Realizing Wardrop equilibria with real-time traffic information

A Wardrop equilibrium for multiple routes from the same origin to the same destination requires equal travel time on each path used. With the advent of real-time traffic data regarding travel times on alternative routes, it becomes important to analyze how best to use the information provided to drivers. In particular, can a Wardrop equilibrium, which is a desired state, be realized? Simulations using a realistic traffic model (the three-phase model) on a two-route example are presented to answer this question. One route (the main line) is a two-lane highway with a stalled vehicle in the right lane and the other route is a low-speed bypass. For a critical incoming flow of vehicles, a phase transition between free flow and congested flow near the stalled vehicle is observed, making this a challenging example. In the first scenario, drivers choose routes selfishly on the basis of current travel times. The result is strong oscillations in travel time because of the inherent delay in the information provided. The second scenario involves a hypothetical control system that limits the number of vehicles on the main line to prevent the free-flow to congested-flow phase transition by diverting sufficient flow to the bypass. The resulting steady state is neither a Wardrop equilibrium nor a system optimum, but an intermediate state in which the main-line travel time is less than on the bypass but the average for all vehicles is close to a minimum. In a third scenario, anticipation is used as a driver-advice system to provide a fair indicator of which route to take. Prediction is based on real-time data comparing the number of vehicles on the main line at the time a vehicle leaves the origin to the actual travel time when it reaches the destination. Steady states that approximate Wardrop equilibria, or at least as close to them as can be expected, are obtained. This approach is also applied to an example with a low-speed boundary condition imposed at the destination in place of a stalled vehicle. The steady state flow approaches a Wardrop equilibrium because there is no abrupt change in travel time due to a phase transition.     

Explaining traffic patterns at on-ramp vicinity by a driver perception model in the framework of three-phase traffic theory

In traffic system, driving behaviors change with the surrounding traffic perceived by drivers, resulting in the complex spatio-temporal traffic patterns. Accordingly, in the majority of traffic models, vehicle accelerations are described by dynamic equations based on driving behavior, system dynamics and some underlying steady-state velocity-gap (bumper-to-bumper spacing) relation in order to guarantee the realistic human behavior. This paper proposes a deterministic car-following model based on a multi-branch fundamental diagram with each branch representing a particular category of driving style. Furthermore, an additional dynamic perception equation is introduced to reflect the driving style adaptation in response to the change in surrounding traffic situations. With simulation based on the proposed “driver perception model” (DP model), empirical findings of traffic breakdown and observed spatio-temporal patterns at on-ramp vicinity are reproduced. Furthermore, comparison results show the consistency between numerical simulation and the real traffic data of Beijing urban freeway.     

Effect of adaptive cruise control systems on mixed traffic flow near an on-ramp

Mixed traffic flow consisting of vehicles equipped with adaptive cruise control (ACC) and manually driven vehicles is analyzed using car-following simulations. Simulations of merging from an on-ramp onto a freeway reported in the literature have not thus far demonstrated a substantial positive impact of ACC. In this paper cooperative merging for ACC vehicles is proposed to improve throughput and increase distance traveled in a fixed time. In such a system an ACC vehicle senses not only the preceding vehicle in the same lane but also the vehicle immediately in front in the other lane. Prior to reaching the merge region, the ACC vehicle adjusts its velocity to ensure that a safe gap for merging is obtained. If on-ramp demand is moderate, cooperative merging produces significant improvement in throughput (20%) and increases up to 3.6 km in distance traveled in 600 s for 50% ACC mixed flow relative to the flow of all-manual vehicles. For large demand, it is shown that autonomous merging with cooperation in the flow of all ACC vehicles leads to throughput limited only by the downstream capacity, which is determined by speed limit and headway time.     

Mitigation of congestion at a traffic bottleneck with diversion and lane restrictions

Mitigation of congestion on a two-lane highway with an off-ramp and an on-ramp is simulated with three-phase traffic theory. Advanced travel information-the average velocity of vehicles near the bottleneck at an on-ramp-is used to divert vehicles at an upstream off-ramp. If enough vehicles divert, previously expanding synchronous flow congestion can be stalled and isolated to the region between the ramps. The introduction of lane restrictions (forbidding lane changing on the portion of highway between the ramps) in addition to diversion substantially reduces and essentially eliminates the congestion, restoring flow to nearly free-flow conditions.     

Phase diagram of a traffic roundabout

We propose a simple cellular automaton model to study the traffic dynamics in a roundabout. Both numerical and analytical results are presented. We are able to obtain exact solutions in the full parameter space. Exact phase diagrams are derived. When the traffic from two directions mixed, there are only five distinct phases. Some of the combinations from naive intuition are strictly forbidden. We also compare the results to a signaled intersection.     

