Synchronized traffic flow from a modified lighthill-whitman model

Multifractality of random walks in the theory of vehicular traffic

We investigate the origin of the experimentally observed multifractal scaling of vehicular traffic flows by studying a hydrodynamic model of traffic. We first extend and apply the formalism of generalized Hurst exponents H(q) to the case of random walkers that not only diffuse but rather also undergo nonlinear convection due to interactions with other walkers. We recover analytically, as expected, that H(q) equals 12 for a single random walker starting at the origin whose probability density function satisfies Burger's equation. Despite this result for a single walker, we find that for a collection of nonlinearly convecting diffusive particles, transient effects can give rise to multiscaling at given time scales for many initial conditions. In the Lighthill-Whitham-Richards hydrodynamic model of traffic, this multiscaling effect becomes more prominent for smaller diffusion constants and larger speed limits. We discuss the relevance of these findings for the realistic scenario of traffic that flows from small roads to large highways and vice versa, where transient effects can be expected to play a significant role.

Analytical results of asymmetric exclusion processes with ramps

We present the analytical results in a simple traffic model describing a single-lane highway with ramps. Both on-ramps and off-ramps are considered. Complete classification of distinct phases is achieved. Exact phase diagrams are derived. In the case of a single ramp (either on-ramp or off-ramp), the bottleneck effect is absent. The traffic conditions of congestion before the ramp and free-flowing after the ramp cannot be realized. In the case of two consecutive ramps, the bottleneck emerges when the on-ramp is placed before the off-ramp and the flow in between the ramps saturates.

Steady-state solutions of hydrodynamic traffic models

We investigate steady-state solutions of hydrodynamic traffic models in the absence of any intrinsic inhomogeneity on roads such as on-ramps. It is shown that typical hydrodynamic models possess seven different types of inhomogeneous steady-state solutions. The seven solutions include those that have been reported previously only for microscopic models. The characteristic properties of wide jam such as moving velocity of its spatiotemporal pattern and/or out-flux from wide jam are shown to be uniquely determined and thus independent of initial conditions of dynamic evolution. Topological considerations suggest that all of the solutions should be common to a wide class of traffic models. The results are discussed in connection with the universality conjecture for traffic models. Also the prevalence of the limit-cycle solution in a recent study of a microscopic model is explained in this approach.

Interpreting the wide scattering of synchronized traffic data by time gap statistics

Based on the statistical evaluation of experimental single-vehicle data, we propose a quantitative interpretation of the erratic scattering of flow-density data in synchronized traffic flows. A correlation analysis suggests that the dynamical flow-density data are well compatible with the so-called jam line characterizing fully developed traffic jams, if one takes into account the variation of their propagation speed due to the large variation of the netto time gaps (the inhomogeneity of traffic flow). The form of the time gap distribution depends not only on the density, but also on the measurement cross section: The most probable netto time gap in congested traffic flow upstream of a bottleneck is significantly increased compared to uncongested freeway sections. Moreover, we identify different power-law scaling laws for the relative variance of netto time gaps as a function of the sampling size. While the exponent is -1 in free traffic corresponding to statistically independent time gaps, the exponent is about -2/3 in congested traffic flow because of correlations between queued vehicles.

Memory effects in microscopic traffic models and wide scattering in flow-density data

By means of microscopic simulations we show that noninstantaneous adaptation of the driving behavior to the traffic situation together with the conventional method to measure flow-density data provides a possible explanation for the observed inverse-lambda shape and the wide scattering of flow-density data in "synchronized" congested traffic. We model a memory effect in the response of drivers to the traffic situation for a wide class of car-following models by introducing an additional dynamical variable (the "subjective level of service") describing the adaptation of drivers to the surrounding traffic situation during the past few minutes and couple this internal state to parameters of the underlying model that are related to the driving style. For illustration, we use the intelligent-driver model (IDM) as the underlying model, characterize the level of service solely by the velocity, and couple the internal variable to the IDM parameter "time gap" to model an increase of the time gap in congested traffic ("frustration effect"), which is supported by single-vehicle data. We simulate open systems with a bottleneck and obtain flow-density data by implementing "virtual detectors." The shape, relative size, and apparent "stochasticity" of the region of the scattered data points agree nearly quantitatively with empirical data. Wide scattering is even observed for identical vehicles, although the proposed model is a time-continuous, deterministic, single-lane car-following model with a unique fundamental diagram.

