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Asymmetric effect of route-length difference and bottleneck on route choice in two-route traffic system
Physica A 428 (2015) 416–425
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Physica A journal homepage: www.elsevier.com/locate/physa
Asymmetric effect of route-length difference and bottleneck on route choice in two-route traffic system Yuki Hino, Takashi Nagatani ∗ Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan
highlights • We extended the symmetric two-route traffic model to the asymmetric case. • We studied the asymmetric effect of route-length difference and bottleneck on the route choice and the traffic behavior. • We clarified the dependence of the travel time and route choice on the route-length difference and bottleneck.
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Article history: Received 20 October 2014 Received in revised form 1 December 2014 Available online 11 February 2015 Keywords: Traffic dynamics Route choice Congestion Asymmetry Complex system
abstract We study the traffic behavior in the asymmetric two-route traffic system with real-time information. In the asymmetric two-route system, the length on route A is different from that on route B and there exists a bottleneck on route A. We extend the symmetric tworoute dynamic model to the asymmetric case. We investigate the asymmetric effects of the route-length difference and bottleneck on the route choice with real-time information. The travel time on each route depends on the road length, bottleneck, and vehicular density. We derive the dependence of the travel time and mean density on the route-length ratio. We show where, when, and how the congestion occurs by the route choice in the asymmetric two-route system. We clarify the effect of the route-length ratio on the traffic behavior in the route choice. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Recently, physicists have devoted much attention to traffic flow [1–5]. The traffic systems include so many factors that it is difficult to discover the essential factors affecting on the traffic behavior. Physicists have proposed the simplified traffic models including a few factors at most to clarify the cause and effect [1–7]. The transportation systems have been investigated with the use of the physical models and concepts [8–44]. For modeling traffic, the various traffic models with different conceptual frameworks have been proposed [1–44]. In the microscopic models, the traffic is treated as a system of interacting particles driven far from equilibrium. In contrast, in the so-called macroscopic models, the traffic is viewed as a compressible fluid [1–5]. Gupta and Sharma have presented the nonlinear analysis of traffic jam in the anisotropic continuum model [38]. Gupta and Dhiman have studied the jamming transitions in the extended lattice hydrodynamic model [39]. Information is a key commodity in traffic system [45]. The information has an important effect on the traffic dynamics [45–50]. In real traffic, advanced traveler information systems provide real-time information about the traffic conditions to road users by means of communication such as variable message signs, radio broadcasts or on-board computers [45]. The real-time information helps the individual road users to minimize their personal travel time.
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Corresponding author. E-mail addresses: [email protected], [email protected] (T. Nagatani).
http://dx.doi.org/10.1016/j.physa.2015.01.086 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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Fig. 1. Schematic illustration of the asymmetric two-route traffic system. The road length on route B is longer than that on route A. Also, the bottleneck is positioned at the center on route A. Two types of vehicles are introduced: dynamic and static vehicles. When vehicles enter the system, a so-called dynamic driver will make a choice on the basis of the travel-time feedback, while a static driver enters route A(B) with probability 1/2 (probability 1/2) ignoring any advice.
Wahle et al. have proposed the dynamic model for the two-route traffic system with real-time information [45]. They have studied the effect of real-time information on the route choice. The route-choice strategy has been extended to the three-route and crossing traffic systems [46–48]. It is important and necessary not only to obtain the real-time information but also to know the control strategy of signals because the city traffic is generally controlled by many signals. Tobita and Nagatani have extended the two-route traffic system with real-time traffic information to that controlled by signals [49]. They have clarified the effect of signals on the two-route traffic flow. Also, Hino and Nagatani have studied the effect of a bottleneck on the route choice in the two-route traffic system with real-time information [50]. In the two-route traffic system studied until now, the route length on route A is the same as that on route B. The tworoute traffic system is symmetric. However, in real traffic, the road length on route A is generally different from that on route B. The two-route traffic system is not symmetric but asymmetric. The traffic behavior in the asymmetric two-route traffic system is definitely different from that in the symmetric case. However, it is not known how the asymmetry of two routes affects the route choice and traffic behavior. It is important and necessary to know the effect of the route-length difference on the route choice. Also, it is important to know the bottleneck effect on the route choice in the asymmetric two-route system. In this paper, we study the effect of the route-length difference on the two-route traffic system with a bottleneck at the travel-time feedback strategy. We extend the dynamic model proposed by Wahle et al. [45] to the asymmetric two-route traffic system with a bottleneck. We investigate how the traffic behavior induced by the asymmetry and bottleneck varies with time due to the real-time information. We study the combined effect of the asymmetry and bottleneck on the route choice. We derive the dependence of the travel time and the density on the route-length ratio. We clarify the effect of the length ratio on the route choice for the asymmetric two-route traffic system with a bottleneck. 2. Asymmetric two-route traffic model We consider the asymmetric two-route traffic system in which the road length on route A is different from that on route B. The route-length difference affects highly the route choice because the travel time depends on both road length and density. Using the scenario with dynamic information, Wahle et al. have proposed the symmetric two-route traffic system in which the road length on route A is the same as that on route B [45]. We extend the symmetric model to the asymmetric two-route system. We study the traffic behavior in the asymmetric two-route traffic system with dynamic information. We apply the travel-time feedback strategy to the route choice according to the strategy proposed by Whale et al. When vehicles enter the system, vehicles move either on route A or B. Fig. 1 shows the schematic illustration of the asymmetric two-route traffic system. The road length on route B is longer than that on route A. Also, we consider the effect of bottleneck on the route choice in the asymmetric two-route traffic system. In the case, the bottleneck is positioned at the center on route A. The bottleneck has the effective effect on the travel time because the traffic jam is induced by the bottleneck. We represent the bottleneck by the blocking probability. Vehicles pass through the bottleneck with probability 1 − pb and are stopped at the bottleneck with probability pb , where pb is the blocking probability. When the blocking probability is zero and the road length on route A is the same as that on route B, the above model represents the original model by Wahle et al. Two types of vehicles are introduced: dynamic and static vehicles. When vehicles enter the system, a so-called dynamic driver will make a choice on the basis of the travel-time feedback, while a static driver enters route A(B) with probability 1/2 (probability 1/2) ignoring any advice. The dynamic driver always chooses the route with the shortest travel time at the entrance. The densities of dynamic and static drivers are Sdyn and 1 − Sdyn respectively. The static driver enters route A(B) with probability 1/2. If the road length on route B is longer than that on route A, the travel time on route B is higher than that on route A for the same density of vehicles. If the travel time on route B is higher than that on route A, static drivers do not change their route but dynamic vehicles change from route B to route A. In time, the travel time on route A is higher than that on
