a Three-Lane Traffic Model Using Cellular Automata A. Karim Daoudia and Najem Moussa LMSPCPV, Dépt. de Physique, FST, B.P. 509, Boutalamine, Errachidia, Morocco (Received February 10, 2003) We study a simple three-lane cellular automaton, based upon the well known Nagel- Schreckenberg model, and examine the effect of slow cars in such a system. We point out the important parameters defining the shape of the fundamental digram for the three-lane model and compare it to that of a two-lane one, showing the new mode of interactions between lanes. It is possible to reduce the influence of slow cars by choosing an adequate version of the symmetry with respect to lanes. PACS numbers: 89.40.+k, 05.70.Fh, 05.65.+b, 05.40.Jc I. INTRODUCTION In recent times many highway traffic models formulated in terms of cellular automata (CA) have been studied, both in one [1-8] and two dimensions [9-14]. For a realistic description of traffic on highways, we must mention in particular the model introduced by Nagel and Schreckenberg (NaSch model), which has been generalized in the past years [12,14] to develop a CA model of two-lane traffic. Several attempts have so far been made in this direction, and different lane-changing procedures have been proposed [9-14]. Chowdhury et al. [14] considered a two-lane ring with slow and fast vehicles (different v max ) evolving in both lanes. The simulation results have shown that, already for small densities, the fast vehicles take on the average free-flow velocity of the slow vehicles, even if only a small fraction of slow vehicles are considered. Recent simulation results of Knospe et al. [12] considered an anticipation effect and found that it reduces the influence of the slow vehicles drastically. In order to look for the systematic slowing of cars caused by trucks, we propose in this work a three-lane version of the NaSch model which takes into account the exchange of vehicles between the different lanes. This corresponds to roads with three lanes that are met, in particular, in many of the European highways. In general, different types of vehicles are present, and from daily experience it is known that slow vehicles have a strong influence on the system s performance. This is obviously most pronounced in single-lane traffic where passing is not possible, and the slow vehicles dominate the dynamics. For multi-lane traffic models the influence of slow vehicles is not so obvious, nevertheless it dominates the average flow at low densities [12-14]. We introduce two kinds of vehicles on the circuit: fast and slow vehicles, e.g. cars and trucks. We study the effect of trucks on the three-lane traffic flow for different traffic http://psroc.phys.ntu.edu.tw/cjp 671 c 2003 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA
672 NUMERICAL SIMULATIONS OF A... VOL. 41 situations. The lane-changing rules are derived from those elaborated by Knospe et al. [12] for two-lane traffic, which can be symmetric or asymmetric with respect to the lanes or the vehicles. Thus, for the asymmetric situation with respect to the lanes, only cars can exchange between all lanes while the trucks always move in the second and third lanes and exchange between these lanes; they are not allowed to change to the first one. For the symmetric situation, both cars and trucks can use all lanes. Our model is an extension of the one-lane NaSch model defined on the three 1-D lattices of 3 N sites with periodic boundary conditions, forming a closed circuit. In the next section, we summarize the definition of the single-lane model and present a simple three-lane extension. Simulation results for the three-lane system compared to the twolane system and the influence of trucks on its dynamics are presented in section III and IV. Finally, we present some conclusions in section V. II. MODEL The NaSch model is a CA model which is described as follows: On a ring with L sites every site can either be empty or occupied by one vehicle with velocity v = 0, 1,..., v max. Let gap be the number of empty sites in front of the car, and v its velocity at time t. At each discrete time step the arrangement of N cars is updated in parallel, according to the following rules: Acceleration: with regard to the vehicle ahead: v min (v + 1, gap, v max ) Noise: with a probability p : v max(v 1, 0) Movement: the car moves v sites ahead. In this work, we consider a three-lane model consisting of three single lanes with periodic boundary conditions, where additional rules defining the exchange of vehicles between the lanes are introduced. It is clarified that this extension can be made without changing the basic properties of the single-lane model. Moreover, lane changes lead to strong correlations between neighboring lanes [12]. We choose the Knospe et al. exchange rules elaborated for the two-lane case [12] which are extended here to the three-lane system. Let gap other ij (resp. gap back ij ) be the number of free cells between the vehicle on lane i and its forward (resp. back) neighboring vehicle on lane j. Note that a vehicle in lane (1) or (3) can change only to the center lane whereas a vehicle in the center lane can change to lane (1) or (3). In this case, several situations arise: if the lane-change conditions of are fulfilled only for one of the two adjacent lanes (lanes (1) and (3)), the driver will change to the latter. On the other hand, if the conditions of lane-changing are fulfilled for the two adjacent lanes, the driver will choose between these lanes according to its speed optimization criteria, e.g., the lane where the parameters gap other and gap back are most significant, or the lane where the predecessor is fastest, etc... The corresponding exchange rules are defined by the following two criteria: first, a vehicle needs an incentive to change a lane. Second, a lane change is only possible if some safety constraints are fulfilled (see Figure 1) : Incentive criterion: 1. v hope > gap, with v hope = min (v + 1, v max ).
