Dynamical model of traffic congestion and numerical simulation

Optimization of congested traffic by controlling stop-and-go waves

We propose a new optimization strategy based on inducing stop-and-go waves on the main road and controlling their wavelength. Using numerical simulations of a recent stochastic car-following model we show that this strategy yields optimization of traffic flow when implemented in systems with a localized periodic inhomogeneity, such as signalized intersections and entry ramps. The optimization process is explained by our finding of a generalized fundamental diagram (GFD) for traffic, namely a flux-density-wavelength relation. Projecting the GFD on the density-flux plane yields a two-dimensional region of stable states, qualitatively similar to that found empirically [Kerner, Phys. Rev. Lett. 81, 3797 (1998)] in synchronized traffic.

Improved stability bound for steady-state flow in a car-following model of road traffic on a circular route

This note revisits a car-following model of road traffic on a circular route that was studied in recent literature, and improves a stability result for steady-state flows that was obtained in this literature. It will be shown through a counter example that the stability bound obtained in the literature only gives a sufficient condition for stability, which only becomes necessary when the number of cars on the route tends to infinity. We will further present a result that gives a necessary and sufficient condition for stability.

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.

Capacity drop due to the traverse of pedestrians

In this paper, we have proposed a simplified model to describe the traffic flow when there are pedestrians traversing the road. The numerical simulation shows that the capacity of the road decreases in the presence of pedestrians. If the traffic flow rate is small, the traffic flow is basically unaffected even if some pedestrians traverse the road. However, if the flow rate exceeds a critical value, the vehicles cannot pass without delay, and a traffic jam appears. We also discuss simplified conditions of the model and accordingly present a modified model, which predicts qualitatively the same results except with a different capacity.

Towards a variational principle for motivated vehicle motion

We deal with the problem of deriving the microscopic equations governing individual car motion based on assumptions about the strategy of driver behavior. We presume the driver behavior to be a result of a certain compromise between the will to move at a speed that is comfortable for him under the surrounding external conditions, comprising the physical state of the road, the weather conditions, etc., and the necessity to keep a safe headway distance between the cars in front of him. Such a strategy implies that a driver can compare the possible ways of further motion and so choose the best one. To describe the driver preferences, we introduce the priority functional whose extremals specify the driver choice. For simplicity we consider a single-lane road. In this case solving the corresponding equations for the extremals we find the relationship between the current acceleration, velocity, and position of the car. As a special case we get a certain generalization of the optimal velocity model similar to the "intelligent driver model" proposed by Treiber and Helbing.

Effect of looking at the car that follows in an optimal velocity model of traffic flow

An extension of an optimal velocity model is proposed. In the new model, a driver looks at the following car as well as the preceding car. We introduce an additional optimal velocity function that depends on the headway of the following car. We investigate the effect of looking back at the car that follows and show that this extension effectively stabilizes the traffic flow.

Headways in traffic flow: remarks from a physical perspective

Traffic flow can be understood as a realization of a broad class of one dimensional physical systems, where a hard core repulsive interaction competes with a longer ranged attraction between the particles. It can be shown rigorously that the statistical properties of such systems in thermal equilibrium are well described by a family of distributions that stems from the random matrix theory. Analyzing the traffic data from different sources, we show that traffic on real roads belongs to that class of random matrix distributions. Also, various traffic simulation models show a similar behavior. It is demonstrated in such a way that the headway distribution of a highway traffic, that serves usually as a paradigm of systems driven far from equilibrium, is reasonably well described by a distribution originating from equilibrium statistical physics.

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.

Bifurcation phenomena in the optimal velocity model for traffic flow

In the optimal velocity model with a time lag, we show that there appear multiple exact solutions in some ranges of car density, describing a metastable uniform flow, a metastable congested flow, and an unstable congested flow. This establishes the presence of subcritical Hopf bifurcations. Our analytical results have implications for continuum traffic flow, such as hysteresis phenomena associated with discontinuous transitions between uniform and congested flow.

On-ramp simulations and solitary waves of a car-following model

An on-ramp simulation of a car-following model reveals qualitatively similar results to previous simulations of continuum models carried out by Helbing et al. [Phys. Rev. Lett. 82, 4360 (1999)] and by Lee, Lee, and Kim[Phys. Rev. E 59, 5101 (1999)]. Here, we discuss the solitary solution type in greater detail. It can be approximated by a Kortweg-de Vries equation derived from the analogous continuum version. Hence, this establishes a further link between these two traffic simulation types and supports the idea that models of either kind lead to similar results when they contain a relaxation term.