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Fig. 2. (a) Plots of dimensionless travel times Tt against time t on symmetric two routes at length ratio LB /LA = 1.0. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on two routes at LB /LA = 1.0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
route B because the route A is congested. When route A is congested, the drivers with the real-time information go to route B. The process is repeated. The traffic flow on each route varies with time. The fluctuating traffic flow changes greatly by the asymmetry of two routes. The traffic flow displays the dynamic behavior different from the original model by Wahle et al. [45]. The simulations are performed in the following steps. First, we set two routes and board empty (see Fig. 1). Second, after the vehicles enter the entrance, they becomes the dynamic driver with probability Sdyn and the static driver with probability 1 − Sdyn . Third, the static driver chooses route A with probability 1/2 and route B with probability 1/2. Finally, the travel time information will be generated, transmitted, and displayed on the board at each time step, according to the travel-time feedback strategy. The dynamic drivers will choose the route with the shortest travel time at the entrance of two routes irrespective of the route length. After entering the route, the vehicle moves through the system according the dynamics of the cellular automaton model proposed by Nagel and Schreckenberg (NS model) [6]. In NS model, the road is subdivided into cells with a length. Each cell is either empty or occupied by only one vehicle with an integer velocity vi where the maximum velocity is vmax . Let be NA(B) the total number of vehicles on route A(B) of length LA(B) . Then, the vehicular density on route A(B) is ρA(B) = NA(B) /LA(B) . The motion of vehicles is updated in parallel. The maximum velocity is set as vmax = 3 throughout this paper. Also, the randomization probability is set as p = 0.25. For generalization, we use the dimensionless travel time Tt = tt vmax /LA where LA is the road length on route A. Road length LA on route A is the characteristic (reference) length. 3. Simulation result without a bottleneck We carry out numerical simulation for the traffic flows on routes A and B in the asymmetric two-route model. We study how the travel time and density on routes A and B vary with time for various values of the route-length ratio. The length ratio is defined as LB /LA . The parameters are set as road length LA = 2000 on route A, maximum velocity vmax = 3, randomization probability p = 0.25, and density Sdyn = 0.5 of dynamic drivers. The road length LB on route B varies from 1000 to 3000. All simulation results shown here are obtained by 2,000,000 iterations. The travel times and numbers of vehicles on two routes are calculated at each time step. The mean values of travel time and density are averaged over 1,000,000 time steps. The dimensionless travel time on route A is defined as Tt ,A = tt ,A vmax /LA where tt ,A is the travel time on route A. The dimensionless travel time on route B is defined as Tt ,B = tt ,B vmax /LA where tt ,B is the travel time on route B. The densities on routes A and B are defined as ρA = NA /LA and ρB = NB /LB respectively, where NA (NB ) is the number of vehicles on route A(B). For comparison, we show the variations of travel times and densities on two routes for the symmetric two-route system in which the length ratio is 1. Fig. 2(a) shows the plots of dimensionless travel times Tt against time t on two routes at LB /LA = 1.0. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. The travel time on each route varies periodically with time. The travel time on two routes varies alternately. Fig. 2(b) shows the plots of densities ρ against time t on two routes at LB /LA = 1.0. Red and black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean density on routes A and B respectively. Density on each route also varies periodically and alternately. The mean values of travel time and density on route A agree with those on route B. We calculate the travel times and densities on two routes for the case in which the length ratio is not 1. We study the effect of the length ratio on travel times and densities. Fig. 3(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.1. Red and black lines indicate travel times on routes A and B respectively.
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Fig. 3. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes at length ratio LB /LA = 1.1. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes at LB /LA = 1.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Plot of vehicular position versus time step at LB /LA = 1.1 corresponding to Fig. 3. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Red and black dashed lines represent the mean travel time on routes A and B respectively. Fig. 3(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.1. Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean density on routes A and B respectively. Fig. 3 is compared with Fig. 2. By changing the length ratio from LB /LA = 1.0 to LB /LA = 1.1, the travel times and densities on two routes change from Fig. 2 to Fig. 3. The peak value of the travel time on route B is higher than that on route A. Also, the mean value of the travel time on route B is higher than that on route A. However, the mean density on route A is higher than that on route B. This is due to the reason that dynamic drivers trend to select the short route A. We derive the traffic pattern to investigate the traffic behavior induced by the road length difference. Fig. 4 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.1 corresponding to Fig. 3. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. The regions with high density are indicated by dense color. The congestion is represented by a shade of color. The shade of color displays the stripe pattern. The region with black stripe represents the cluster of dynamic and static drivers, while the region with light and red stripe indicates the cluster of static drivers. The width of black stripe on route A is wider than that on route B. Dynamic drivers trend to select route A than route B because the travel time on route A is shorter than that on route B. When the dynamic drivers select route A, there are in time no dynamic drivers on route B. Then, the travel time on route A increases due to the high density. When route A is congested, the dynamic drivers on route A go to route B. In the result, the travel time on route B increases. The process is repeated. The travel times and densities oscillate. The traffic pattern on route A is definitely different from that on route B. Fig. 5(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. Fig. 5(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.2 Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the
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Fig. 5. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes at length ratio LB /LA = 1.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes at LB /LA = 1.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Plot of vehicular position versus time step at LB /LA = 1.2 corresponding to Fig. 5. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 6 is compared with Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
