Nagel-Schreckenberg Model Research Articles

A study on an improved Nagel-Schreckenberg traffic flow -model with open boundary conditions

Using the improved Nagel-Schreckenberg traffic model with open boundary conditions,we present the fundamental diagrams of traffic flow under different parameters by numerical simulation.It turns out that the traffic flow of the improved Nagel-Schrec-kenberg model is greater than that of the Nagel-Schreckenberg model,then we analyse the characters and critical points of the transition from free flow phase to maximum current phase and the transition from jamming phase to free flow phase.

Optimizing traffic lights in a cellular automaton model for city traffic.

We study the impact of global traffic light control strategies in a recently proposed cellular automaton model for vehicular traffic in city networks. The model combines basic ideas of the Biham-Middleton-Levine model for city traffic and the Nagel-Schreckenberg model for highway traffic. The city network has a simple square lattice geometry. All streets and intersections are treated equally, i.e., there are no dominant streets. Starting from a simple synchronized strategy, we show that the capacity of the network strongly depends on the cycle times of the traffic lights. Moreover, we point out that the optimal time periods are determined by the geometric characteristics of the network, i.e., the distance between the intersections. In the case of synchronized traffic lights, the derivation of the optimal cycle times in the network can be reduced to a simpler problem, the flow optimization of a single street with one traffic light operating as a bottleneck. In order to obtain an enhanced throughput in the model, improved global strategies are tested, e.g., green wave and random switching strategies, which lead to surprising results.

A VARIED FORM OF THE NAGEL–SCHRECKENBERG MODEL

By making slight changes of the Nagel–Schreckenberg model, we propose a varied model for a realistic description of the traffic flow on single-lane highways. While the NaSch model fails to cover the "takeover", this realistic situation gets a good description in our model. Moreover, from the fundamental diagrams achieved in our model, we can divide the forms of traffic flow into five different modes, which are obviously observed in their corresponding time-space diagrams. We also make a comparison of the results of a simple kind of mean-field approximation and the cellular automaton approach.

SIMULATING TRAFFIC ON GERMAN HIGHWAYS BASED ON THE NAGEL–SCHRECKENBERG-MODEL

The Nagel–Schreckenberg model (NaSch model) proved to be a realistic approach to describe traffic flow on single-lane streets. In this paper, we add some lane changing rules for two-lane traffic and discuss the influence of the variation on the mean velocity and the flow.

Random walk theory of jamming in a cellular automaton model for traffic flow

The jamming behavior of a single lane traffic model based on a cellular automaton approach is studied. Our investigations concentrate on the so-called VDR model which is a simple generalization of the well-known Nagel–Schreckenberg model. In the VDR model one finds a separation between a free flow phase and jammed vehicles. This phase separation allows to use random walk like arguments to predict the resolving probabilities and lifetimes of jam clusters or disturbances. These predictions are in good agreement with the results of computer simulations and even become exact for a special case of the model. Our findings allow a deeper insight into the dynamics of wide jams occuring in the model.

Cellular automaton traffic flow model between the Fukui-Ishibashi and Nagel-Schreckenberg models.

We propose and study a one-dimensional traffic flow cellular automaton model of high-speed vehicles with the Fukui-Ishibashi-type acceleration for all cars, and the Nagel-Schreckenberg-type (NS) stochastic delay only for cars following the trail of the car ahead. The main difference in the delay scenario between our model and the NS model is that a car with spacing ahead longer than the velocity limit M may not be delayed in our model. By using a car-oriented mean-field theory, we analytically derive fundamental diagrams of the average speed as a function of the car density. Our theoretical results are in excellent agreement with numerical simulations.

Empirical evidence for a boundary-induced nonequilibrium phase transition

A recently developed theory for boundary-induced phenomena in nonequilibrium systems predicts the existence of various steady-state phase transitions induced by the motion of a shock wave. We provide direct empirical evidence that a phase transition between a free flow and a congested phase occurring in traffic flow on highways in the vicinity of on- and off-ramps can be interpreted as an example of such a boundary-induced phase transition of first order. We analyse the empirical traffic data and give a theoretical interpretation of the transition in terms of the macroscopic current. Additionally we support the theory with computer simulations of the Nagel-Schreckenberg model of vehicular traffic on a road segment which also exhibits the expected second-order transition. Our results suggest ways to predict and to some extent to optimize the capacity of a general traffic network.