Long-lived states in synchronized traffic flow: empirical prompt and dynamical trap model

The present paper proposes an interpretation of the widely scattered states (called synchronized traffic) stimulated by Kerner's hypothesis about the existence of a multitude of metastable states in the fundamental diagram. Using single-vehicle data collected at the German highway A1, temporal velocity patterns have been analyzed to show a collection of certain fragments with approximately constant velocities and sharp jumps between them. The particular velocity values in these fragments vary in a wide range. In contrast, the flow rate is more or less constant because its fluctuations are mainly due to the discreteness of traffic flow. Subsequently, we develop a model for synchronized traffic that can explain these characteristics. Following previous work [I. A. Lubashevsky and R. Mahnke, Phys. Rev. E 62, 6082 (2000)] the vehicle flow is specified by car density, mean velocity, and additional order parameters h and a that are due to the many-particle effects of the vehicle interaction. The parameter h describes the multilane correlations in the vehicle motion. Together with the car density it determines directly the mean velocity. The parameter a, in contrast, controls the evolution of h only. The model assumes that a fluctuates randomly around the value corresponding to the car configuration optimal for lane changing. When it deviates from this value the lane change is depressed for all cars forming a local cluster. Since exactly the overtaking maneuvers of these cars cause the order parameter a to vary, the evolution of the car arrangement becomes frozen for a certain time. In other words, the evolution equations form certain dynamical traps responsible for the long-time correlations in the synchronized mode.

Empirical macroscopic features of spatial-temporal traffic patterns at highway bottlenecks

Results of an empirical study of congested patterns measured during 1995-2001 at German highways are presented. Based on this study, various types of congested patterns at on and off ramps have been identified, their macroscopic spatial-temporal features have been derived, and an evolution of those patterns and transformations between different types of the patterns over time has been found out. It has been found that at an isolated bottleneck (a bottleneck that is far enough from other effective bottlenecks) either the general pattern (GP) or the synchronized flow pattern (SP) can be formed. In GP, synchronized flow occurs and wide moving jams spontaneously emerge in that synchronized flow. In SP, no wide moving jams emerge, i.e., SP consists of synchronized flow only. An evolution of GP into SP when the flow rate to the on ramp decreases has been found and investigated. Spatial-temporal features of complex patterns that occur if two or more effective bottlenecks exist on a highway have been found out. In particular, the expanded pattern where synchronized flow covers two or more effective bottlenecks can be formed. It has been found that the spatial-temporal structure of congested patterns possesses predictable, i.e., characteristic, unique, and reproducible features, for example, the most probable types of patterns that are formed at a given bottleneck. According to the empirical investigations the cases of the weak and the strong congestion should be distinguished. In contrast to the weak congestion, the strong congestion possesses the following characteristic features: (i) the flow rate in synchronized flow is self-maintaining near a limit flow rate; (ii) the mean width of the region of synchronized flow in GP does not depend on traffic demand; (iii) there is a correlation between the parameters of synchronized flow and wide moving jams: the higher the flow rate out from a wide moving jam is, the higher is the limit flow rate in the synchronized flow. The strong congestion often occurs in GP whereas the weak congestion is usual for SP. The weak congestion is often observed at off ramps whereas the strong congestion much more often occurs at on ramps. Under the weak congestion diverse transformations between different congested patterns can occur.

Highway on-ramp control

We study the phase transition on a highway induced by the fluctuations of on-ramp flow. The highway traffic is provided by a hydrodynamical model. We analyze the characteristics of perturbations to induce congestion near on ramp. The phase boundary is obtained. A scaling relation is revealed. We also analyze the time evolution of the local density profile. Conventional control mechanisms to regulate the on-ramp flow are examined. A control scheme is proposed to suppress the congestion.

Human behavior as origin of traffic phases

It is shown that the desire for smooth and comfortable driving is directly responsible for the occurrence of synchronized traffic in highway traffic. This desire goes beyond the avoidance of accidents, which so far has been the main focus of microscopic modeling and that is mainly responsible for the other two phases observed empirically, free flow and wide moving jams. These features have been incorporated into a microscopic model based on stochastic cellular automata by means of event-driven anticipation. The results of computer simulations are compared with empirical data. It turns out that anticipation effects are responsible for the stabilization of the traffic phases and even reproduce the empirically observed coexistence of wide moving jams with both free flow and synchronized traffic.

Macroscopic traffic models from microscopic car-following models

We present a method to derive macroscopic fluid-dynamic models from microscopic car-following models via a coarse-graining procedure. The method is first demonstrated for the optimal velocity model. The derived macroscopic model consists of a conservation equation and a momentum equation, and the latter contains a relaxation term, an anticipation term, and a diffusion term. Properties of the resulting macroscopic model are compared with those of the optimal velocity model through numerical simulations, and reasonable agreement is found although there are deviations in the quantitative level. The derivation is also extended to general car-following models.

Standing localized cluster in a continuum traffic model

We study the emergence of standing localized cluster in a continuum traffic model. The local-density profile, local velocity profile, and the phase boundary are obtained. The effect of the on ramp is discussed. The indication of the boundary induced phase transition is also discussed.