mean density on routes A and B respectively. Fig. 5 is compared with Fig. 3. By changing the length ratio from LB /LA = 1.1 to LB /LA = 1.2, the travel times and densities on two routes change from Fig. 3 to Fig. 5. The travel time and density on each route oscillate with a period longer than that in Fig. 3. Fig. 6 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.2 corresponding to Fig. 5. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 6 is compared with Fig. 4. The width of black stripe is wider than that in Fig. 4 because more dynamic drivers go to route A. The density on route A is higher than that on route B. The density on route A is higher than that on route A in Figs. 3 and 4. Thus, the width of black stripe increases with increasing the length ratio. This is due to the reason that more dynamic drivers go to route A with increasing the length ratio. If the length ratio increases furthermore, the travel time, density, and traffic pattern change greatly. Fig. 7(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.4. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. Fig. 7(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.4. Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean density on routes A and B respectively. Fig. 7 is compared with Fig. 3 and 5. By changing the length ratio from LB /LA = 1.1, through LB /LA = 1.2, to LB /LA = 1.4, the travel times and densities on two routes change from Fig. 3, through Fig. 5, to Fig. 7. The travel time on route B does not vary highly but the travel time on route A fluctuates irregularly. Densities on routes A and B do not vary highly with time but fluctuate bit by bit with time. The fluctuation of the travel time on route A is due to the occurrence of spontaneous jam. Fig. 8 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.4 corresponding to Fig. 7. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 8 is compared with Fig. 6. All dynamic drivers go to
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Fig. 7. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes at length ratio LB /LA = 1.4. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes at LB /LA = 1.4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Plot of vehicular position versus time step at LB /LA = 1.4 corresponding to Fig. 7. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 8 is compared with Fig. 6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
route A, there are no dynamic drivers on route B, and there are only static drivers on route B because the travel time on route B is always higher than that on route A. Such stripe patterns as Figs. 4 and 6 do not appear but the homogeneous patterns occur on routes A and B. Thus, the crossover phenomenon from the stripe pattern to the homogeneous pattern occurs with increasing the length ratio. We derive the relationship between the mean travel time and the length ratio. Also, we study the dependence of the mean density on the length ratio. Fig. 9(a) shows the plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean travel times on routes A and B. Fig. 9(b) shows the plots of mean density ⟨ρ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean densities on routes A and B. For LB /LA > 1, the mean travel time on route B is higher than that on route A and increases with increasing the length ratio. However, the mean density on route B is less than that on route A for LB /LA > 1. For LB /LA > 1, the mean density on route A increases with increasing the length ratio, while the mean density route B decreases with increasing the length ratio. The mean density on route A saturates near LB /LA = 1.4. The saturation is due to the upper limit of vehicular inflow. The crossover phenomenon occurs at the saturation point from the stripe pattern to the homogeneous pattern. 4. Simulation result with a bottleneck We study the effect of the bottleneck on the asymmetric two-route traffic system with real-time information. The combined effect of the bottleneck with the asymmetry will have the important effect on the route choice. We carry out simulation for the asymmetric two-route system with a bottleneck. The bottleneck is positioned at the center on route A. The parameter values of the two-route traffic system are the same as those in Section 3. The blocking probability representing the bottleneck’s strength is taken as pb = 0.2.
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Fig. 9. (a) Plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean travel times for routes A and B. (b) Plots of mean density ⟨ρ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean densities for routes A and B.
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Fig. 10. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.2 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.2 for blocking probability pb = 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
We study the traffic behavior in the asymmetric two-route system with a bottleneck on route A. We calculate the travel times and densities on routes A and B by varying the route length ratio. Fig. 10(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.2 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel times on routes A and B respectively. Fig. 10(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.2 for blocking probability pb = 0.2. Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean densities on routes A and B respectively. Fig. 10 is compared with Fig. 5 without bottlenecks. By introducing the bottleneck into the asymmetric two routes, the travel times and densities on two routes change from Fig. 5 to Fig. 10. The bottleneck affects greatly the time-dependent behaviors of the travel time and density. By introducing the bottleneck, the travel times and densities vary with the shorter period than those in Fig. 5. This is due to the reason that the travel time increases faster with increasing inflow. Fig. 11 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.2 corresponding to Fig. 10. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 11 is compared with Fig. 6 without bottlenecks. The width of black stripe is narrower than that in Fig. 6 because dynamic drivers change frequently their route. The traffic jam with a triangular shape is formed at the center on route A. The jam appears, disappears in time, and this is repeated. Thus, the traffic behavior changes greatly by the bottleneck’ effect. With increasing the length ratio from length ratio LB /LA = 1.2, through LB /LA = 1.4, and to LB /LA = 1.9, the travel time changes from Fig. 10(a), through Fig. 12(a), to Fig. 14(a). Also, the density changes from Fig. 10(b), through Fig. 12(b), to Fig. 14(b). The traffic pattern changes from Fig. 11, through Fig. 13, to Fig. 15 where the traffic patterns in Figs. 13 and 15 correspond to those in Figs. 12 and 14 respectively. The jam in Figs. 11 and 13 oscillates with time. The jam in Fig. 15
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Fig. 11. Plot of vehicular position versus time step at length ratio LB /LA = 1.2 for blocking probability (bottleneck’s strength) pb = 0.2, corresponding to Fig. 10. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B.
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Fig. 12. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.4 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.4 for blocking probability pb = 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 13. Plot of vehicular position versus time step at length ratio LB /LA = 1.4 for blocking probability (bottleneck’s strength) pb = 0.2, corresponding to Fig. 12. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B.
is not oscillating but is stationary. The jam in Fig. 15 extends to the entrance. With increasing length ratio, the traffic jam formed at the bottleneck changes from the oscillating jam to the stationary jam. Thus, the traffic behavior for the asymmetric two-route traffic system changes greatly by introducing both bottleneck and asymmetry.
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Fig. 14. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.9 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.9 for blocking probability pb = 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 15. Plot of vehicular position versus time step at length ratio LB /LA = 1.9 for blocking probability (bottleneck’s strength) pb = 0.2, corresponding to Fig. 14. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B.