Mixed traffic flow model considering illegal lane-changing behavior: Simulations in the framework of Kerner’s three-phase theory

This paper studies the mixed motorized vehicle (m$m$-vehicle) and non-motorized vehicle (nm$nm$-vehicle) traffic flow in the m$m$-vehicle lane. We study the formation mechanism of the nm$nm$-vehicle illegal lane-changing behavior (NILB) by considering the overtaking motivation and the traffic safety awareness. In the framework of Kerner’s three-phase theory, we propose a model for the mixed traffic flow by introducing a new set of rules. A series of simulations are carried out in order to reveal the formation, travel process and influence of the mixed traffic flow. The simulation results show that the proposed model can be used to study not only the travel characteristic of the mixed traffic flow, but also some complex traffic problems such as traffic breakdown, moving synchronized flow pattern (MSP) and moving jam. Moreover, the results illustrate that the proposed model reflects the phenomenon of the mixed flow and the influence of the MSP caused by the NILB, which is consistent with the actual traffic system, and thus this work is helpful for the management of the mixed traffic flow.     

Vehicular motion in counter traffic flow through a series of signals controlled by a phase shift

We study the dynamical behavior of counter traffic flow through a sequence of signals (traffic lights) controlled by a phase shift. There are two lanes for the counter traffic flow: the first lane is for east-bound vehicles and the second lane is for west-bound vehicles. The green-wave strategy is studied in the counter traffic flow where the phase shift of signals in the second lane has opposite sign to that in the first lane. A nonlinear dynamic model of the vehicular motion is presented by nonlinear maps at a low density. There is a distinct difference between the traffic flow in the first lane and that in the second lane. The counter traffic flow exhibits very complex behavior on varying the cycle time, the phase difference, and the split. Also, the fundamental diagram is derived by the use of the cellular automaton (CA) model. The dependence of east-bound and west-bound vehicles on cycle time, phase difference, and density is clarified.     

Effect of driver over-acceleration on traffic breakdown in three-phase cellular automaton traffic flow models

Based on simulations with cellular automaton (CA) traffic flow models, a generic physical feature of the three-phase models studied in the paper is disclosed. The generic feature is a discontinuous character of driver over-acceleration caused by a combination of two qualitatively different mechanisms of over-acceleration: (i) Over-acceleration through lane changing to a faster lane, (ii) over-acceleration occurring in car-following without lane changing. Based on this generic feature a new three-phase CA traffic flow model is developed. This CA model explains the set of the fundamental empirical features of traffic breakdown in real heterogeneous traffic flow consisting of passenger vehicles and trucks. The model simulates also quantitative traffic pattern characteristics as measured in real heterogeneous flow.     

A cellular automaton model based on empirical observations of a driver’s oscillation behavior reproducing the findings from Kerner’s three-phase traffic theory

In this paper, we propose a new cellular automaton model based on the well known three-phase traffic theory. The model takes into account the mechanism of a driver’s oscillation behavior obtained from engineering experiments in real traffic conditions. This mechanism shows the inner competition between speed adaptation and distance adjustment effects. The speed adaptation effect leads to synchronized flow, whereas a pinch region emerges, associated with the spontaneous occurrence of wide moving jams, due to distance over-adjustment. Numerical simulations are carried out both with periodic and with open boundary conditions in order to investigate the spatiotemporal features of traffic flow. The results indicate that our model is able to reproduce the three distinct traffic phases and exhibit the four congested patterns upstream of an isolated on-ramp, which is in good consistency with the results predicted from the three-phase theory.     

The effect of restricted velocity in the two-lane on-ramp system

Usually there are multi-lane on the main road of the on-ramp system. The drivers may decelerate for more safety when they are near the on-ramp. In addition, the car velocity may be restricted according to the traffic regulation. In this paper, we study phenomenon using the cellular automata traffic flow model. We find that: (i) the phase diagram of the two-lane on-ramp system appears a new region, in which the traffic of the on-ramp reaches maximum flow. (ii) The introduction of restricted velocity region will decrease capacity of the on-ramp, but reduce the drastic velocity fluctuation near the on-ramp.     