VOL. 41 A. KARIM DAOUDIA AND NAJEM MOUSSA 673 FIG. 1: The quantities relevant for lane changing rules in the three-lane system. The hatched cells are occupied by vehicles. Of interest to us is the vehicle noted by v: a) in the first lane, and b) in the center lane. This criterion is valid for whichever lane contains the considered vehicle. For the safety criteria, two cases are presented: i) The vehicle is in the lane i = 1 or 3: 2. gap other i2 > gap i, 3. gap back i2 v max. The vehicle changes from lane (i) to lane (2). ii) The vehicle is in lane (2): 2. gap other2j > gap 2, 3. gap back2j v max. The vehicle changes from lane (2) to lane (j). We adopt the following temporal treatment: in the first sub-step, the vehicles in the first lane may move effectively by the forward movement rules or change lanes in parallel following the lane changing rules, in the second (resp. third) sub-step the vehicles in the second (resp. third) lane are considered. Two situations are examined with these lane changing rules: the symmetric case where the lane-changing criteria are applied equally to both cars and trucks, and the asymmetric case where the lane-changing rules are applied to cars for all lanes and to trucks for lanes (2) and (3) only, i.e. the trucks are constrained to move in the second and third lanes and exchange between these lanes, but are not allowed to change to the first lane. In order to present the simulation results for the model explained above, we set down some definitions: we denote by ρ the total density of vehicles in the three lanes. In the initial state, the vehicles are randomly distributed at sites of the system, and the proportion of trucks is 10%. We consider two different situations: a homogeneous case, where the threelane system is occupied by cars only, and an inhomogeneous case, where both types of vehicles, i.e. cars and trucks evolve in the system. The results are obtained from numerical
674 NUMERICAL SIMULATIONS OF A... VOL. 41 FIG. 2: Individual lanes in the three-lane model for homogeneous systems compared with the corresponding two lane homogeneous systems, for v max = 5 and p = 0.4. The inset shows a comparison of the average flows for both categories of models. simulations on a lattice of 6 10 3 sites with a hundred random initial configurations of vehicles. For each initial configuration results are obtained by averaging over 5 10 4 time steps after the first 1 10 4 time-steps, so that the system reaches a stationary state. III. HOMOGENEOUS SYSTEM As a first step towards a realistic three-lane model for highway traffic we studied the behavior of fast vehicles in the homogeneous case (no trucks in the circuit). The lane changing rules are applied equally in the three lanes (symmetric case with respect to the lanes). The rules set are relevant for the traffic in towns, where overtaking on all lanes is allowed. The fundamental diagram of the individual lanes is given in Figure 2, where we have included curves corresponding to the case of a two-lane system with braking noise p = 0.4 and v max = 5. It is shown, first, that the average flow per lane in the first and third lanes is fairly similar and is similar to the corresponding flow in the two-lane system [13] (lanes (1) and (3) reproduce two-lane traffic systems.). Second, the simulations show an increase of the maximum flow in the central lane compared to the flow in the adjacent ones. Moreover, the behavior of the flow shows a maximum at low densities in both the two-lane and three-lane models : the flows reaches a maximum at ρ j max 0.1, which is at or near a sharp bend of the flow curves. A comparison of the average flows between the two categories of models shows an increase in the three-lane flow, essentially for densities
VOL. 41 A. KARIM DAOUDIA AND NAJEM MOUSSA 675 FIG. 3: Density dependence of the frequency of lane-changes per car for a homogeneous system in the two-lane and three-lane models, with v max = 5 and p = 0.4. The inset shows a comparison of the ping-pong change frequency for both categories of models. ρ ρ j max. Another interesting quantity to look at in the lane changing dynamics is the frequency of lane changes at different densities. Hence, in Figure 3 the variations of the lane-changing frequency per car against the total density ρ of cars in the three-lane model is compared to the corresponding case in the two-lane model. It is shown, first, that for the two categories of models the maximum number of lane changes occurs at densities much higher than ρ j max. Second, the frequency of lane changes is less important in the three-lane model. This is due to differences in correlations between lanes in the two types of models. In particular, the three-lane model offers to drivers in the center lane the opportunity to change to the lane where the speed is most optimal, therefore the lane where it will have less possible changes to carry out. The effect of ping-pong lane changes has already been observed [10-12], for twolane models. The frequency of ping-pong lane changes can be determined as follows: a car makes two lane changes in two consecutive iterations. Figure 3 shows the density dependance of the number of ping-pong lane changes for both two-lane and three-lane models. It is clear that these curves naturally have similar behavior for the frequency of lane-changing. Yet, the number of ping-pong lane changes is smaller in the three-lane version compared with the two-lane one: this corresponds more closely to real traffic flow, where the ping-pong phenomenon is rare, even non-existent.