Car accidents on a single-lane highway.

We study the occurrence of car accidents in the Nagel-Schreckenberg model. Both the effects of stochastic braking and speed limit are analyzed. An approximate scaling relation is observed with the varying of speed limit. The stochastic noise will enhance the probability for accidents in the low density region and suppress that in the high density region. The probability for car accidents to occur becomes a broadened distribution over a wide range of density.

Low-density limit of the Nagel-Schreckenberg model

We study the low-density limit of the Nagel-Schreckenberg model with a large speed limit. Three distinct regions in the fundamental diagram are identified. Analytical approximations are obtained. The dependence on three parameters (speed limit ${v}_{\mathrm{max}},$ random braking p, and lattice size $L)$ is discussed. The relation to finite-size effect is also discussed.

Reaction-diffusion models describing a two-lane traffic flow

A unidirectional two-lane road is approximated by a set of two parallel closed one-dimensional chains. Two types of cars, i.e., slow and fast ones are considered in the system. Based on the Nagel-Schreckenberg model of traffic flow [K. Nagel and M. Schreckenberg, J. Phys. 2, 2221 (1992)], a set of reaction-diffusion processes is introduced to simulate the behavior of the cars. Fast cars can pass the slow ones using the passing lane. We write and solve the mean-field rate equations for the density of slow and fast cars, respectively. We also investigate the properties of the model through computer simulations and obtain the fundamental diagrams. A comparison between our results and the v(max)=2 version of the Nagel-Schreckenberg model is made.

THE INFLUENCE OF TRUCKS ON TRAFFIC FLOW — AN INVESTIGATION ON THE NAGEL–SCHRECKENBERG-MODEL

A few years ago Nagel and Schreckenberg introduced a model for traffic flow,1 which, though quite simple, is able to reproduce the basic properties of traffic. Meanwhile many variations of the model have been developed, part of them concentrate on introducing an inhomogenous fleet of cars with different maximum velocities. Here we also want to study the effect of introducing a certain fraction of vehicles with a lower maximum velocity ("trucks") to the Nagel–Schreckenberg-model. We give a detailed description of this effect and focus on the time development of the influence of trucks on traffic flow.

Steady-states and kinetics of ordering in bus-route models: connection with the Nagel-Schreckenberg model

A Bus Route Model (BRM) can be defined on a one-dimensional lattice, where buses are represented by “particles” that are driven forward from one site to the next with each site representing a bus stop. We replace the random sequential updating rules in an earlier BRM by parallel updating rules. In order to elucidate the connection between the BRM with parallel updating (BRMPU) and the Nagel-Schreckenberg (NaSch) model, we propose two alternative extensions of the NaSch model with space-/time-dependent hopping rates. Approximating the BRMPU as a generalization of the NaSch model, we calculate analytically the steady-state distribution of the time headways (TH) which are defined as the time intervals between the departures (or arrivals) of two successive particles (i.e., buses) recorded by a detector placed at a fixed site (i.e., bus stop) on the model route. We compare these TH distributions with the corresponding results of our computer simulations of the BRMPU, as well as with the data from the simulation of the two extended NaSch models. We also investigate interesting kinetic properties exhibited by the BRMPU during its time evolution from random initial states towards its steady-states.

EFFECTS OF ON- AND OFF-RAMPS IN CELLULAR AUTOMATA MODELS FOR TRAFFIC FLOW

We present results on the modeling of on- and off-ramps in cellular automata for traffic flow, especially the Nagel–Schreckenberg model. We study two different types of on-ramps that cause qualitatively the same effects. In a certain density regime ρ low < ρ < ρ high one observes plateau formation in the fundamental diagram. The plateau value depends on the input-rate of cars at the on-ramp. The on-ramp acts as a local perturbation that separates the system into two regimes: A regime of free flow and another one where only jammed states exist. This phase separation is the reason for the plateau formation and implies a behavior analogous to that of stationary defects. This analogy allows to perform very fast simulations of complex traffic networks with a large number of on- and off-ramps because one can parametrise on-ramps in an exceedingly easy way.

Reply to “Comment on ‘Critical behavior of a traffic flow model’ ”

The use of the dynamical structure factor in order to investigate the transition of the Nagel-Schreckenberg model from free to congested traffic is defended [see the preceding Comment by Chowdhury et al., Phys. Rev. E 61, 3270 (2000)].

