- Email: [email protected]
Discontinuity at edge of traffic jam induced by slowdown
ARTICLE IN PRESS
Physica A 364 (2006) 464–472 www.elsevier.com/locate/physa
Discontinuity at edge of traffic jam induced by slowdown Ryoichi Nagai, Hirotoshi Hanaura, Katsunori Tanaka, Takashi Nagatani Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan Received 6 September 2005 Available online 19 October 2005
Abstract We study the traffic states and jams induced by a slowdown of vehicles in a single lane highway. We use an extended one of the optimal velocity model to take into account the slowdown in a section of highway. The fundamental (flow-density) diagram is calculated. The flow (current) increases linearly with density, saturates at a critical density, and then decreases with density. When the flow saturates, the discontinuous front (stationary shock wave) appears at the end of traffic jam which begins just before the section of slowdown. The position of discontinuous front moves to the upstream with increasing density. It is found that the relationship between the densities holds before and after the discontinuity, while the flow (current) keeps the saturated value. The region map of distinct jams is shown. r 2005 Elsevier B.V. All rights reserved. Keywords: Vehicular dynamics; Traffic state; Traffic jam; Phase transition; Many-particle system
1. Introduction Traffic flow is a kind of many-body system of strongly interacting vehicles. Transportation problems have attracted considerable attention in the field of physics [1–5]. Traffic jams are typical signature of the complex behavior of traffic flow. Traffic jams have been studied by several traffic models: car-following models, cellular automaton (CA) models, gas kinetic models, and hydrodynamic models [6–23]. Recent studies reveal physical phenomena such as the nonequilibrium phase transitions and the nonlinear waves [1–5]. It has been shown that the jamming transition is very similar to the conventional phase transitions and critical phenomena even if the traffic flow is a nonequilibrium system [1]. Mobility is nowadays one of the most significant ingredients of a modern society. The traffic accident often occurs in city traffic networks [22]. Also, traffic networks often exceed the capacity. The city traffic is controlled by traffic lights and speed limit for security and priority for a road [21,23]. Such speed limit as slowdown often induces traffic jams when a density of vehicles is high. One is interested in the structure and formation of traffic jams induced by slowdown. When a density of vehicles is low, vehicles move freely with no jams. If the density is higher than a critical value, the traffic jam is formed just before the section of slowdown. The speed of vehicles within the jam becomes lower than the speed limit of slowdown. The traffic jam ends Corresponding author. Fax: +81 53 478 1048.
E-mail address: [email protected] (T. Nagatani). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.09.055
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with forming a queue of slow vehicles. A discontinuous front appears at the end (edge) of traffic jam. Before and after the discontinuity, the traffic state changes abruptly. However, it is little known about the discontinuity. How does the traffic state change near the discontinuous front? How does the traffic flow change by introducing a slowdown? Does such relationship as Rankine–Hugoniot equations of shock wave hold for the discontinuity? The discontinuity induced by the slowdown have little been investigated by the use of the dynamic models. In this paper, we study the traffic states and discontinuous front induced by the slowdown, by using the extended version of the optimal velocity model to take into account the slowdown. We clarify the dynamical states of traffic and the characteristic of discontinuous front. We present the fundamental diagram in the traffic flow including the slowdown. We show how the traffic state changes with increasing density of vehicles and with a degree of slowdown. 2. Model We consider the traffic of vehicles flowing on the single-lane roadway. Vehicles flow with no passing on the single-lane roadway under periodic boundary condition. We assume that vehicles are forced to slow down when they enter into the section of the slowdown. Fig. 1 shows the schematic illustration of the traffic model for the single-lane highway with the section of slowdown. Vehicles move with low speed in the section of slowdown, while they move with the normal velocity except for the section of slowdown. We apply the optimal velocity model to the traffic flow. The optimal velocity model is described by the following equation of motion of vehicle i: d2 xi dxi ¼ a V ðDx Þ , (1) i dt2 dt where V ðDxi Þ is the optimal velocity, xi ðtÞ is the position of vehicle i at time t, Dxi ðtÞð¼ xiþ1 ðtÞ xi ðtÞÞ is the headway of vehicle i at time t, and a is the sensitivity (the inverse of the delay time). A driver adjusts the vehicle speed to approach the optimal velocity determined by the observed headway. The sensitivity a allows for the time lag t ¼ 1=a that it takes the vehicle speed to reach the optimal velocity when the traffic is varying. Generally, it is necessary that the optimal velocity function has the following properties: it is a monotonically increasing function and it has an upper bound (maximal velocity). In the region of normal speed, the optimal velocity of vehicles is given by vmax V ðDxi Þ ¼ ½tanh ðDxi xc Þ þ tanh ðxc Þ, (2) 2 where vmax is the maximal velocity of vehicles and xc is the safety distance of vehicles. In the section of slowdown, vehicles move with forced low speed. The speed should be lower than the speed limit of slowdown. When vehicles enter into the section of slowdown, the optimal velocity is given by vs V ðDxi Þ ¼ ½tanh ðDxi xcs Þ þ tanh ðxcs Þ, (3) 2 where vs is the speed limit of slowdown, vs ovmax , and xcs is the safety distance in the section of slowdown.
