Traffic jam and discontinuity induced by slowdown in two-stage optimal-velocity model

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Abstract

We study the traffic states and jams induced by a slowdown of vehicles in a single-lane highway. The two-stage optimal velocity model is used in which the optimal velocity function has two turning points. The fundamental (flow-density) diagrams are calculated. At low density, the flow (current) increases linearly with density, while it saturates at some values of intermediate density. When the flow saturates, the discontinuous front (stationary shock wave) appears before or within the section of slowdown. The values of saturated flow are determined by the extreme values of theoretical current curves. The relationship between the densities is derived before and after the discontinuity.

Introduction

Traffic flow is a kind of many-body system of strongly interacting vehicles [1], [2], [3], [4], [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], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27]. Traffic jams are classified into two kinds of jams: (1) spontaneous jam which propagates backward as the stop- and go-wave and (2) stationary jam which is induced by slowdown or blockage at a section of roadway. If sensitivity of driver is lower than a critical value, the spontaneous jam occurs. The jamming transition is very similar to the conventional phase transitions and critical phenomena [1], [26]. When the sensitivity is higher than the critical value, the spontaneous jam does not appear, while the stationary jam induced by slowdown occurs [1], [27].

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 speed limit and traffic lights for security and priority for a road [21], [23], [27]. Such speed limit as slowdown often induces traffic jams when the density of vehicles is high. One is interested in the structure and formation of traffic jams induced by slowdown. In the previous paper [27], we have studied the traffic jam induced by slowdown at a section of roadway. We have shown the following. When the 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 with forming a queue of slow vehicles. A discontinuous front appears at the end (edge) of traffic jam. The relationships between the headways and velocities before and after the discontinuity have been derived.

However, real traffic is very complex. The traffic jams induced by slowdown are also complex. In this paper, we extend the conventional optimal-velocity model to the two-stage optimal-velocity model in which the optimal-velocity function has two turning points. We study the traffic states and discontinuous front induced by the slowdown, by using the extended version of the optimal velocity model. 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, with a degree of slowdown, and by two-stage optimal-velocity function.

Section snippets

Model

We consider the vehicular traffic flowing on the single-lane roadway. Vehicles move 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

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), (3), (4) by using fourth-order Runge–Kutta method where the time interval is Δt=1128.

We carry out simulation by varying the initial headway, slowdown's velocity vs,max for 500 vehicles, sensitivity a=2.5 and maximal

Summary

We have investigated the occurrence of traffic jams in a single-lane highway with the slowdown section for high sensitivity, by using the two-stage optimal velocity model. We have presented the fundamental (flow-density) diagram. We have shown that the discontinuous front (stationary shock wave) appears when the flow saturates. We have derived the position which the discontinuous front appears. We have clarified that the characteristics of the discontinuous front depends highly on the local

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