Simulation of vehicular traffic: a statistical physics perspective

Computer simulation of vehicular traffic can give us insight into a wide variety of phenomena observed in real traffic flow, thereby leading to a greater qualitative understanding of its basic principles. These simulations also help us explore the effects of implementing new traffic rules and control systems without doing potentially hazardous experiments with real traffic. For the sake of computational efficiency, many of the microscopic traffic models developed in recent years have been formulated using the language of cellular automata. These models of vehicular traffic belong to a class of nonequilibrium systems called driven-diffusive lattice gases (B. Schmittmann and R.K.P. Zia, 1995; G. Schutz, 2000). Over the last few years, several groups have been doing extensive computer simulations to understand the nature of the steady states and fluctuations, as well as the kinetics of evolution toward these steady states. The starting points of our discussion are the Nagel-Schreckenberg model (K. Nagel and M. Schreckenberg, 1992) and the Biham-Middleton-Levine model (O. Biham et al., 1992).

The Nagel-Schreckenberg model revisited

The Nagel-Schreckenberg model is a simple cellular automaton for a realistic description of single-lane traffic on highways. For the case υmax=1 the properties of the stationary state can be obtained exactly. For the more relevant case υmax > 1, however, one has to rely on Monte Carlo simulations or approximative methods. Here we study several analytical approximations and compare with the results of computer simulations. The role of the braking parameter p is emphasized. It is shown how the local structure of the stationary state depends on the value of p. This is done by combining the results of computer simulations with those of the approximative methods.

Self-organization of traffic jams in cities: Effects of stochastic dynamics and signal periods

We propose a cellular automata model for vehicular traffic in cities by combining (and appropriately modifying) ideas borrowed from the Biham-Middleton-Levine (BML) model of city traffic and the Nagel-Schreckenberg (NS) model of highway traffic. We demonstrate a phase transition from the "free-flowing" dynamical phase to the completely "jammed" phase at a vehicle density which depends on the time periods of the synchronized signals and the separation between them. The intrinsic stochasticity of the dynamics, which triggers the onset of jamming, is similar to that in the NS model, while the phenomenon of complete jamming through self-organization as well as the final jammed configurations are similar to those in the BML model. Using our new model, we have made an investigation of the time-dependence of the average speeds of the cars in the "free-flowing" phase as well as the dependence of flux and jamming on the time period of the signals.

Spatio-temporal organization of vehicles in a cellular automata model of traffic with `slow-to-start' rule

The spatio-temporal organizations of vehicular traffic in cellular-automata models with `slow-to-start' rules are qualitatively different from those in the Nagel-Schreckenberg (NaSch) model of highway traffic. Here we study the effects of such a slow-to-start rule, introduced by Benjamin, Johnson and Hui (BJH), on the distributions of the distance-headways, time-headways, jam sizes and sizes of the gaps between successive jams by a combination of approximate analytical calculations and extensive computer simulations. We compare these results for the BJH model with the corresponding results for the NaSch model and interpret the qualitative differences in the nature of the spatio-temporal organizations of traffic in these two models in terms of a phase separation of the traffic caused by the slow-to-start rule in the BJH model.

Metastable states in cellular automata for traffic flow

Measurements on real traffic have revealed the existence of metastable states with very high flow. Such states have not been observed in the Nagel-Schreckenberg (NaSch) model which is the basic cellular automaton for the description of traffic. Here we propose a simple generalization of the NaSch model by introducing a velocity-dependent randomization. We investigate a special case which belongs to the so-called slow-to-start rules. It is shown that this model exhibits metastable states, thus sheding some light on the prerequisites for the occurance of hysteresis effects in the flow-density relation.

Distributions of time- and distance-headways in the Nagel-Schreckenberg model of vehicular traffic: effects of hindrances

In the Nagel-Schreckenberg model of vehicular traffic on single-lane highways vehicles are modelled as particles which hop forward from one site to another on a one dimensional lattice and the inter-particle interactions mimic the manner in which the real vehicles influence each other's motion. In this model the number of empty lattice sites in front of a particle is taken to be a measure of the corresponding distance-headway(DH). The time-headway(TH) is defined as the time interval between the departures (or arrivals) of two successive particles recorded by a detector placed at a fixed position on the model highway. We investigate the effects of spatial inhomogeneities of the highway (static hindrances) on the DH and TH distributions in the steady-state of this model.