3L/4
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Fig. 1. Schematic illustration of the traffic model for the single-lane highway with the section of slowdown. Vehicles move with low speed in the section of slowdown, while they move with the normal velocity except for the section of slowdown.
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Thus, when the density of vehicles is very low, vehicles move at the maximal velocity vmax except for the section of slowdown, while they move at the forced low speed vs in the section of slowdown. With increasing density, vehicles interact with each other. The dynamics is determined by Eqs. (1)–(3). Then, various dynamic states of traffic appear and traffic jams may occur. We study the dynamic states and traffic jams in the traffic flow described by model in Fig. 1. 3. Simulation and result We perform computer simulation for the traffic model shown in Fig. 1. We simulate the traffic flow under the periodic boundary condition. The simulation is performed until the traffic flow reaches a steady state. We solve numerically Eq. (1) with optimal velocity functions (2) and (3) by using fourth-order Runge–Kutta method where the time interval is Dt ¼ 1=128. We carry out simulation by varying the initial headway, sensitivity, and velocity ratio vs =vmax for 200 and 2000 vehicles, safety distances xc ¼ xcs ¼ 4:0, and maximal velocity vmax ¼ 2:0. Initially, we put all vehicles on the single-lane highway with the same headway Dxint . The density r is given by the inverse of the headway. The length L of highway varies with the initial headway. The section of slowdown is set on the downstream position L=4 of the highway. Fig. 2(a)–(d) show the plots of traffic currents against density for (a) sensitivity a ¼ 2:0, (b) a ¼ 1:3, (c) a ¼ 1:0, and (d) a ¼ 0:7, where vmax ¼ 2:0, vs ¼ 1:0, and 200 vehicles. The traffic current is obtained by averaging the current from t ¼ 2000 to 5000. The open circles indicate the traffic current with no slowdown. 0.5
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Fig. 2. Plots of traffic currents against density for (a) sensitivity a ¼ 2:0, (b) a ¼ 1:3, (c) a ¼ 1:0, and (d) a ¼ 0:7, where vmax ¼ 2:0, vs ¼ 1:0, and 200 vehicles. The open circles indicate the traffic current with no slowdown. The open triangles indicate the traffic current for velocity ratio vs =vmax ¼ 0:5 of the slowdown. For comparison, the traffic currents for two cases vmax ¼ 2:0 and vmax ¼ 1:0 without traffic jams are shown by two solid lines.
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The open triangles indicate the traffic current for velocity ratio vs =vmax ¼ 0:5 of the slowdown. For comparison, the traffic currents for two cases vmax ¼ 2:0 and vmax ¼ 1:0 without traffic jams are shown by two solid lines. The current is called as the theoretical current curve. It is given by J ¼ V ðDxÞ=Dx, where Dx is the average value of headway and V ðDxÞ is the optimal velocity. The case of no slowdown corresponds to the conventional traffic flow. When sensitivity is higher than critical value 2.0, no traffic jams occur and the current agrees with the theoretical current curve (see Fig. 2(a)). If sensitivity is lower than the critical value, traffic jams occur and current deviates from the theoretical current curve in the region of the density at which traffic jams appear (see Fig. 2(b)–(d)). In the case of slowdown, the current is shown by triangles. The current increases linearly with density at low density, but is lower than the current of no slowdown. Then, the current saturates at the first critical density and keeps a constant value until the second critical density. When the density is higher than the second critical density, the current decreases with increasing density. The first critical density does not depend on the sensitivity but the second critical density depends highly on the sensitivity. The value of second critical density increases with decreasing sensitivity. Also, the saturated value of current is consistent with the maximal value of the theoretical current curve for vmax ¼ 1:0. In the region of saturated current, a traffic jam appears just before the section of slowdown. We study the headway and velocity profiles for the traffic flow with jam induced by the slowdown. Fig. 3(a) shows the plot of headway against position of vehicles for sensitivity a ¼ 2:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. Fig. 3(b) shows the plot of velocity against position of vehicles, corresponding to the headway profile in Fig. 3(a). The section of slowdown begins at x ¼ 675 and ends at
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Fig. 3. (a) Plot of headway against position of vehicles for sensitivity a ¼ 2:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. (b) Plot of velocity against position of vehicles, corresponding to the headway profile. The traffic jam begins just before the section of slowdown and ends at x ¼ 510. The discontinuous front appears just after the edge of traffic jam.
