Schreckenberg Model Research Articles

Simulation of Vehicular Traffic using Macro- and Microscopic Models

To effectively regulate traffic on highways and networks of modern megacities, it is necessary to introduce Intelligent Transport Systems, which include many innovative solutions, in particular, mathematical models for describing the dynamics of traffic flows.The article is devoted to a brief description of the current state in this area in its development — from the simplest macroscopic and microscopic models that have become classic to modern developments.Special attention is paid to the original multilane models developed by the authors of the article within both approaches.The macroscopic model is based on the quasigasdynamic approach, while the microscopic one uses the ideology of cellular automata and constitutes a generalization of the Nagel-Schreckenberg model for the multilane case.The difference in the representation method and the mathematical apparatus for the mac-roscopic and microscopic description of traffic flows is briefly described, followed by the review of the main models at different stages of their development, presented by foreign and Russian authors.Special attention is paid to the three-phase theory of Boris Kerner and models built in the framework of this theory.Examples of modern software for traffic modeling are given.The original quasigasdynamic model of traffic flows, which uses the continuum approximation and is constructed by analogy with the well-known model of gas dynamics, is briefly described. Due to the introduction of the lateral speed, the model is generalized to the multilane case.An original microscopic model based on the cellular automata theory and representing a generalization of Nagel- Schreckenberg model for the multilane case is described. The model has been further developed by taking into account various driving strategies and behavioral aspects.The article presents a brief overview of the state of the art in the field of mathematical modeling of traffic flows, as well as original macroscopic and microscopic models developed by the authors for the case of multilane traffic.

Effect of pedestrian traffic light on traffic flow accompany with pedestrian crossing

In reality, vehicles and pedestrians are delayed for their interference with each other at the crosswalk, or by the unreasonable allocation of traffic light time. Especially under the popular rule of vehicles yield to pedestrians in front of the crosswalk, traffic flow are seriously affected. In this paper, the effects of pedestrian traffic light and the related parameters on traffic flow are studied. On the road, the Nagel–Schreckenberg (NaSch) model is conducted to simulate the scenario with and without pedestrian traffic light. Results show that, vehicle traffic flow can be effectively improved with pedestrian traffic light. Moreover, the optimization of pedestrian waiting time with different vehicle and pedestrian arrival rates are obtained from comprehensive simulations. Finally, effects of some other parameters are briefly discussed.

Extreme events in Nagel–Schreckenberg model of traffic flow on complex networks

The Nagel–Schreckenberg vehicular traffic model is extended to analyse a system of vehicles moving on a scale-free network to study extreme events in them. The dependence of the free-flow and congestion states on the density of vehicles is determined. In particular, a free-flow state at low vehicular density is observed and it gradually transitions to a congested state at higher density. Using detrended fluctuation analysis, it is shown that the flux of vehicles are long range correlated in a regime of low density. At higher densities, the flux becomes uncorrelated. We study the recurrence interval distribution for extreme events and the probability for its occurrence at low and high vehicular density regimes. It is shown that the occurrence probability for extreme events is independent of the degree of the node or the threshold used for defining extreme events

Social dilemma in traffic with heterogeneous drivers

There is a “tragedy of the traffic” analogous to the “tragedy of the commons” that can be caused by overtaking. We analyze the effect of overtaking in a minimal model of vehicular traffic, the Nagel–Schreckenberg model, with two types of strategies: vehicles that overtake and vehicles that do not. We show that, under certain circumstances, overtaking is good because it maximize the flux of vehicles and minimizes the vehicle’s mean time spent on the road. However, when these conditions are not met, overtaking is harmful to all. More specifically, we found that a social dilemma emerges in the vicinity of the transition to the congested traffic if the probability of random deceleration is low, which can also happen in more realistic single-lane models. The essential mechanism creating the social dilemma is the abrupt deceleration when the overtaking vehicle returns to its lane. We analyze how the payoffs depend on the frequency of strategies in the population to conclude that the vehicles that overtake are defectors and the ones that do not are cooperators, analogous to the strategies in “tragedy of the commons” class of games.