We derive the relationship between the mean travel time and the length ratio for the asymmetric two-route traffic system with a bottleneck. Also, we study the dependence of the mean density on the length ratio. Fig. 16(a) shows the plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA at blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean travel times on routes A and B. Fig. 16(b) shows the plots of mean density ⟨ρ⟩ against length ratio LB /LA blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean densities on routes A and B. The relationship between the mean travel time and the length ratio changes from Fig. 9(a) to Fig. 16(a) by introducing the bottleneck. The relationship between the mean density and the length ratio changes from Fig. 9(b) to Fig. 16(b). Thus, the mean travel time and the mean density change greatly by both bottleneck and the route-length difference. 5. Summary In real city traffic network, there are some routes with various road lengths. Generally, the travel time on a route depends on the road length and vehicular density. Drivers try to select the shortest route. However, the travel time may be high due to the road congestion even if the route is short. On the basis of the travel-time feedback, drivers try to choice the route with the shortest travel time. For the route choice of drivers, it is important and necessary to know the effect of the routelength difference on the travel time. We have extended the symmetric two-route traffic model proposed by Wahle et al. to take into account the asymmetric road length. The traffic behavior changes greatly by the competition between the short route and driver’s avoidance to the congestion. We have investigated where, when, and how the congestion occurs. We have clarified the effect of the route-length difference on the route choice in the asymmetric two-route traffic system with real-time information. We have derived the dependence of the mean travel time and mean density on the road length ratio. We have shown that the crossover phenomenon occurs from the stripe pattern to the homogeneous pattern with increasing the route-length ratio. Also, we have studied the combined effect of the route-length difference and bottleneck on the route
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Fig. 16. (a) Plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA at blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean travel times on routes A and B. (b) Plots of mean density ⟨ρ⟩ against length ratio LB /LA blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean densities on routes A and B.
choice. We have shown that the bottleneck has the important effect on the traffic behavior in the asymmetric two-route traffic system with real-time information. The study of the route-length difference and bottleneck effects on the route choice is the first and this study will be useful for the route choice in city traffic network. 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] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]
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. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.X. Ge, S.Q. Dai, L.Y. Dong, Y. Xue, Phys. Rev. E 70 (2004) 066134. Z.P. Li, Y.C. Liu, Eur. Phys. J. B 53 (2006) 367. C. Chen, J. Chen, X. Guo, Physica A 389 (2010) 141. W.X. Zhu, Internat. J. Modern Phys. C 19 (2008) 727. A.K. Gupta, V.K. Katiyar, J. Phys. A 38 (2005) 4069. A.K. Gupta, V.K. Katiyar, Physica A 368 (2006) 551. A.K. Gupta, S. Sharma, Chin. Phys. B 19 (2010) 110503. A.K. Gupta, V.K. Katiyar, Physica A 371 (2006) 674. A.K. Gupta, P. Redhu, Physica A 392 (2013) 5622. G.H. Peng, X.H. Cai, C.Q. Liu, B.F. Cao, Phys. Lett. A 375 (2011) 2153. G.H. Peng, X.H. Cai, B.F. Cao, C.Q. Liu, Phys. Lett. A 375 (2011) 2823. G.H. Peng, F.Y. Nie, B.F. Cao, C.Q. Liu, Nonlinear Dynam. 67 (2012) 1811. G.H. Peng, D.H. Sun, Phys. Lett. A 374 (2010) 1694. G.H. Peng, X.H. Cai, C.Q. Liu, B.F. Cao, M.X. Tuo, Phys. Lett. A 375 (2011) 3973. G.H. Peng, R.J. Cheng, Physica A 392 (2013) 3563. T.Q. Tang, H.J. Huang, S.G. Zhao, H.Y. Shang, Phys. Lett. A 373 (2009) 2461. T.Q. Tang, H.J. Huang, G. Xu, Physica A 387 (2008) 6845. E. Brockfeld, R. Barlovic, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 64 (2001) 056132. M. Sasaki, T. Nagatani, Physica A 325 (2002) 531. S. Kurata, T. Nagatani, Physica A 318 (2003) 537. K. Nagel, D.E. Wolf, P. Wagner, P. Simon, Phys. Rev. E 58 (1998) 1425. S. Lammer, D. Helbing, J. Stat. Mech. Theory Exp. (2008) P04019. M. Bando, K. Hasebe, A. Nakayama, Y. Sugiyama, Phys. Rev. E 51 (1995) 1035. B.A. Toledo, E. Cerda, J. Rogan, V. Munoz, C. Tenreino, R. Zarama, J.A. Valdivia, Phys. Rev. E 75 (2007) 189701. T. Nagatani, Physica A 377 (2007) 651. D. Helbing, B. Tilch, Phys. Rev. E 58 (1998) 133. R. Jiang, Q. Wu, Z. Zhu, Phys. Rev. E 64 (2001) 017101. H.X. Ge, R.J. Cheng, Z.P. Li, Physica A 387 (2008) 5239. T. Nagatani, S. Yonekura, Physica A 404 (2014) 171. N. Sugiyama, T. Nagatani, Physica A 392 (2013) 1848. A.K. Gupta, S. Sharma, Chin. Phys. B 21 (2012) 015201. A.K. Gupta, I. Dhiman, Internat. J. Modern Phys. C 25 (2014) 1450045. A.K. Gupta, P. Redhu, Commun. Nonlinear Sci. Simul. 19 (2014) 1600. A.K. Gupta, P. Redhu, Phys. Lett. A 377 (2013) 2027. A.K. Gupta, Internat. J. Modern Phys. C 25 (2013) 1350018. A.K. Gupta, I. Dhiman, Nonlinear Dynam. 79 (2015) 663. A.K. Gupta, S. Sharma, P. Redhu, Commun. Theor. Phys. 62 (2014) 393. J. Wahle, A. Lucia, C. Bazzan, F. Klugl, M. Schreckenberg, Physica A 287 (2000) 669. Y. Yokoya, Phys. Rev. E 69 (2004) 016121. C. Dong, X. Ma, B. Wang, X. Sun, Physica A 389 (2010) 3274. M. Fukui, K. Nishinari, Y. Yokoya, Y. Ishibashi, Physica A 388 (2009) 1207. K. Tobita, T. Nagatani, Physica A 391 (2012) 6137. Y. Hino, T. Nagatani, Physica A 395 (2014) 425.