Full velocity difference model for a car-following theory

In this paper, we present a full velocity difference model for a car-following theory based on the previous models in the literature. To our knowledge, the model is an improvement over the previous ones theoretically, because it considers more aspects in car-following process than others. This point is verified by numerical simulation. Then we investigate the property of the model using both analytic and numerical methods, and find that the model can describe the phase transition of traffic flow and estimate the evolution of traffic congestion.

Enhancement and stabilization of traffic flow by moving in groups

We study the traffic behavior of vehicles moving in groups analytically and numerically. A car-following model of traffic is extended to take into account a binary mixture of vehicles. It is shown that the movement in groups stabilizes the traffic flow. The jamming transition among the free traffic, the inhomogeneous traffic, and the homogeneous congested traffic occurs at a higher density than the threshold of the original model. The traffic current is highly enhanced at a high-density region by keeping a short headway without jam. The jamming transition is analyzed by using the linear stability method. It is found that the theoretical neutral stability curve agrees with the transition line obtained by the simulation.

Collective directional transport in coupled nonlinear oscillators without external bias

Directed collective motion in a circular array of unidirectionally coupled oscillators with symmetric potential is obtained numerically in the absence of external bias. This striking feature is interpreted as the effect of the spontaneous breaking of temporal symmetry of the coupling. It is revealed that a proper match of various control parameters is important in generating an optimal coherent global transport. Noise-sustained directed transport is also observed, and the related stochastic resonance in an autonomous system is identified.

Bunching transition in a time-headway model of a bus route

A time-headway model is presented to mimic bus behavior on the bus route. The motion of a bus is described in terms of the time headway between its bus and the bus in front. We study the bunching behavior of buses induced by interacting with other buses and passengers. It is shown that the dynamical phase transitions among the inhomogeneous bunching phase, the homogeneous free phase, the coexisting phase, and the homogeneous congested phase occur with varying the initial time headway. We study the effect of not stopping at bus stops on the time-headway profile. It is found that the bunching transition lines are consistent with the neutral stability curves obtained by the linear stability analysis.

Traveling waves in an optimal velocity model of freeway traffic

Car-following models provide both a tool to describe traffic flow and algorithms for autonomous cruise control systems. Recently developed optimal velocity models contain a relaxation term that assigns a desirable speed to each headway and a response time over which drivers adjust to optimal velocity conditions. These models predict traffic breakdown phenomena analogous to real traffic instabilities. In order to deepen our understanding of these models, in this paper, we examine the transition from a linear stable stream of cars of one headway into a linear stable stream of a second headway. Numerical results of the governing equations identify a range of transition phenomena, including monotonic and oscillating travelling waves and a time- dependent dispersive adjustment wave. However, for certain conditions, we find that the adjustment takes the form of a nonlinear traveling wave from the upstream headway to a third, intermediate headway, followed by either another traveling wave or a dispersive wave further downstream matching the downstream headway. This intermediate value of the headway is selected such that the nonlinear traveling wave is the fastest stable traveling wave which is observed to develop in the numerical calculations. The development of these nonlinear waves, connecting linear stable flows of two different headways, is somewhat reminiscent of stop-start waves in congested flow on freeways. The different types of adjustments are classified in a phase diagram depending on the upstream and downstream headway and the response time of the model. The results have profound consequences for autonomous cruise control systems. For an autocade of both identical and different vehicles, the control system itself may trigger formations of nonlinear, steep wave transitions. Further information is available [Y. Sugiyama, Traffic and Granular Flow (World Scientific, Singapore, 1995), p. 137].

Phase diagram of congested traffic flow: An empirical study

We analyze traffic data from a highway section containing one effective on-ramp. Based on two criteria, local velocity variation patterns and expansion (or nonexpansion) of congested regions, three distinct congested traffic states are identified. These states appear at different levels of the upstream flux and the on-ramp flux, thereby generating a phase digram of the congested traffic flow. Observed traffic states are compared with recent theoretical analyses and both agreeing and disagreeing features are found.

Multibunch solutions of the differential-difference equation for traffic flow

The Newell-Whitham type of car-following model, with a hyperbolic tangent as the optimal velocity function, has a finite number of exact steady traveling wave solutions that can be expressed in terms of elliptic theta functions. Each such solution describes a density wave with a definite number of car bunches on a circuit. In our numerical simulations, we observe a transition process from uniform flow to congested flow described by a one-bunch analytic solution, which appears to be an attractor of the system. In this process, the system exhibits a series of transitions through which it comes to assume configurations closely approximating multibunch solutions with successively fewer bunches.

