It is a well known fact that metastable states of very high throughput and hysteresis effects exist in traffic flow, which the simple cellular automaton model of traffic flow and its continuous generalization fail to reproduce. It is shown that the model can be generalized to give a one-parametric family of models, a part of which reproduces the metastable states and the hysteresis. The models that have that property and those that do not that are separated by a transition that can be clearly identified.

Microscopic modeling of multilane traffic is usually done by applying heuristic lane changing rules and often with unsatisfying results. Recently, a cellular automaton model for two-lane traffic was able to overcome some of these problems and to generate the density inversion found in many European countries at densities somewhat below the maximum flow density. In this paper we summarize different approaches to lane changing and their results and propose a general scheme, according to which realistic lane changing rules can be developed. We test this scheme by applying it to several different lane changing rules, which, in spite of their differences, generate similar and realistic results. We thus conclude that, for producing realistic results, the logical structure of the lane changing rules, as proposed here, is at least as important as the microscopic details of the rules.

Certain aspects of traffic flow measurements imply the existence of a phase transition. Models known from chaos and fractals, such as nonlinear analysis of coupled differential equations, cellular automata, or coupled maps, can generate behavior which indeed resembles a phase transition in the flow behavior. Other measurements point out that the same behavior could be generated by geometrical constraints of the scenario. This paper looks at some of the empirical evidence, but mostly focuses on different modeling approaches. The theory of traffic jam dynamics is reviewed in some detail, starting from the well-established theory of kinematic waves and then veering into the area of phase transitions. One aspect of the theory of phase transitions is that, by changing one single parameter, a system can be moved from displaying a phase transition to not displaying a phase transition. This implies that models for traffic can be tuned so that they display a phase transition or not. This paper focuses on microscopic modeling, i.e., coupled differential equations, cellular automata, and coupled maps. The phase transition behavior of these models, as far as it is known, is discussed. Similarly, fluid-dynamical models for the same questions are considered. A large portion of this paper is given to the discussion of extensions and open questions, which makes clear that the question of traffic jam dynamics is, albeit important, only a small part of an interesting and vibrant field. As our outlook shows, the whole field is moving away from a rather static view of traffic toward a dynamic view, which uses simulation as an important tool.

Microscopic simulation models are becoming increasingly important tools in modeling transport systems. A large number of models are used in many countries. The most difficult stage in the development and use of such models is the calibration and validation of the microscopic submodels such as the car-following and gap acceptance models. This difficulty results from the lack of suitable methods for adapting models to empirical data. Recent progress has been made in calibrating a number of microscopic traffic flow models. Ten very different models have been tested with data collected via Differential Global Positioning System-equipped cars on a test track in Japan. To calibrate the models, the data of the leading car are fed into the model under consideration, and the model is used to compute the headway time series of the following car. The deviations between the measured and the simulated headways are then used to calibrate and validate the models. The calibration results agree with earlier studies as there are errors of 12% to 17% for all models, and no model can be denoted to be the best. The differences between individual drivers are larger than the differences between different models. The validation process gives acceptable errors from 17% to 22%. But for special data sets with validation errors up to 60%, the calibration process has reached what is known as "overfitting": because of the adaptation to a particular situation, the models are not capable of generalizing to other situations.

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.