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Physica A journal homepage: www.elsevier.com/locate/physa
Asymmetric effect of route-length difference and bottleneck on route choice in two-route traffic system Yuki Hino, Takashi Nagatani ∗ Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan
highlights • We extended the symmetric two-route traffic model to the asymmetric case. • We studied the asymmetric effect of route-length difference and bottleneck on the route choice and the traffic behavior. • We clarified the dependence of the travel time and route choice on the route-length difference and bottleneck.
article
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Article history: Received 20 October 2014 Received in revised form 1 December 2014 Available online 11 February 2015 Keywords: Traffic dynamics Route choice Congestion Asymmetry Complex system
abstract We study the traffic behavior in the asymmetric two-route traffic system with real-time information. In the asymmetric two-route system, the length on route A is different from that on route B and there exists a bottleneck on route A. We extend the symmetric tworoute dynamic model to the asymmetric case. We investigate the asymmetric effects of the route-length difference and bottleneck on the route choice with real-time information. The travel time on each route depends on the road length, bottleneck, and vehicular density. We derive the dependence of the travel time and mean density on the route-length ratio. We show where, when, and how the congestion occurs by the route choice in the asymmetric two-route system. We clarify the effect of the route-length ratio on the traffic behavior in the route choice. © 2015 Elsevier B.V. All rights reserved.
1. Introduction Recently, physicists have devoted much attention to traffic flow [1–5]. The traffic systems include so many factors that it is difficult to discover the essential factors affecting on the traffic behavior. Physicists have proposed the simplified traffic models including a few factors at most to clarify the cause and effect [1–7]. The transportation systems have been investigated with the use of the physical models and concepts [8–44]. For modeling traffic, the various traffic models with different conceptual frameworks have been proposed [1–44]. In the microscopic models, the traffic is treated as a system of interacting particles driven far from equilibrium. In contrast, in the so-called macroscopic models, the traffic is viewed as a compressible fluid [1–5]. Gupta and Sharma have presented the nonlinear analysis of traffic jam in the anisotropic continuum model [38]. Gupta and Dhiman have studied the jamming transitions in the extended lattice hydrodynamic model [39]. Information is a key commodity in traffic system [45]. The information has an important effect on the traffic dynamics [45–50]. In real traffic, advanced traveler information systems provide real-time information about the traffic conditions to road users by means of communication such as variable message signs, radio broadcasts or on-board computers [45]. The real-time information helps the individual road users to minimize their personal travel time.
∗
Corresponding author. E-mail addresses: [email protected], [email protected] (T. Nagatani).
http://dx.doi.org/10.1016/j.physa.2015.01.086 0378-4371/© 2015 Elsevier B.V. All rights reserved.
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Fig. 1. Schematic illustration of the asymmetric two-route traffic system. The road length on route B is longer than that on route A. Also, the bottleneck is positioned at the center on route A. Two types of vehicles are introduced: dynamic and static vehicles. When vehicles enter the system, a so-called dynamic driver will make a choice on the basis of the travel-time feedback, while a static driver enters route A(B) with probability 1/2 (probability 1/2) ignoring any advice.
Wahle et al. have proposed the dynamic model for the two-route traffic system with real-time information [45]. They have studied the effect of real-time information on the route choice. The route-choice strategy has been extended to the three-route and crossing traffic systems [46–48]. It is important and necessary not only to obtain the real-time information but also to know the control strategy of signals because the city traffic is generally controlled by many signals. Tobita and Nagatani have extended the two-route traffic system with real-time traffic information to that controlled by signals [49]. They have clarified the effect of signals on the two-route traffic flow. Also, Hino and Nagatani have studied the effect of a bottleneck on the route choice in the two-route traffic system with real-time information [50]. In the two-route traffic system studied until now, the route length on route A is the same as that on route B. The tworoute traffic system is symmetric. However, in real traffic, the road length on route A is generally different from that on route B. The two-route traffic system is not symmetric but asymmetric. The traffic behavior in the asymmetric two-route traffic system is definitely different from that in the symmetric case. However, it is not known how the asymmetry of two routes affects the route choice and traffic behavior. It is important and necessary to know the effect of the route-length difference on the route choice. Also, it is important to know the bottleneck effect on the route choice in the asymmetric two-route system. In this paper, we study the effect of the route-length difference on the two-route traffic system with a bottleneck at the travel-time feedback strategy. We extend the dynamic model proposed by Wahle et al. [45] to the asymmetric two-route traffic system with a bottleneck. We investigate how the traffic behavior induced by the asymmetry and bottleneck varies with time due to the real-time information. We study the combined effect of the asymmetry and bottleneck on the route choice. We derive the dependence of the travel time and the density on the route-length ratio. We clarify the effect of the length ratio on the route choice for the asymmetric two-route traffic system with a bottleneck. 2. Asymmetric two-route traffic model We consider the asymmetric two-route traffic system in which the road length on route A is different from that on route B. The route-length difference affects highly the route choice because the travel time depends on both road length and density. Using the scenario with dynamic information, Wahle et al. have proposed the symmetric two-route traffic system in which the road length on route A is the same as that on route B [45]. We extend the symmetric model to the asymmetric two-route system. We study the traffic behavior in the asymmetric two-route traffic system with dynamic information. We apply the travel-time feedback strategy to the route choice according to the strategy proposed by Whale et al. When vehicles enter the system, vehicles move either on route A or B. Fig. 1 shows the schematic illustration of the asymmetric two-route traffic system. The road length on route B is longer than that on route A. Also, we consider the effect of bottleneck on the route choice in the asymmetric two-route traffic system. In the case, the bottleneck is positioned at the center on route A. The bottleneck has the effective effect on the travel time because the traffic jam is induced by the bottleneck. We represent the bottleneck by the blocking probability. Vehicles pass through the bottleneck with probability 1 − pb and are stopped at the bottleneck with probability pb , where pb is the blocking probability. When the blocking probability is zero and the road length on route A is the same as that on route B, the above model represents the original model by Wahle et al. Two types of vehicles are introduced: dynamic and static vehicles. When vehicles enter the system, a so-called dynamic driver will make a choice on the basis of the travel-time feedback, while a static driver enters route A(B) with probability 1/2 (probability 1/2) ignoring any advice. The dynamic driver always chooses the route with the shortest travel time at the entrance. The densities of dynamic and static drivers are Sdyn and 1 − Sdyn respectively. The static driver enters route A(B) with probability 1/2. If the road length on route B is longer than that on route A, the travel time on route B is higher than that on route A for the same density of vehicles. If the travel time on route B is higher than that on route A, static drivers do not change their route but dynamic vehicles change from route B to route A. In time, the travel time on route A is higher than that on