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x ¼ 900. The traffic jam begins just before the section of slowdown and ends at x ¼ 510. The discontinuous front appears just after the edge of traffic jam. Fig. 4(a) shows the plot of headway against position of vehicles for sensitivity a ¼ 1:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. Fig. 4(b) shows the plot of velocity against position of vehicles, corresponding to the headway profile in Fig. 4(a). The oscillatory jam begins just before the section of slowdown and ends at x ¼ 489. The discontinuous front appears just after the edge of traffic jam. When the value of sensitivity is low, the oscillatory jam occurs. We study the characteristic properties of the discontinuity. We derive, numerically, the headways before and after discontinuous front by varying velocity ratio vs =vmax . Fig. 5(a) shows the plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 2:0 where the jam is not oscillatory but uniform. Open circle indicates the value of headway just before the continuous front. Open square indicates the value of headway just after the continuous front. Open triangle indicates the value of average headway within the section of slowdown. Fig. 5(b) shows the plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 1:0 where the jam is uniform or oscillatory. Full square indicates the mean value of headway just after the discontinuous front when the oscillatory jam occurs. When the velocity ratio is not low, the oscillatory jam appears. The solid lines indicate the headways obtained from the theoretical analysis later. We derive the headways before and after discontinuous front analytically. Fig. 6 shows two theoretical current curves for vmax ¼ 2:0 and vmax ¼ 1:0. The solid and dotted lines represent, respectively, the theoretical curves for vmax ¼ 2:0 and vmax ¼ 1:0. At the steady state, the traffic current is uniform over the highway irrespective of slowdown section. When the traffic current saturates, the traffic current throughout the section 10
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Fig. 4. (a) Plot of headway against position of vehicles for sensitivity a ¼ 1:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. (b) Plot of velocity against position of vehicles, corresponding to the headway profile. The oscillatory jam begins just before the section of slowdown and ends at x ¼ 489. The discontinuous front appears just after the edge of traffic jam.
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Fig. 5. (a) Plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 2:0. Open circle indicates the value of headway just before the continuous front. Open square indicates the value of headway just after the continuous front. Open triangle indicates the value of average headway within the section of slowdown. The solid lines indicate the headways obtained from the theoretical analysis. (b) Plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 1:0. Full square indicates the mean value of headway just after the discontinuous front when the oscillatory jam occurs.
of slowdown becomes maximal value of the theoretical current curve of vmax ¼ 1:0. If no traffic jam occurs, the maximal current is given by the maximum point of the theoretical current curve with optimal velocity function (3). The maximum point is indicated by the full triangle in Fig. 6. Because the traffic current is uniform over the highway, the crossing points of solid curve (theoretical current curve of vmax ¼ 2:0) with the horizontal line on the maximal point of full triangle give such densities that are allowed to take on the highway except for the slowdown section. The values of densities are indicated by full circle and square. Thus, one obtains the headways (the inverse of densities) before and after the discontinuous front for various values of velocity ratio. The headways obtained from the theoretical analysis are shown by the solid lines in Fig. 5. For a ¼ 2:0 in Fig. 5(a), the theoretical result agrees with the simulation result. For a ¼ 1:0 in Fig. 5(b), the theoretical result agrees with the simulation result when velocity ratio vs =vmax is low. However, when velocity ratio is not low and traffic jam is oscillatory, the simulation value deviates a little from the theoretical value. We derive the region map of distinct jams by studying the distinct states of traffic. We find three distinct jams: (1) the uniform jam, (2) the oscillatory jam, and (3) the propagating jam. The uniform jam is stationary in the highway and the headway and velocity take constant values within the jam. The headway and velocity profiles are shown in Fig. 3. In the oscillatory jam, the headway and velocity oscillate. The jam propagates from the front of slowdown section to the discontinuous front. The headway and velocity profiles are shown in Fig. 4. Propagating jam (3) is shown in Fig. 7. Fig. 7(a) shows the trajectories of all vehicles where sensitivity a ¼ 0:7, 200 vehicles, average headway Dx0 ¼ 4:0, and velocity ratio vs =vmax ¼ 0:9. Fig. 7(b) shows the headway profile at t ¼ 10 000. A single pulse jam propagates backward over the whole highway.