Spatiotemporal structure of traffic flow in a system with an open boundary

The spatiotemporal structure of a traffic flow pattern is investigated under the open boundary condition using the optimal velocity model. The parameter region where the uniform solution is convectively unstable is determined. It is found that a localized perturbation triggers a linearly unstable oscillatory solution out of the linearly unstable uniform state, and it is shown that the oscillatory solution is also convectively stabilized. It is demonstrated that the observed traffic pattern near an on-ramp can be interpreted as the noise sustained structure in the open flow system.

Congested traffic states in empirical observations and microscopic simulations

We present data from several German freeways showing different kinds of congested traffic forming near road inhomogeneities, specifically lane closings, intersections, or uphill gradients. The states are localized or extended, homogeneous or oscillating. Combined states are observed as well, like the coexistence of moving localized clusters and clusters pinned at road inhomogeneities, or regions of oscillating congested traffic upstream of nearly homogeneous congested traffic. The experimental findings are consistent with a recently proposed theoretical phase diagram for traffic near on-ramps [D. Helbing, A. Hennecke, and M. Treiber, Phys. Rev. Lett. 82, 4360 (1999)]. We simulate these situations with a continuous microscopic single-lane model, the "intelligent driver model," using empirical boundary conditions. All observations, including the coexistence of states, are qualitatively reproduced by describing inhomogeneities with local variations of one model parameter. We show that the results of the microscopic model can be understood by formulating the theoretical phase diagram for bottlenecks in a more general way. In particular, a local drop of the road capacity induced by parameter variations has essentially the same effect as an on-ramp.

Density waves in traffic flow

Density waves are investigated in the car-following model analytically and numerically. This work is a continuation of our previous investigation of traffic flow in the metastable and unstable regions [Phys. Rev. E 58, 4271 (1998); 60, 180 (1999)]. The Burgers equation is derived for the density wave in the stable region of traffic flow by the use of nonlinear analysis. It is shown, numerically, that the triangular shock wave appears as the density wave at the late stage in the stable region. The decay rate of the shock wave is calculated and compared with the analytical result. It is shown that the density waves out of the coexisting curve, near the spinodal line, and within the spinodal line appear, respectively, as the triangular shock wave, the soliton, and the kink-antikink wave. The density waves are described, respectively, by the Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations.

Traffic jams induced by fluctuation of a leading car

We present a phase diagram of the different kinds of congested traffic triggered by fluctuation of a leading car in an open system without sources and sinks. Traffic states and density waves are investigated numerically by varying the amplitude of fluctuation using a car following model. The phase transitions among the free traffic, oscillatory congested traffic, and homogeneous congested traffic occur by fluctuation of a leading car. With increasing the amplitude of fluctuation, the transition between the free traffic and oscillatory traffic occurs at lower density and the transition between the homogeneous congested traffic and the oscillatory traffic occurs at higher density. The oscillatory congested traffic corresponds to the coexisting phase. Also, the moving localized clusters appear just above the transition lines.

Delayed stochastic systems

Noise and time delay are two elements that are associated with many natural systems, and often they are sources of complex behaviors. Understanding of this complexity is yet to be explored, particularly when both elements are present. As a step to gain insight into such complexity for a system with both noise and delay, we investigate such delayed stochastic systems both in dynamical and probabilistic perspectives. A Langevin equation with delay and a random-walk model whose transition probability depends on a fixed time-interval past (delayed random walk model) are the subjects of in depth focus. As well as considering relations between these two types of models, we derive an approximate Fokker-Planck equation for delayed stochastic systems and compare its solution with numerical results.

Continuum approach to car-following models

A continuum version of the car-following Bando model is developed using a series expansion of the headway in terms of the density. This continuum model obeys the same stability criterion as its discrete counterpart. To compare both models we show that traveling wave solutions of the Bando model are very similar to those of the continuum model in the limit of small changes of headway. As the change of headway across the wave increases the solutions gradually diverge. Our transformation relating headway to density enables predictions of the global impact and characteristics of any car-following model using the analogous continuum model. In contrast, we show that the conventional continuum models which account for effects of pressure and dispersion predict behavior which is distinct from the global behavior of discrete models.

Presence of many stable nonhomogeneous states in an inertial car-following model

We present a single lane car- following model of traffic flow which is inertial and free of collisions. It demonstrates observed features of traffic flow such as existence of three regimes: free, nonhomogeneous congested (NHC) or synchronized, and homogeneous congested (HC) or jammed flow; bistability of free and NHC flow states in a range of densities, hysteresis in transitions between these states; jumps in the density-flux plane in the NHC regime; gradual spatial transition from synchronized to free flow; long survival time of jams in the HC regime. The model predicts that in the NHC regime there exist many stable states with different wavelengths, and noise can cause transitions between them.