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Fig. 2. (a) Plots of dimensionless travel times Tt against time t on symmetric two routes at length ratio LB /LA = 1.0. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on two routes at LB /LA = 1.0. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
route B because the route A is congested. When route A is congested, the drivers with the real-time information go to route B. The process is repeated. The traffic flow on each route varies with time. The fluctuating traffic flow changes greatly by the asymmetry of two routes. The traffic flow displays the dynamic behavior different from the original model by Wahle et al. [45]. The simulations are performed in the following steps. First, we set two routes and board empty (see Fig. 1). Second, after the vehicles enter the entrance, they becomes the dynamic driver with probability Sdyn and the static driver with probability 1 − Sdyn . Third, the static driver chooses route A with probability 1/2 and route B with probability 1/2. Finally, the travel time information will be generated, transmitted, and displayed on the board at each time step, according to the travel-time feedback strategy. The dynamic drivers will choose the route with the shortest travel time at the entrance of two routes irrespective of the route length. After entering the route, the vehicle moves through the system according the dynamics of the cellular automaton model proposed by Nagel and Schreckenberg (NS model) [6]. In NS model, the road is subdivided into cells with a length. Each cell is either empty or occupied by only one vehicle with an integer velocity vi where the maximum velocity is vmax . Let be NA(B) the total number of vehicles on route A(B) of length LA(B) . Then, the vehicular density on route A(B) is ρA(B) = NA(B) /LA(B) . The motion of vehicles is updated in parallel. The maximum velocity is set as vmax = 3 throughout this paper. Also, the randomization probability is set as p = 0.25. For generalization, we use the dimensionless travel time Tt = tt vmax /LA where LA is the road length on route A. Road length LA on route A is the characteristic (reference) length. 3. Simulation result without a bottleneck We carry out numerical simulation for the traffic flows on routes A and B in the asymmetric two-route model. We study how the travel time and density on routes A and B vary with time for various values of the route-length ratio. The length ratio is defined as LB /LA . The parameters are set as road length LA = 2000 on route A, maximum velocity vmax = 3, randomization probability p = 0.25, and density Sdyn = 0.5 of dynamic drivers. The road length LB on route B varies from 1000 to 3000. All simulation results shown here are obtained by 2,000,000 iterations. The travel times and numbers of vehicles on two routes are calculated at each time step. The mean values of travel time and density are averaged over 1,000,000 time steps. The dimensionless travel time on route A is defined as Tt ,A = tt ,A vmax /LA where tt ,A is the travel time on route A. The dimensionless travel time on route B is defined as Tt ,B = tt ,B vmax /LA where tt ,B is the travel time on route B. The densities on routes A and B are defined as ρA = NA /LA and ρB = NB /LB respectively, where NA (NB ) is the number of vehicles on route A(B). For comparison, we show the variations of travel times and densities on two routes for the symmetric two-route system in which the length ratio is 1. Fig. 2(a) shows the plots of dimensionless travel times Tt against time t on two routes at LB /LA = 1.0. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. The travel time on each route varies periodically with time. The travel time on two routes varies alternately. Fig. 2(b) shows the plots of densities ρ against time t on two routes at LB /LA = 1.0. Red and black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean density on routes A and B respectively. Density on each route also varies periodically and alternately. The mean values of travel time and density on route A agree with those on route B. We calculate the travel times and densities on two routes for the case in which the length ratio is not 1. We study the effect of the length ratio on travel times and densities. Fig. 3(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.1. Red and black lines indicate travel times on routes A and B respectively.
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Fig. 3. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes at length ratio LB /LA = 1.1. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes at LB /LA = 1.1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 4. Plot of vehicular position versus time step at LB /LA = 1.1 corresponding to Fig. 3. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Red and black dashed lines represent the mean travel time on routes A and B respectively. Fig. 3(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.1. Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean density on routes A and B respectively. Fig. 3 is compared with Fig. 2. By changing the length ratio from LB /LA = 1.0 to LB /LA = 1.1, the travel times and densities on two routes change from Fig. 2 to Fig. 3. The peak value of the travel time on route B is higher than that on route A. Also, the mean value of the travel time on route B is higher than that on route A. However, the mean density on route A is higher than that on route B. This is due to the reason that dynamic drivers trend to select the short route A. We derive the traffic pattern to investigate the traffic behavior induced by the road length difference. Fig. 4 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.1 corresponding to Fig. 3. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. The regions with high density are indicated by dense color. The congestion is represented by a shade of color. The shade of color displays the stripe pattern. The region with black stripe represents the cluster of dynamic and static drivers, while the region with light and red stripe indicates the cluster of static drivers. The width of black stripe on route A is wider than that on route B. Dynamic drivers trend to select route A than route B because the travel time on route A is shorter than that on route B. When the dynamic drivers select route A, there are in time no dynamic drivers on route B. Then, the travel time on route A increases due to the high density. When route A is congested, the dynamic drivers on route A go to route B. In the result, the travel time on route B increases. The process is repeated. The travel times and densities oscillate. The traffic pattern on route A is definitely different from that on route B. Fig. 5(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. Fig. 5(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.2 Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the
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Fig. 5. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes at length ratio LB /LA = 1.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes at LB /LA = 1.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Plot of vehicular position versus time step at LB /LA = 1.2 corresponding to Fig. 5. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 6 is compared with Fig. 4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