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Fig. 6. Two theoretical current curves for vmax ¼ 2:0 and vmax ¼ 1:0. The solid and dotted lines represent, respectively, the theoretical curves for vmax ¼ 2:0 and vmax ¼ 1:0. The maximum point of dotted curve is indicated by the full triangle. The crossing points of solid curve (theoretical current curve of vmax ¼ 2:0) with the horizontal line through the maximal point of full triangle are indicated by full circle and square. 9500
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Fig. 7. A single pulse jam propagating backward over the whole highway where sensitivity a ¼ 0:7, 200 vehicles, average headway Dx0 ¼ 4:0, and velocity ratio vs =vmax ¼ 0:9. (a) Trajectories of all vehicles. (b) Headway profile at t ¼ 10 000.
Fig. 8 shows the region map obtained for 2000 vehicles and Dx0 ¼ 4:0. Here, cross points represent uniform jam (1). Circles indicate oscillatory jam (2). Triangles represent propagating jam (3). The uniform jam appears for high values of sensitivity, while the oscillatory jam occurs for low values of sensitivity. When the sensitivity is low and the velocity ratio is high, the propagating jam appears only in the narrow region of Fig. 8. The
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Fig. 8. Region map obtained for 2000 vehicles and Dx0 ¼ 4:0. Here, cross points represent uniform jam. Circles indicate oscillatory jam. Triangles represent propagating jam. The curve indicated by the dotted line represents the neutral stability line.
Traffic jam length ratio
1.0 0.8 0.6 a = 0.7 a = 1.0 a = 1.3 a = 2.0
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Density Fig. 9. Plots of jam length ratio against density for sensitivities a ¼ 2:0; 1:3; 1:0; 0:7, where the number of vehicles is 200.
curve indicated by the dotted line represents the neutral stability line. We derived the neutral stability curve by studying the linear stability for the uniform traffic jam shown by the full square in Fig. 6. The neutral stability line is almost consistent with the boundary between the uniform and oscillatory jams. The boundary obtained from the simulation of 200 vehicles gives low values of sensitivity, but the boundary approaches to the neutral stability line with increasing number of vehicles. We study how the length of traffic jam varies with density. Fig. 9 shows the plots of jam length ratio against density for sensitivities a ¼ 2:0; 1:3; 1:0; 0:7 where the number of vehicles is 200. The jam length ratio is defined by the jam length divided by the length of normal speed section. When the jam reaches the position x ¼ 0(starting point of highway), the value of ratio takes one. Under a constant value of sensitivity, the jam length increases linearly with density. When the uniform jam appears (a ¼ 2:0; 1:3), the jam length does not depend on the sensitivity. However, the jam length depends highly on the sensitivity for the oscillatory jam. Until the traffic jam reaches the position x ¼ 0, the current saturates and takes a constant value. If the traffic jam passes over x ¼ 0, the current decreases and becomes lower than the saturated current. Such density that the jam length becomes one is consistent with such value that the traffic current begins to decrease from the saturated current in Fig. 2. 4. Summary We have investigated the traffic jams in a single-lane highway with the slowdown section by using an extended one of the optimal velocity model. We have shown that the discontinuous front (stationary shock
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wave) appears at the end of traffic jam and the flow then saturates. We have clarified the characteristic properties of discontinuous front appearing at the edge of traffic jam. We have presented the fundamental (flow-density) diagram. We have shown that the fundamental diagram depends highly on sensitivity and slowdown velocity. We have found that three distinct jams occur by varying sensitivity and velocity ratio. We have derived the region map for the distinct jams. We have shown that the boundary between distinct jams is consistent with the neutral stability curve obtained from the linear stability analysis. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
T. Nagatani, Rep. Prog. Phys. 65 (2002) 1331. D. Helbing, Rev. Mod. Phys. 73 (2001) 1067. D. Chowdhury, L. Santen, A. Schadscheider, Phys. Rep. 329 (2000) 199. B.S. Kerner, Networks Spatial Econ. 1 (2001) 35. D. Helbing, H.J. Herrmann, M. Schreckenberg, D.E. Wolf (Eds.), Traffic and Granular Flow ‘99, Springer, Heidelberg, 2000. K. Nagel, M. Schreckenberg, J. Phys. I France 2 (1992) 2221. E. Ben-Naim, P.L. Krapivsky, S. Redner, Phys. Rev. E 50 (1994) 822. E. Tomer, L. Safonov, S. Havlin, Phys. Rev. Lett. 84 (2000) 382. M. Treiber, A. Hennecke, D. Helbing, Phys. Rev. E 62 (2000) 1805. H.K. Lee, H.-W. Lee, D. Kim, Phys. Rev. E 64 (2001) 056126. I. Lubashevsky, S. Kalenkov, R. Mahnke, Phys. Rev. E 65 (2002) 036140. I. Lubashevsky, R. Mahnke, P. Wagner, S. Kalenkov, Phys. Rev. E 66 (2002) 016117. F. Weifeng, Y. Lizhong, F. Weicheng, Physica A 321 (2003) 633. S. Maniccam, Physica A 321 (2003) 653. O.J. O’loan, M.R. Evans, M.E. Cates, Phys. Rev. E 58 (1998) 1404. T. Nagatani, Phys. Rev. E 63 (2001) 036116. H.J.C. Huijberts, Physica A 308 (2002) 489. T. Nagatani, Phys. Rev. E 66 (2002) 046103. L.A. Safonov, E. Tomer, V.V. Strygin, Y. Ashkenazy, S. Havlin, Chaos 12 (2002) 1006. 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.