Stabilization and enhancement of traffic flow by the next-nearest-neighbor interaction

The car-following model of traffic is extended to take into account the car interaction before the next car ahead (the next-nearest-neighbor interaction). The traffic behavior of the extended car-following model is investigated numerically and analytically. It is shown that the next-nearest-neighbor interaction stabilizes the traffic flow. The jamming transition between the freely moving and jammed phases occurs at a higher density than the threshold of the original car-following model. By increasing the maximal velocity, the traffic current is enhanced without jam by the stabilization effect. The jamming transition is analyzed with the use of the linear stability and nonlinear perturbation methods. The traffic jam is described by the kink solution of the modified Korteweg-de Vries equation. The theoretical coexisting curve is in good agreement with the simulation result.

Coupled map car-following model and its delayed-feedback control

This paper proposes a coupled map car-following traffic model, which describes a dynamical behavior of a group of road vehicles running in a single lane without overtaking. This model consists of a lead vehicle and following vehicles, which have a piecewise linear optimal velocity function. When the lead-vehicle speed is varied, we can observe a traffic jam in the group of the vehicles. We derive a condition under which the traffic jam never occurs in our model. Furthermore, in order to suppress the traffic jam, for each vehicle we use a dynamic version of decentralized delayed-feedback control proposed in [Konishi, Hirai, and Kokame, Phys. Rev. E 58, 3055 (1998)], and provide a systematic procedure for designing the controller.

Chaotic jam and phase transition in traffic flow with passing

The lattice hydrodynamic model is presented to take into account the passing effect in one-dimensional traffic flow. When the passing constant gamma is small, the conventional jamming transition occurs between the uniform traffic and kink density wave flows. When passing constant gamma is larger than the critical value, the jamming transitions occur from the uniform traffic flow, through the chaotic density wave flow, to the kink density wave flow, with an increasing delay time. The chaotic region increases with passing constant gamma. The neutral stability line is derived from the linear stability analysis. The neutral stability line coincides with the transition line between the uniform traffic and density wave flows. The modified Korteweg-de Vries equation describing the kink jam is derived for small values of gamma by use of a nonlinear analysis.

Soliton and kink jams in traffic flow with open boundaries

Soliton density wave is investigated numerically and analytically in the optimal velocity model (a car-following model) of a one-dimensional traffic flow with open boundaries. Soliton density wave is distinguished from the kink density wave. It is shown that the soliton density wave appears only at the threshold of occurrence of traffic jams. The Korteweg-de Vries (KdV) equation is derived from the optimal velocity model by the use of the nonlinear analysis. It is found that the traffic soliton appears only near the neutral stability line. The soliton solution is analytically obtained from the perturbed KdV equation. It is shown that the soliton solution obtained from the nonlinear analysis is consistent with that of the numerical simulation.

Dynamic states of a continuum traffic equation with on-ramp

We study the phase diagram of the continuum traffic flow model of a highway with an on-ramp. Using an open boundary condition, traffic states and metastabilities are investigated numerically for several representative values of the upstream boundary flux f(up) and for the whole range of the on-ramp flux f(rmp). An inhomogeneous but time-independent traffic state (standing localized cluster state) is found and related to a recently measured traffic state. Due to the density gradient near the on-ramp, a traffic jam can occur even when the downstream density is below the critical density of the usual traffic jam formation in homogeneous highways, and its structure varies qualitatively with f(rmp). The free flow, the recurring hump (RH) state, and the traffic jam can all coexist in a certain metastable region where the free flow can undergo phase transitions either to the RH state or to the traffic jam state. We also find two nontrivial analytic solutions. These solutions correspond to the standing localized cluster state and the homogeneous congested traffic state (one form of the traffic jam), which are observed in numerical simulations.

Jamming transition in a two-dimensional traffic flow model

Phase transition and critical phenomenon are investigated in the two-dimensional traffic flow numerically and analytically. The one-dimensional lattice hydrodynamic model of traffic is extended to the two-dimensional traffic flow in which there are two types of cars (northbound and eastbound cars). It is shown that the phase transition among the freely moving phase, the coexisting phase, and the uniformly congested phase occurs below the critical point. Above the critical point, no phase transition occurs. The value a(c) of the critical point decreases as increasing fraction c of the eastbound cars for c<or=0.5. The linear stability theory is applied. The neutral stability lines are found. The time-dependent Ginzburg-Landau (TDGL) equation is derived by the use of nonlinear analysis. The phase separation lines, the spinodal lines, and the critical point are calculated from the TDGL equation.