mean density on routes A and B respectively. Fig. 5 is compared with Fig. 3. By changing the length ratio from LB /LA = 1.1 to LB /LA = 1.2, the travel times and densities on two routes change from Fig. 3 to Fig. 5. The travel time and density on each route oscillate with a period longer than that in Fig. 3. Fig. 6 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.2 corresponding to Fig. 5. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 6 is compared with Fig. 4. The width of black stripe is wider than that in Fig. 4 because more dynamic drivers go to route A. The density on route A is higher than that on route B. The density on route A is higher than that on route A in Figs. 3 and 4. Thus, the width of black stripe increases with increasing the length ratio. This is due to the reason that more dynamic drivers go to route A with increasing the length ratio. If the length ratio increases furthermore, the travel time, density, and traffic pattern change greatly. Fig. 7(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.4. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. Fig. 7(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.4. Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean density on routes A and B respectively. Fig. 7 is compared with Fig. 3 and 5. By changing the length ratio from LB /LA = 1.1, through LB /LA = 1.2, to LB /LA = 1.4, the travel times and densities on two routes change from Fig. 3, through Fig. 5, to Fig. 7. The travel time on route B does not vary highly but the travel time on route A fluctuates irregularly. Densities on routes A and B do not vary highly with time but fluctuate bit by bit with time. The fluctuation of the travel time on route A is due to the occurrence of spontaneous jam. Fig. 8 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.4 corresponding to Fig. 7. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 8 is compared with Fig. 6. All dynamic drivers go to
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Fig. 7. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes at length ratio LB /LA = 1.4. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes at LB /LA = 1.4. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 8. Plot of vehicular position versus time step at LB /LA = 1.4 corresponding to Fig. 7. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 8 is compared with Fig. 6. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
route A, there are no dynamic drivers on route B, and there are only static drivers on route B because the travel time on route B is always higher than that on route A. Such stripe patterns as Figs. 4 and 6 do not appear but the homogeneous patterns occur on routes A and B. Thus, the crossover phenomenon from the stripe pattern to the homogeneous pattern occurs with increasing the length ratio. We derive the relationship between the mean travel time and the length ratio. Also, we study the dependence of the mean density on the length ratio. Fig. 9(a) shows the plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean travel times on routes A and B. Fig. 9(b) shows the plots of mean density ⟨ρ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean densities on routes A and B. For LB /LA > 1, the mean travel time on route B is higher than that on route A and increases with increasing the length ratio. However, the mean density on route B is less than that on route A for LB /LA > 1. For LB /LA > 1, the mean density on route A increases with increasing the length ratio, while the mean density route B decreases with increasing the length ratio. The mean density on route A saturates near LB /LA = 1.4. The saturation is due to the upper limit of vehicular inflow. The crossover phenomenon occurs at the saturation point from the stripe pattern to the homogeneous pattern. 4. Simulation result with a bottleneck We study the effect of the bottleneck on the asymmetric two-route traffic system with real-time information. The combined effect of the bottleneck with the asymmetry will have the important effect on the route choice. We carry out simulation for the asymmetric two-route system with a bottleneck. The bottleneck is positioned at the center on route A. The parameter values of the two-route traffic system are the same as those in Section 3. The blocking probability representing the bottleneck’s strength is taken as pb = 0.2.
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Fig. 9. (a) Plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean travel times for routes A and B. (b) Plots of mean density ⟨ρ⟩ against length ratio LB /LA . Circles and squares indicate, respectively, the mean densities for routes A and B.
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Fig. 10. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.2 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.2 for blocking probability pb = 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
We study the traffic behavior in the asymmetric two-route system with a bottleneck on route A. We calculate the travel times and densities on routes A and B by varying the route length ratio. Fig. 10(a) shows the plots of dimensionless travel times Tt against time t on two routes at length ratio LB /LA = 1.2 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel times on routes A and B respectively. Fig. 10(b) shows the plots of densities ρ against time t on two routes at length ratio LB /LA = 1.2 for blocking probability pb = 0.2. Red and Black lines indicate densities on routes A and B respectively. Red and black dashed lines represent the mean densities on routes A and B respectively. Fig. 10 is compared with Fig. 5 without bottlenecks. By introducing the bottleneck into the asymmetric two routes, the travel times and densities on two routes change from Fig. 5 to Fig. 10. The bottleneck affects greatly the time-dependent behaviors of the travel time and density. By introducing the bottleneck, the travel times and densities vary with the shorter period than those in Fig. 5. This is due to the reason that the travel time increases faster with increasing inflow. Fig. 11 shows the plot of vehicular position versus time step at length ratio LB /LA = 1.2 corresponding to Fig. 10. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B. Vehicles with the dynamic and static drivers are indicated by black and red dots respectively. Fig. 11 is compared with Fig. 6 without bottlenecks. The width of black stripe is narrower than that in Fig. 6 because dynamic drivers change frequently their route. The traffic jam with a triangular shape is formed at the center on route A. The jam appears, disappears in time, and this is repeated. Thus, the traffic behavior changes greatly by the bottleneck’ effect. With increasing the length ratio from length ratio LB /LA = 1.2, through LB /LA = 1.4, and to LB /LA = 1.9, the travel time changes from Fig. 10(a), through Fig. 12(a), to Fig. 14(a). Also, the density changes from Fig. 10(b), through Fig. 12(b), to Fig. 14(b). The traffic pattern changes from Fig. 11, through Fig. 13, to Fig. 15 where the traffic patterns in Figs. 13 and 15 correspond to those in Figs. 12 and 14 respectively. The jam in Figs. 11 and 13 oscillates with time. The jam in Fig. 15
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Fig. 11. Plot of vehicular position versus time step at length ratio LB /LA = 1.2 for blocking probability (bottleneck’s strength) pb = 0.2, corresponding to Fig. 10. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B.
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Fig. 12. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.4 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.4 for blocking probability pb = 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 13. Plot of vehicular position versus time step at length ratio LB /LA = 1.4 for blocking probability (bottleneck’s strength) pb = 0.2, corresponding to Fig. 12. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B.
is not oscillating but is stationary. The jam in Fig. 15 extends to the entrance. With increasing length ratio, the traffic jam formed at the bottleneck changes from the oscillating jam to the stationary jam. Thus, the traffic behavior for the asymmetric two-route traffic system changes greatly by introducing both bottleneck and asymmetry.
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Fig. 14. (a) Plots of dimensionless travel times Tt against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.9 for blocking probability (bottleneck’s strength) pb = 0.2. Red and black lines indicate travel times on routes A and B respectively. Red and black dashed lines represent the mean travel time on routes A and B respectively. (b) Plots of densities ρ against time t on asymmetric two routes with a bottleneck at length ratio LB /LA = 1.9 for blocking probability pb = 0.2. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 15. Plot of vehicular position versus time step at length ratio LB /LA = 1.9 for blocking probability (bottleneck’s strength) pb = 0.2, corresponding to Fig. 14. Diagrams (a) and (b) represent, respectively, the traffic patterns (trajectories of vehicles) on routes A and B.