676 NUMERICAL SIMULATIONS OF A... VOL. 41 FIG. 4: Individual lanes in the symmetric three-lane model for inhomogeneous systems, with 10% trucks for v max = 5 and p cars = p trucks = 0.4. The inset shows the three-lane average flow compared with the corresponding two lane average flow. IV. A HETEROGENEOUS SYSTEM While most investigations have considered homogeneous systems with one type of vehicle, real traffic is in many respects inhomogeneous. In this section, we discuss a three-lane inhomogeneous system containing 10% trucks with vmax trucks = 3 and 90% cars, with vmax cars = 5, along with the same braking noise for both types of vehicles (p cars = p trucks = 0.4). We consider two different situations : a) the symmetric case where the fast as well as the slow vehicles may use all three lanes, i.e. both categories of vehicles are treated equally with respect to the lane changing rules; b) the asymmetric case where trucks are allowed only in the second and third lanes: trucks are constrained to move or exchange between these lanes and are not allowed to change to the first lane, whereas the fast vehicles may use all three lanes. For the single-lane or two-lane model, the introduction of slow vehicles into the system, even for a weak proportion, induces a considerable disruption of traffic. To reduce this disruption, several attempts have been introduced into the NaSch traffic model, such as anticipation and sequential updates, which have lead to slightly increased values of the maximum flow [12]. Another way of minimizing these effects is to initially have all trucks in the second lane only, and not allow them to change lanes [14].
VOL. 41 A. KARIM DAOUDIA AND NAJEM MOUSSA 677 FIG. 5: Individual lanes in the asymmetric three-lane model for an inhomogeneous systems with 10% trucks, for v max = 5 and p cars = p trucks = 0.4. The inset shows the three-lane average flow compared with the corresponding two lane average flow. IV-1. Flow behavior In order to quantify the effects of slow vehicles in the three-lane system, we first examine inhomogeneous flows for the symmetric version of the lane-changing rules. In this configuration, the general evolution of lane flows is similar to the homogeneous case, in the sense that the center lane remains the fast lane and that the adjacent lanes have similar behaviors (figure 4). The symmetry of the lane changing update rules is reflected in the fundamental diagram: lanes (1) and (3) show the same flow density relationship, in particular, they have the same maximum flow J max at the same density ρ max, whereas the center lane is the seat of the most significant flow. Comparing the average flow in a three-lane inhomogeneous system with the corresponding flow for a two-lane system, one sees that the three-lane flow is always higher than the two-lane flow, particularly at low densities. As a result of asymmetry of the lane changing rules, the flow in the second and third lanes is dominated by trucks, while the flow in the first lane is higher (fast lane). Let us note for this case the hierarchy in the behavior of the individual flow in the different lanes. Figure 5 displays this asymmetry: lane (1) is faster than lane (2), which is even faster than lane (3). This behavior is easily explained if the following considerations are made: a) The slow vehicles are contained in lanes (2) and (3) (minor lanes), with only the fast vehicles in lane (1); the first lane is for this reason more fluid than both the others; b) The center lane, being in connection with the fast lane on the one hand and with a minor lane on the other hand, is crossed by a middle flow. In Figure 5, the average three-lane
678 NUMERICAL SIMULATIONS OF A... VOL. 41 FIG. 6: Comparison of the average flow of the homogeneous systems with the corresponding symmetric and asymmetric inhomogeneous average flow with 10% of trucks, for v max = 5 and p cars = p trucks = 0.4. flow is compared with the corresponding flow in the two-lane system. It is shown that, starting from relatively low densities, the three-lane average flow is more significant than that of the two-lane, but for very low densities the reverse occurs, i.e. a two-lane system is most advantageous The differences between average flows ceases for high densities. Figure 6 shows the inhomogeneous average flows in the symmetric and the asymmetric version of the three-lane model, compared to the homogeneous average flow. It is clear that, for relatively small densities, the asymmetric flow largely exceeds the