Physica A 364 (2006) 464–472 www.elsevier.com/locate/physa
Discontinuity at edge of traffic jam induced by slowdown Ryoichi Nagai, Hirotoshi Hanaura, Katsunori Tanaka, Takashi Nagatani Department of Mechanical Engineering, Division of Thermal Science, Shizuoka University, Hamamatsu 432-8561, Japan Received 6 September 2005 Available online 19 October 2005
Abstract We study the traffic states and jams induced by a slowdown of vehicles in a single lane highway. We use an extended one of the optimal velocity model to take into account the slowdown in a section of highway. The fundamental (flow-density) diagram is calculated. The flow (current) increases linearly with density, saturates at a critical density, and then decreases with density. When the flow saturates, the discontinuous front (stationary shock wave) appears at the end of traffic jam which begins just before the section of slowdown. The position of discontinuous front moves to the upstream with increasing density. It is found that the relationship between the densities holds before and after the discontinuity, while the flow (current) keeps the saturated value. The region map of distinct jams is shown. r 2005 Elsevier B.V. All rights reserved. Keywords: Vehicular dynamics; Traffic state; Traffic jam; Phase transition; Many-particle system
1. Introduction Traffic flow is a kind of many-body system of strongly interacting vehicles. Transportation problems have attracted considerable attention in the field of physics [1–5]. Traffic jams are typical signature of the complex behavior of traffic flow. Traffic jams have been studied by several traffic models: car-following models, cellular automaton (CA) models, gas kinetic models, and hydrodynamic models [6–23]. Recent studies reveal physical phenomena such as the nonequilibrium phase transitions and the nonlinear waves [1–5]. It has been shown that the jamming transition is very similar to the conventional phase transitions and critical phenomena even if the traffic flow is a nonequilibrium system [1]. Mobility is nowadays one of the most significant ingredients of a modern society. The traffic accident often occurs in city traffic networks [22]. Also, traffic networks often exceed the capacity. The city traffic is controlled by traffic lights and speed limit for security and priority for a road [21,23]. Such speed limit as slowdown often induces traffic jams when a density of vehicles is high. One is interested in the structure and formation of traffic jams induced by slowdown. When a density of vehicles is low, vehicles move freely with no jams. If the density is higher than a critical value, the traffic jam is formed just before the section of slowdown. The speed of vehicles within the jam becomes lower than the speed limit of slowdown. The traffic jam ends Corresponding author. Fax: +81 53 478 1048.
E-mail address: [email protected] (T. Nagatani). 0378-4371/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physa.2005.09.055
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with forming a queue of slow vehicles. A discontinuous front appears at the end (edge) of traffic jam. Before and after the discontinuity, the traffic state changes abruptly. However, it is little known about the discontinuity. How does the traffic state change near the discontinuous front? How does the traffic flow change by introducing a slowdown? Does such relationship as Rankine–Hugoniot equations of shock wave hold for the discontinuity? The discontinuity induced by the slowdown have little been investigated by the use of the dynamic models. In this paper, we study the traffic states and discontinuous front induced by the slowdown, by using the extended version of the optimal velocity model to take into account the slowdown. We clarify the dynamical states of traffic and the characteristic of discontinuous front. We present the fundamental diagram in the traffic flow including the slowdown. We show how the traffic state changes with increasing density of vehicles and with a degree of slowdown. 2. Model We consider the traffic of vehicles flowing on the single-lane roadway. Vehicles flow with no passing on the single-lane roadway under periodic boundary condition. We assume that vehicles are forced to slow down when they enter into the section of the slowdown. Fig. 1 shows the schematic illustration of the traffic model for the single-lane highway with the section of slowdown. Vehicles move with low speed in the section of slowdown, while they move with the normal velocity except for the section of slowdown. We apply the optimal velocity model to the traffic flow. The optimal velocity model is described by the following equation of motion of vehicle i: d2 xi dxi ¼ a V ðDx Þ , (1) i dt2 dt where V ðDxi Þ is the optimal velocity, xi ðtÞ is the position of vehicle i at time t, Dxi ðtÞð¼ xiþ1 ðtÞ xi ðtÞÞ is the headway of vehicle i at time t, and a is the sensitivity (the inverse of the delay time). A driver adjusts the vehicle speed to approach the optimal velocity determined by the observed headway. The sensitivity a allows for the time lag t ¼ 1=a that it takes the vehicle speed to reach the optimal velocity when the traffic is varying. Generally, it is necessary that the optimal velocity function has the following properties: it is a monotonically increasing function and it has an upper bound (maximal velocity). In the region of normal speed, the optimal velocity of vehicles is given by vmax V ðDxi Þ ¼ ½tanh ðDxi xc Þ þ tanh ðxc Þ, (2) 2 where vmax is the maximal velocity of vehicles and xc is the safety distance of vehicles. In the section of slowdown, vehicles move with forced low speed. The speed should be lower than the speed limit of slowdown. When vehicles enter into the section of slowdown, the optimal velocity is given by vs V ðDxi Þ ¼ ½tanh ðDxi xcs Þ þ tanh ðxcs Þ, (3) 2 where vs is the speed limit of slowdown, vs ovmax , and xcs is the safety distance in the section of slowdown.