Uncertainty Quantification of Load Effects under Stochastic Traffic Flows

Load effect characterization under traffic flow has received tremendous attention in bridge engineering, and uncertainty quantification (UQ) of load effect is critical in the inference process. Bayesian probabilistic approach is developed to overcome the unreliable issue caused by negligence of uncertainty of parametric and modeling aspects. Stochastic traffic load simulation is conducted by embedding the random inflow component into the Nagel–Schreckenberg (NS) model, and load effects are calculated by stochastic traffic load samples and influence lines. Two levels of UQ are performed for traffic load effect characterization: at parametric level of UQ, not only the optimal parameter values but also the associated uncertainties are identified; at model level of UQ, rather than using a single prescribed probability model for load effects, a set of probability distribution model candidates is proposed, and model probability of each candidate is evaluated for selecting the most suitable/plausible probability distribution model. Analytic work was done to give closed-form solutions for the expression involved in both parametric and model UQ. In the simulated examples, the efficiency and robustness of the proposed approach are firstly validated, and UQ are performed to different load effect data achieved by varying the structural span length under the changing total traffic volume. It turns out that the uncertainties of load effects are traffic-specific and response-specific, so it is important to conduct UQ of load effects under different traffic scenarios by using the developed approach.

Impact of traffic lights on car accidents at intersections

Using the cellular automata Nagel–Schreckenberg (NaSch) model, we numerically study the impact of traffic lights on the probability of car accidents ([Formula: see text]) at the intersection of two roads. It is found that, the probability [Formula: see text] is more stable with variation of the green light ([Formula: see text]) when the symmetric lights are adopted. Moreover, simulation results show the existence of a critical time [Formula: see text], below which ([Formula: see text]) [Formula: see text] increases as the injection rate ([Formula: see text]) increases, however, above which ([Formula: see text]) the growing of [Formula: see text] has for effect the decrease of [Formula: see text]. Furthermore, the decrease of [Formula: see text] is almost always accompanied by a loss of the flux, especially with asymmetrical lights. To overcome this problem, we proposed a strategy that can greatly increase the flux and keep the probability [Formula: see text] as small as possible, especially for the low and high injection rates.

Continuously variable spreading exponents in the absorbing Nagel–Schreckenberg model

I study the critical behavior of a traffic model with an absorbing state. The model is a variant of the Nagel–Schreckenberg (NS) model, in which drivers do not decelerate if their speed is smaller than their headway, the number of empty sites between them and the car ahead. This makes the free-flow state (i.e. all vehicles traveling at the maximum speed, vmax, and with all headways greater than vmax) absorbing; such states are possible for densities ρ smaller than a critical value . Drivers with nonzero velocity, and with headway equal to velocity, decelerate with probability p. This absorbing Nagel–Schreckenberg (ANS) model, introduced in Iannini and Dickman (2017 Phys. Rev. E 95 022106), exhibits a line of continuous absorbing-state phase transitions in the ρ-p plane. Here I study the propagation of activity from a localized seed, and find that the active cluster is compact, as is the active region at long times, starting from uniformly distributed activity. The critical exponents δ (governing the decay of the survival probability) and η (governing the growth of activity) vary continuously along the critical curve. The exponents satisfy a hyperscaling relation associated with compact growth.

An analysis on the traffic processing efficiency of a combination of serial and parallel bottlenecks

By means of the Nagel–Schreckenberg model, we have investigated into the maximum vehicular flow rate of traffic processing bottlenecks. The evaluated analytical form of this flow rate is found to give quantitative insights into the underlying physics of collective vehicular motions constrained by these bottlenecks. Our analysis shows that for large-scale expansion, a new class of processing bottleneck known as the serial bottleneck is more efficient than the conventional parallel bottleneck in the absence of human driving behavior. When characteristics such as slow-to-start is considered in the model, the consequential delay due to human reaction time not only degrade the overall efficiency, it also diminishes the efficacy of serial processing such that a serial bottleneck is no longer tenable for traffic processing. These results point to the fundamental importance of optimizing traffic efficiency, which we illustrate by elucidating the detailed mechanisms with which vehicles interact collectively in the bottlenecks. In particular, we demonstrate that by constructing combinations of serial and parallel bottlenecks, optimal efficiencies are achieved via configurations with few (many) lanes of a large (small) number of serial units when the processing time is short (long). A direct implication of these results is that autonomous self-driving vehicles could serve to improve the transportation capacity for the densely populated urban cities of the future, due to the intrinsically more efficient collective vehicular motions through these bottlenecks.

Behaviour of traffic on a link with traffic light boundaries

This paper considers a single link with traffic light boundary conditions at both ends, and investigates the traffic evolution over time with various signal and system configurations. A hydrodynamic model and a modified stochastic domain wall theory are proposed to describe the local density variation. The Nagel–Schreckenberg model (NaSch), an agent based stochastic model, is used as a benchmark. The hydrodynamic model provides good approximations over short time scales. The domain wall model is found to reproduce the time evolution of local densities, in good agreement with the NaSch simulations for both short and long time scales. A systematic investigation of the impact of network parameters, including system sizes, cycle lengths, phase splits and signal offsets, on traffic flows suggests that the stationary flow is dominated by the boundary with the smaller split. Nevertheless, the signal offset plays an important role in determining the flow. Analytical expressions of the flow in relation to those parameters are obtained for the deterministic domain wall model and match the deterministic NaSch simulations. The analytic results agree qualitatively with the general stochastic models. When the cycle is sufficiently short, the stationary state is governed by effective inflow and outflow rates, and the density profile is approximately linear and independent of time.