symmetric flow, but for other densities these flows take at the same values. Thus the asymmetric version is used for the highway traffic whereas the symmetric version is reserved for urban traffic, where the densities are generally higher. Finally, we notice a decrease of the inhomogeneous total flow compared to that of the homogeneous case, for both versions of the lane changing rules symmetry. IV-2. Lane-changing behavior Figures 7 illustrates the simulation results of the density dependence of the lane changing frequency for inhomogeneous systems with 10% of trucks. We notice, first, that the introduction of trucks into the system induces an increase in the lane changing frequency. This is easily recognizable, since the presence of trucks constrains drivers, forcing them to more often change lanes to keep an optimal speed. Second, by comparing this quantity for both symmetric and asymmetric situations, it is shown that the curves are marginally different and that the maximum lane changing frequency in the two examined versions is
VOL. 41 A. KARIM DAOUDIA AND NAJEM MOUSSA 679 FIG. 7: Density dependence of the three-lane frequency of lane-changes per lane for inhomogeneous systems for both symmetric and asymmetric versions, with v max = 5 and p cars = p trucks = 0.4. The inset shows the same quantities for the two-lane inhomogeneous systems. reached at values close to ρ j max. In contrast, for the two-lane model, the profile of the lane changing frequency is different according to whether the lane changing rules are symmetric or asymmetric. Finally, we study ping-pong lane changes in an inhomogeneous system (Figure 8). The asymmetry of the lane changing rules leads to an increase of ping-pong lane changes for densities close to ρ j max. This characteristic is also verified for the two-lane model and results from the fact that in the asymmetric version, with trucks being not allowed to change to the fast lane, cars multiply their attempts at lane changes. Finally, we point out that the introduction of trucks drastically increases the lane change frequency, as well as the number of ping-pong lane changes, in both symmetry versions. V. CONCLUSION In summary, we have presented an extension of a deterministic cellular automaton model to traffic flow in a three-lane roadway, with two species of vehicles and for two versions of symmetry. A systematic comparison was made with the two-lane model. First, we showed that even in three-lane systems, where fast vehicles have more possibilities to overtake slow ones, particle disorder may dominate the behavior of the whole system at low densities. Second, for asymmetric lane changing rules, trucks influence the systems performance less than in the symmetric case, as in the two-lane model. Moreover, the
680 NUMERICAL SIMULATIONS OF A... VOL. 41 FIG. 8: Density dependence of the three-lane ping-pong change frequency for inhomogeneous systems for both symmetric and asymmetric versions, with v max = 5 and p cars = p trucks = 0.4. The inset shows the same quantities for the two-lane inhomogeneous systems. reduction of the total flow due to the trucks has a larger magnitude only when close to the maximum. On the other hand, the investigation of lane changes and the ping-pong frequencies in the three-lane model shows a general decrease of these quantities, compared to the corresponding behaviour in the two-lane model (especially in the asymmetric version). As a consequence of this behavior, three-lane traffic is found to be more fluid compared to two-lane traffic. References [1] N.H. Gartner and N.H.M. Wilson (eds), Transportation and Traffic Theory, (New York: Elsevier, 1987) [2] S. Wolfram, Theory and Application of Cellular Automata, (World Scientific, Singapore, 1986) [3] K. Nagel and M. Schreckenberg, J.Physique I, 2, 2221 (1992). [4] M. Fukui and Y. Ishibashi, J.Phys. Soc. Jpn. 62 (1993). [5] P. Wagner, K. Nagel, and D.E. Wolf, Physica A 234, 687 (1997). [6] K. Nagel, D.E. Wolf, P. Wagner, and P. Simon, Phys. Rev. E 58, 1425 (1998). [7] X. Zhang and G. Hu, Phys. Rev. E 52, 4664 (1995). [8] Lei Wang, Bing-Hong Wang, and Bambi Hu; Phys. Rev. E 63, 056117 (2001). [9] D. Chowdhury, D.E. Wolf, and M. Schreckenberg, Physica A 235, 417 (1997). [10] T. Nagatani, J. Phys. A : Math.Gen. 26, L781 (1993). [11] M. Rickert, K. Nagel, M. Schreckenberg, and A. Latour, Physica A 231, 534 (1996). [12] W. Knospe, L. Santen, A. Schadschneider, and M. Schreckenberg, Physica A 265, 614 (1999).
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