We derive the relationship between the mean travel time and the length ratio for the asymmetric two-route traffic system with a bottleneck. Also, we study the dependence of the mean density on the length ratio. Fig. 16(a) shows the plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA at blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean travel times on routes A and B. Fig. 16(b) shows the plots of mean density ⟨ρ⟩ against length ratio LB /LA blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean densities on routes A and B. The relationship between the mean travel time and the length ratio changes from Fig. 9(a) to Fig. 16(a) by introducing the bottleneck. The relationship between the mean density and the length ratio changes from Fig. 9(b) to Fig. 16(b). Thus, the mean travel time and the mean density change greatly by both bottleneck and the route-length difference. 5. Summary In real city traffic network, there are some routes with various road lengths. Generally, the travel time on a route depends on the road length and vehicular density. Drivers try to select the shortest route. However, the travel time may be high due to the road congestion even if the route is short. On the basis of the travel-time feedback, drivers try to choice the route with the shortest travel time. For the route choice of drivers, it is important and necessary to know the effect of the routelength difference on the travel time. We have extended the symmetric two-route traffic model proposed by Wahle et al. to take into account the asymmetric road length. The traffic behavior changes greatly by the competition between the short route and driver’s avoidance to the congestion. We have investigated where, when, and how the congestion occurs. We have clarified the effect of the route-length difference on the route choice in the asymmetric two-route traffic system with real-time information. We have derived the dependence of the mean travel time and mean density on the road length ratio. We have shown that the crossover phenomenon occurs from the stripe pattern to the homogeneous pattern with increasing the route-length ratio. Also, we have studied the combined effect of the route-length difference and bottleneck on the route
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Fig. 16. (a) Plots of mean travel time ⟨Tt ⟩ against length ratio LB /LA at blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean travel times on routes A and B. (b) Plots of mean density ⟨ρ⟩ against length ratio LB /LA blocking probability pb = 0.2. Circles and squares indicate, respectively, the mean densities on routes A and B.
choice. We have shown that the bottleneck has the important effect on the traffic behavior in the asymmetric two-route traffic system with real-time information. The study of the route-length difference and bottleneck effects on the route choice is the first and this study will be useful for the route choice in city traffic network. 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] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50]
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. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.X. Ge, S.Q. Dai, L.Y. Dong, Y. Xue, Phys. Rev. E 70 (2004) 066134. Z.P. Li, Y.C. Liu, Eur. Phys. J. B 53 (2006) 367. C. Chen, J. Chen, X. Guo, Physica A 389 (2010) 141. W.X. Zhu, Internat. J. Modern Phys. C 19 (2008) 727. A.K. Gupta, V.K. Katiyar, J. Phys. A 38 (2005) 4069. A.K. Gupta, V.K. Katiyar, Physica A 368 (2006) 551. A.K. Gupta, S. Sharma, Chin. Phys. B 19 (2010) 110503. A.K. Gupta, V.K. Katiyar, Physica A 371 (2006) 674. A.K. Gupta, P. Redhu, Physica A 392 (2013) 5622. G.H. Peng, X.H. Cai, C.Q. Liu, B.F. Cao, Phys. Lett. A 375 (2011) 2153. G.H. Peng, X.H. Cai, B.F. Cao, C.Q. Liu, Phys. Lett. A 375 (2011) 2823. G.H. Peng, F.Y. Nie, B.F. Cao, C.Q. Liu, Nonlinear Dynam. 67 (2012) 1811. G.H. Peng, D.H. Sun, Phys. Lett. A 374 (2010) 1694. G.H. Peng, X.H. Cai, C.Q. Liu, B.F. Cao, M.X. Tuo, Phys. Lett. A 375 (2011) 3973. G.H. Peng, R.J. Cheng, Physica A 392 (2013) 3563. T.Q. Tang, H.J. Huang, S.G. Zhao, H.Y. Shang, Phys. Lett. A 373 (2009) 2461. T.Q. Tang, H.J. Huang, G. Xu, Physica A 387 (2008) 6845. E. Brockfeld, R. Barlovic, A. Schadschneider, M. Schreckenberg, Phys. Rev. E 64 (2001) 056132. M. Sasaki, T. Nagatani, Physica A 325 (2002) 531. S. Kurata, T. Nagatani, Physica A 318 (2003) 537. K. Nagel, D.E. Wolf, P. Wagner, P. Simon, Phys. Rev. E 58 (1998) 1425. S. Lammer, D. Helbing, J. Stat. Mech. Theory Exp. (2008) P04019. M. Bando, K. Hasebe, A. Nakayama, Y. Sugiyama, Phys. Rev. E 51 (1995) 1035. B.A. Toledo, E. Cerda, J. Rogan, V. Munoz, C. Tenreino, R. Zarama, J.A. Valdivia, Phys. Rev. E 75 (2007) 189701. T. Nagatani, Physica A 377 (2007) 651. D. Helbing, B. Tilch, Phys. Rev. E 58 (1998) 133. R. Jiang, Q. Wu, Z. Zhu, Phys. Rev. E 64 (2001) 017101. H.X. Ge, R.J. Cheng, Z.P. Li, Physica A 387 (2008) 5239. T. Nagatani, S. Yonekura, Physica A 404 (2014) 171. N. Sugiyama, T. Nagatani, Physica A 392 (2013) 1848. A.K. Gupta, S. Sharma, Chin. Phys. B 21 (2012) 015201. A.K. Gupta, I. Dhiman, Internat. J. Modern Phys. C 25 (2014) 1450045. A.K. Gupta, P. Redhu, Commun. Nonlinear Sci. Simul. 19 (2014) 1600. A.K. Gupta, P. Redhu, Phys. Lett. A 377 (2013) 2027. A.K. Gupta, Internat. J. Modern Phys. C 25 (2013) 1350018. A.K. Gupta, I. Dhiman, Nonlinear Dynam. 79 (2015) 663. A.K. Gupta, S. Sharma, P. Redhu, Commun. Theor. Phys. 62 (2014) 393. J. Wahle, A. Lucia, C. Bazzan, F. Klugl, M. Schreckenberg, Physica A 287 (2000) 669. Y. Yokoya, Phys. Rev. E 69 (2004) 016121. C. Dong, X. Ma, B. Wang, X. Sun, Physica A 389 (2010) 3274. M. Fukui, K. Nishinari, Y. Yokoya, Y. Ishibashi, Physica A 388 (2009) 1207. K. Tobita, T. Nagatani, Physica A 391 (2012) 6137. Y. Hino, T. Nagatani, Physica A 395 (2014) 425.