3L/4
L/4
Normal speed
Slow down
Fig. 1. Schematic illustration of the traffic model for the single-lane highway with the section of slowdown. Vehicles move with low speed in the section of slowdown, while they move with the normal velocity except for the section of slowdown.
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Thus, when the density of vehicles is very low, vehicles move at the maximal velocity vmax except for the section of slowdown, while they move at the forced low speed vs in the section of slowdown. With increasing density, vehicles interact with each other. The dynamics is determined by Eqs. (1)–(3). Then, various dynamic states of traffic appear and traffic jams may occur. We study the dynamic states and traffic jams in the traffic flow described by model in Fig. 1. 3. Simulation and result We perform computer simulation for the traffic model shown in Fig. 1. We simulate the traffic flow under the periodic boundary condition. The simulation is performed until the traffic flow reaches a steady state. We solve numerically Eq. (1) with optimal velocity functions (2) and (3) by using fourth-order Runge–Kutta method where the time interval is Dt ¼ 1=128. We carry out simulation by varying the initial headway, sensitivity, and velocity ratio vs =vmax for 200 and 2000 vehicles, safety distances xc ¼ xcs ¼ 4:0, and maximal velocity vmax ¼ 2:0. Initially, we put all vehicles on the single-lane highway with the same headway Dxint . The density r is given by the inverse of the headway. The length L of highway varies with the initial headway. The section of slowdown is set on the downstream position L=4 of the highway. Fig. 2(a)–(d) show the plots of traffic currents against density for (a) sensitivity a ¼ 2:0, (b) a ¼ 1:3, (c) a ¼ 1:0, and (d) a ¼ 0:7, where vmax ¼ 2:0, vs ¼ 1:0, and 200 vehicles. The traffic current is obtained by averaging the current from t ¼ 2000 to 5000. The open circles indicate the traffic current with no slowdown. 0.5
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Fig. 2. Plots of traffic currents against density for (a) sensitivity a ¼ 2:0, (b) a ¼ 1:3, (c) a ¼ 1:0, and (d) a ¼ 0:7, where vmax ¼ 2:0, vs ¼ 1:0, and 200 vehicles. The open circles indicate the traffic current with no slowdown. The open triangles indicate the traffic current for velocity ratio vs =vmax ¼ 0:5 of the slowdown. For comparison, the traffic currents for two cases vmax ¼ 2:0 and vmax ¼ 1:0 without traffic jams are shown by two solid lines.