Modeling and Analysis of Traffic Flow Considering Automatic Gap Control

SUMMARYIn recent years, the number of vehicles with automatic gap control (AGC) (e.g., adaptive cruise control) has increased with the development of vehicle control technology. AGC mainly helps to ease driver fatigue on highways. Vehicles equipped with AGC can maintain an appropriate distance between two vehicles. Therefore, the traffic flow can be improved by AGC‐equipped vehicles. In this research, we develop a traffic flow model that considers AGC‐equipped vehicles, and is based on the Nagel–Schreckenberg model, which is a probabilistic cellular automaton model. The Nagel–Schreckenberg model can reproduce the basic features of traffic flow. On analyzing the mathematical model for the traffic including AGC‐equipped vehicles, we can obtain analytical solutions of the traffic flow and the average velocity.

Occurrence of synchronized flow due to overtaking strategy in the Nagel–Schreckenberg model

The Nagel–Schreckenberg model with overtaking strategy (NSOS) is proposed, and numerical simulations are performed for both closed and open boundary conditions. The fundamental diagram, space-time diagram and spatial-temporal distribution of speed are investigated. In order to identify the synchronized flow state, both the correlation functions (autocorrelation and cross-correlation) and the one-minute average flow rate versus density diagram are studied. All the results verify that synchronized flow does occur in our model.

The investigation of the reentrance phenomenon in cellular automaton traffic flow model

Abstract We investigate analytically and by numerical simulations, the reentrant low density phase for high injection rate in the Nagel–Schreckenberg model. It is found that the reentrance phenomenon is a direct consequence of our injection strategy. Indeed, by adopting our injection rule, an injection rate exists, above which the in-flow begins to decrease by increasing the injection rate. In addition, we have studied the extraction rate interval at which the reentrance of the low density phase appears. It is found that this interval increases with increasing the maximal velocity. For the non deterministic case, the reentrance interval shifts to higher values of extraction rate.

Influence of bus stop with left-turn lines between two adjacent signalized intersections**Project supported by the National Natural Science Foundation of China (Grant No. 50478088) and the Natural Science Foundation of Hebei Province, China (Grant No. E2015202266).

Based on the symmetric two-lane Nagel–Schreckenberg (STNS) model, a three-lane cellular automaton model between two intersections containing a bus stop with left-turning buses is established in which model the occurrences of vehicle accidents are taken into account. The characteristics of traffic flows with different ratios of left-turn lines are discussed via the simulation experiments. The results indicate that the left-turn lines have more negative effects on capacity, accident rate as well as delay if the stop is located close to the intersections, where the negative effect in a near-side stop is more severe than that in a far-side one. The range of appropriate position for a bus stop without the bottleneck effect becomes more and more narrow with the increase of the ratio of left-turn bus lines. When the inflow is small, a short signal cycle and a reasonable offset are beneficial. When the inflow reaches or exceeds the capacity, a longer signal cycle is helpful. But if the stop position is inappropriate, the increase of cycle fails in reducing the negative effect of left-turning buses and the effectiveness of offset is weakened.

Effects of heterogeneous traffic with speed limit zone on the car accidents

Using the extended Nagel–Schreckenberg (NS) model, we numerically study the impact of the heterogeneity of traffic with speed limit zone (SLZ) on the probability of occurrence of car accidents [Formula: see text]. SLZ in the heterogeneous traffic has an important effect, typically in the mixture velocities case. In the deterministic case, SLZ leads to the appearance of car accidents even in the low densities, in this region [Formula: see text] increases with increasing of fraction of fast vehicles [Formula: see text]. In the nondeterministic case, SLZ decreases the effect of braking probability [Formula: see text] in the low densities. Furthermore, the impact of multi-SLZ on the probability [Formula: see text] is also studied. In contrast with the homogeneous case [X. Li, H. Kuang, Y. Fan and G. Zhang, Int. J. Mod. Phys. C 25 (2014) 1450036], it is found that in the low densities the probability [Formula: see text] without SLZ [Formula: see text] is low than [Formula: see text] with multi-SLZ [Formula: see text]. However, the existence of multi-SLZ in the road decreases the risk of collision in the congestion phase.