The theoretical analysis of the lattice hydrodynamic models for traffic flow theory

The lattice hydrodynamic model is not only a simplified version of the macroscopic hydrodynamic model, but also connected with the microscopic car following model closely. The modified Korteweg-de Vries (mKdV) equation related to the density wave in a congested traffic region has been derived near the critical point since Nagatani first proposed it. But the Korteweg-de Vries (KdV) equation near the neutral stability line has not been studied, which has been investigated in detail for the car following model. We devote ourselves to obtaining the KdV equation from the original lattice hydrodynamic models and the KdV soliton solution to describe the traffic jam. Especially, we obtain the general soliton solution of the KdV equation and the mKdV equation. We review several lattice hydrodynamic models, which were proposed recently. We compare the modified models and carry out some analysis. Numerical simulations are conducted to demonstrate the nonlinear analysis results.     

Effect of speed fluctuations on a green-light path in a 2d traffic network controlled by signals

When a vehicle moves through a series of green lights, avoiding red signals in a two-dimensional (2d) city traffic network, the vehicle describes a characteristic trajectory (green-light path) and the travel time has a minimal value. The green-light path depends on the cycle time, split, signal-control strategy, and fluctuations of vehicular speed. We clarify the effect of speed fluctuations on a green-light path in a 2d traffic network controlled by signals. Even if an extremely small quantity of speed fluctuation is added, the green-light path changes greatly. It is shown that the root-mean square (RMS) of the deviation from the mean path depends highly on the cycle time. Also, the dependence of the green-light path on the speed-fluctuation strength is shown under a constant value of cycle time.     

The application of Hölder exponent to traffic congestion warning

In this paper, we applied multifractal modeling techniques to analyze the traffic data collected from the Beijing Yuquanying. The results indicated that multifractal characteristics obviously exist in the traffic system; the degree of fractality of these traffic data tends to increase as the traffic system becomes congested; the Hölder exponent that measures the local rate of fractality may be used as indicators to predict the presence of the traffic congestion.     

Freezing transition in a four-directional traffic model for facing and crossing pedestrian flow

We study the traffic behavior in the facing and crossing traffic of pedestrians numerically and analytically. There are four kinds of walkers, those moving to east, to west, to north, and to south. We present the mean-field approximation (MFA) model for the four-directional traffic. The model is described in terms of four nonlinear difference equations. The excluded-volume effect and directionality are taken into account. The fundamental diagrams (current-density diagrams) are derived. When pedestrian density is higher than a critical value, the dynamical phase transition occurs from the free flow to the frozen (stopping) state. The critical density is derived by using the linear stability analysis. The velocity and current (flow) at the steady state are derived analytically. The analytical result is consistent with that obtained by the numerical simulation.     

Phase diagrams of noisy traffic states in the presence of a bottleneck

The traffic states are investigated for the noisy traffic flow in the presence of a bottleneck under the open boundary condition. The phase diagrams of the different kinds of congested traffic are presented for different values of sensitivity (the inverse of delay time). It is shown that the five different types of traffic states appear below the critical point: the free traffic with a pinned localized cluster, the weak standing shock, the moving localized cluster, the oscillatory congested traffic, and the strong standing shock. Above the critical point, the moving localized cluster and oscillatory congested traffic do not occur. Also, in the traffic flow without noise, the moving localized cluster and oscillatory congested traffic do not appear.     

Clustering and maximal flow in vehicular traffic through a sequence of traffic lights

We study the maximal current (maximum traffic capacity) of vehicular traffic through a sequence of traffic lights on a highway, where all signals turn on and off synchronously. The dynamical model of vehicular traffic controlled by signals is expressed in terms of a nonlinear map, where the excluded-volume effect is taken into account. The dynamical behaviors of vehicles are clarified by analyzing traffic patterns. The clustering of vehicles varies with the cycle time of signals. The maximum current is closely connected to vehicular clustering. Clustering of vehicles is controlled by varying both split and cycle time of signals. The dependence of the maximal current on both split and cycle time is derived.     

A framework for a relativistic theory of state reduction

The problem of formulating a relativistic theory of state-vector reduction is considered. A stochastic process is associated in a subjectively invariant way with each element of the proper orthochronous Poincaré group. The procedure yields a class of invariant stochastic differential equations in Hilbert space. Various aspects of the proposed method are discussed.     

A new lattice hydrodynamic model accounting for the traffic interruption probability on a gradient highway

In this paper, a novel lattice hydrodynamic model is presented by accounting for the traffic interruption probability on a gradient highway. The stability condition can be obtained by the use of linear analysis. Linear analysis demonstrates that the traffic interruption probability and the slope will affect the stability region. Through nonlinear analysis, the mKdV equation is derived to describe the phase transition of traffic flow. Furthermore, the numerical simulation is carried out, and the results are consistent with the analytical results. Numerical results demonstrate that the traffic flow can be efficiently improved by accounting for the traffic interruption probability on a gradient highway.