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The open triangles indicate the traffic current for velocity ratio vs =vmax ¼ 0:5 of the slowdown. For comparison, the traffic currents for two cases vmax ¼ 2:0 and vmax ¼ 1:0 without traffic jams are shown by two solid lines. The current is called as the theoretical current curve. It is given by J ¼ V ðDxÞ=Dx, where Dx is the average value of headway and V ðDxÞ is the optimal velocity. The case of no slowdown corresponds to the conventional traffic flow. When sensitivity is higher than critical value 2.0, no traffic jams occur and the current agrees with the theoretical current curve (see Fig. 2(a)). If sensitivity is lower than the critical value, traffic jams occur and current deviates from the theoretical current curve in the region of the density at which traffic jams appear (see Fig. 2(b)–(d)). In the case of slowdown, the current is shown by triangles. The current increases linearly with density at low density, but is lower than the current of no slowdown. Then, the current saturates at the first critical density and keeps a constant value until the second critical density. When the density is higher than the second critical density, the current decreases with increasing density. The first critical density does not depend on the sensitivity but the second critical density depends highly on the sensitivity. The value of second critical density increases with decreasing sensitivity. Also, the saturated value of current is consistent with the maximal value of the theoretical current curve for vmax ¼ 1:0. In the region of saturated current, a traffic jam appears just before the section of slowdown. We study the headway and velocity profiles for the traffic flow with jam induced by the slowdown. Fig. 3(a) shows the plot of headway against position of vehicles for sensitivity a ¼ 2:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. Fig. 3(b) shows the plot of velocity against position of vehicles, corresponding to the headway profile in Fig. 3(a). The section of slowdown begins at x ¼ 675 and ends at
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Fig. 3. (a) Plot of headway against position of vehicles for sensitivity a ¼ 2:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. (b) Plot of velocity against position of vehicles, corresponding to the headway profile. The traffic jam begins just before the section of slowdown and ends at x ¼ 510. The discontinuous front appears just after the edge of traffic jam.
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x ¼ 900. The traffic jam begins just before the section of slowdown and ends at x ¼ 510. The discontinuous front appears just after the edge of traffic jam. Fig. 4(a) shows the plot of headway against position of vehicles for sensitivity a ¼ 1:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. Fig. 4(b) shows the plot of velocity against position of vehicles, corresponding to the headway profile in Fig. 4(a). The oscillatory jam begins just before the section of slowdown and ends at x ¼ 489. The discontinuous front appears just after the edge of traffic jam. When the value of sensitivity is low, the oscillatory jam occurs. We study the characteristic properties of the discontinuity. We derive, numerically, the headways before and after discontinuous front by varying velocity ratio vs =vmax . Fig. 5(a) shows the plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 2:0 where the jam is not oscillatory but uniform. Open circle indicates the value of headway just before the continuous front. Open square indicates the value of headway just after the continuous front. Open triangle indicates the value of average headway within the section of slowdown. Fig. 5(b) shows the plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 1:0 where the jam is uniform or oscillatory. Full square indicates the mean value of headway just after the discontinuous front when the oscillatory jam occurs. When the velocity ratio is not low, the oscillatory jam appears. The solid lines indicate the headways obtained from the theoretical analysis later. We derive the headways before and after discontinuous front analytically. Fig. 6 shows two theoretical current curves for vmax ¼ 2:0 and vmax ¼ 1:0. The solid and dotted lines represent, respectively, the theoretical curves for vmax ¼ 2:0 and vmax ¼ 1:0. At the steady state, the traffic current is uniform over the highway irrespective of slowdown section. When the traffic current saturates, the traffic current throughout the section 10
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Fig. 4. (a) Plot of headway against position of vehicles for sensitivity a ¼ 1:0, average(initial) headway Dx0 ¼ 4:5, velocity ratio vs =vmax ¼ 0:8, and 200 vehicles. (b) Plot of velocity against position of vehicles, corresponding to the headway profile. The oscillatory jam begins just before the section of slowdown and ends at x ¼ 489. The discontinuous front appears just after the edge of traffic jam.
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Fig. 5. (a) Plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 2:0. Open circle indicates the value of headway just before the continuous front. Open square indicates the value of headway just after the continuous front. Open triangle indicates the value of average headway within the section of slowdown. The solid lines indicate the headways obtained from the theoretical analysis. (b) Plot of headways before and after the discontinuous front against the velocity ratio for sensitivity a ¼ 1:0. Full square indicates the mean value of headway just after the discontinuous front when the oscillatory jam occurs.
of slowdown becomes maximal value of the theoretical current curve of vmax ¼ 1:0. If no traffic jam occurs, the maximal current is given by the maximum point of the theoretical current curve with optimal velocity function (3). The maximum point is indicated by the full triangle in Fig. 6. Because the traffic current is uniform over the highway, the crossing points of solid curve (theoretical current curve of vmax ¼ 2:0) with the horizontal line on the maximal point of full triangle give such densities that are allowed to take on the highway except for the slowdown section. The values of densities are indicated by full circle and square. Thus, one obtains the headways (the inverse of densities) before and after the discontinuous front for various values of velocity ratio. The headways obtained from the theoretical analysis are shown by the solid lines in Fig. 5. For a ¼ 2:0 in Fig. 5(a), the theoretical result agrees with the simulation result. For a ¼ 1:0 in Fig. 5(b), the theoretical result agrees with the simulation result when velocity ratio vs =vmax is low. However, when velocity ratio is not low and traffic jam is oscillatory, the simulation value deviates a little from the theoretical value. We derive the region map of distinct jams by studying the distinct states of traffic. We find three distinct jams: (1) the uniform jam, (2) the oscillatory jam, and (3) the propagating jam. The uniform jam is stationary in the highway and the headway and velocity take constant values within the jam. The headway and velocity profiles are shown in Fig. 3. In the oscillatory jam, the headway and velocity oscillate. The jam propagates from the front of slowdown section to the discontinuous front. The headway and velocity profiles are shown in Fig. 4. Propagating jam (3) is shown in Fig. 7. Fig. 7(a) shows the trajectories of all vehicles where sensitivity a ¼ 0:7, 200 vehicles, average headway Dx0 ¼ 4:0, and velocity ratio vs =vmax ¼ 0:9. Fig. 7(b) shows the headway profile at t ¼ 10 000. A single pulse jam propagates backward over the whole highway.
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Fig. 6. Two theoretical current curves for vmax ¼ 2:0 and vmax ¼ 1:0. The solid and dotted lines represent, respectively, the theoretical curves for vmax ¼ 2:0 and vmax ¼ 1:0. The maximum point of dotted curve is indicated by the full triangle. The crossing points of solid curve (theoretical current curve of vmax ¼ 2:0) with the horizontal line through the maximal point of full triangle are indicated by full circle and square. 9500
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Fig. 7. A single pulse jam propagating backward over the whole highway where sensitivity a ¼ 0:7, 200 vehicles, average headway Dx0 ¼ 4:0, and velocity ratio vs =vmax ¼ 0:9. (a) Trajectories of all vehicles. (b) Headway profile at t ¼ 10 000.
Fig. 8 shows the region map obtained for 2000 vehicles and Dx0 ¼ 4:0. Here, cross points represent uniform jam (1). Circles indicate oscillatory jam (2). Triangles represent propagating jam (3). The uniform jam appears for high values of sensitivity, while the oscillatory jam occurs for low values of sensitivity. When the sensitivity is low and the velocity ratio is high, the propagating jam appears only in the narrow region of Fig. 8. The
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Fig. 8. Region map obtained for 2000 vehicles and Dx0 ¼ 4:0. Here, cross points represent uniform jam. Circles indicate oscillatory jam. Triangles represent propagating jam. The curve indicated by the dotted line represents the neutral stability line.
Traffic jam length ratio
1.0 0.8 0.6 a = 0.7 a = 1.0 a = 1.3 a = 2.0
0.4 0.2 0.0 0.1
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Density Fig. 9. Plots of jam length ratio against density for sensitivities a ¼ 2:0; 1:3; 1:0; 0:7, where the number of vehicles is 200.
curve indicated by the dotted line represents the neutral stability line. We derived the neutral stability curve by studying the linear stability for the uniform traffic jam shown by the full square in Fig. 6. The neutral stability line is almost consistent with the boundary between the uniform and oscillatory jams. The boundary obtained from the simulation of 200 vehicles gives low values of sensitivity, but the boundary approaches to the neutral stability line with increasing number of vehicles. We study how the length of traffic jam varies with density. Fig. 9 shows the plots of jam length ratio against density for sensitivities a ¼ 2:0; 1:3; 1:0; 0:7 where the number of vehicles is 200. The jam length ratio is defined by the jam length divided by the length of normal speed section. When the jam reaches the position x ¼ 0(starting point of highway), the value of ratio takes one. Under a constant value of sensitivity, the jam length increases linearly with density. When the uniform jam appears (a ¼ 2:0; 1:3), the jam length does not depend on the sensitivity. However, the jam length depends highly on the sensitivity for the oscillatory jam. Until the traffic jam reaches the position x ¼ 0, the current saturates and takes a constant value. If the traffic jam passes over x ¼ 0, the current decreases and becomes lower than the saturated current. Such density that the jam length becomes one is consistent with such value that the traffic current begins to decrease from the saturated current in Fig. 2. 4. Summary We have investigated the traffic jams in a single-lane highway with the slowdown section by using an extended one of the optimal velocity model. We have shown that the discontinuous front (stationary shock
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wave) appears at the end of traffic jam and the flow then saturates. We have clarified the characteristic properties of discontinuous front appearing at the edge of traffic jam. We have presented the fundamental (flow-density) diagram. We have shown that the fundamental diagram depends highly on sensitivity and slowdown velocity. We have found that three distinct jams occur by varying sensitivity and velocity ratio. We have derived the region map for the distinct jams. We have shown that the boundary between distinct jams is consistent with the neutral stability curve obtained from the linear stability analysis. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23]
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