Cellular automata model simulating traffic car accidents in the on-ramp system

In this paper, using Nagel–Schreckenberg model we study the on-ramp system under the expanded open boundary condition. The phase diagram of the two-lane on-ramp system is computed. It is found that the expanded left boundary insertion strategy enhances the flow in the on-ramp lane. Furthermore, we have studied the probability of the occurrence of car accidents. We distinguish two types of car accidents: the accident at the on-ramp site (Prc) and the rear-end accident in the main road (Pac). It is shown that car accidents at the on-ramp site are more likely to occur when traffic is free on road A. However, the rear-end accidents begin to occur above a critical injecting rate αc1. The influence of the on-ramp length (LB) and position (xC0) on the car accidents probabilities is studied. We found that large LB or xC0 causes an important decrease of the probability Prc. However, only large xC0 provokes an increase of the probability Pac. The effect of the stochastic randomization is also computed.

The effect of traffic light on accident probability in open and periodic boundaries system

In this paper we numerically study the dependence of car accident probability Pac, per site and per time step on cycle time T of traffic light, both in open and periodic boundaries system. In this study one traffic light is placed in the middle of the system. This work is based on Nagel and Schreckenberg (NaSch) model (Nagel and Schreckenberg (1992)) in parallel dynamics. The Pac dependence on T and the (α,β) phase diagrams are established. α and β are the injecting and extracting rates of cars in the traffic lane respectively. The increase of the cycle time light T causes an important decrease of the accident probability Pac both in the open and periodic cases.

Simulation of three lanes one-way freeway in low visibility weather by possible traffic accidents

The aim of this work is to investigate the traffic impact of low visibility weather on a freeway including the fraction of real vehicle rear-end accidents and road traffic capacity. Based on symmetric two-lane Nagel–Schreckenberg (STNS) model, a cellular automaton model of three-lane freeway mainline with the real occurrence of rear-end accidents in low visibility weather, which considers delayed reaction time and deceleration restriction, was established with access to real-time traffic information of intelligent transportation system (ITS). The characteristics of traffic flow in different visibility weather were discussed via the simulation experiments. The results indicate that incoming flow control (decreasing upstream traffic volume) and inputting variable speed limits (VSL) signal are effective in accident reducing and road actual traffic volume’s enhancing. According to different visibility and traffic demand the appropriate control strategies should be adopted in order to not only decrease the probability of vehicle accidents but also avoid congestion.

Simulation study of car accidents at the intersection of two roads in the mixed traffic flow

Using cellular automata (CA) Nagel–Schreckenberg (NaSch) model, we numerically study the probability P ac of the occurrence of car accidents at nonsignalized intersection when drivers do not respect the priority rules. We also investigated the impact of mixture lengths and velocities of vehicles on this probability. It is found that in the first case, where vehicles distinguished only by their lengths, the car accidents start to occur above a critical density ρc. Furthermore, the increase of the fraction of long vehicles (FL) delays the occurrence of car accidents (increasing ρc) and increases the risk of collisions when ρ > ρc. In other side, the mixture of maximum velocities (with same length for all vehicles) leads to the appearance of accidents at the intersection even in the free flow regime. Moreover, the increase of the fraction of fast vehicles (Ff) reduces the accident probability (P ac ). The influence of roads length is also studied. We found that the decrease of the roads length enhance the risk of collision.

Emergence of bistable states and phase diagrams of traffic flow at an unsignalized intersection

This paper studies phase diagrams of traffic states induced by the bottleneck of an unsignalized intersection which consists of two perpendicular one-lane roads. Parallel updates rules are employed for both roads. At the crossing point, in order to avoid colliding, the consideration of yield dynamics may be suitable herein. Different from previous studies, the deterministic Nagel and Schreckenberg model is adopted in this work. Based on theoretical analysis and computer simulations, the phase diagrams of traffic flow have been presented and the flow formulas in all regions have been derived in the phase diagram. The results of theoretical analysis are in good agreement with computer simulation ones. One finds an interesting phenomenon: there exist bistable states in some regions of the phase diagrams.

Simulation study of traffic car accidents in single-lane highway

In this paper we numerically study the probability Pac of the occurrence of car accidents in the extended Nagel–Schreckenberg (NS) model in the case of mixture of fast (Vmax1=5) and slow vehicles (Vmax2=1) by taking also to the risky overtaking of fast vehicles. In comparison with previous existing models, we find that accidents can occur in the free traffic phase and/or congested one depending on the overtaking rate of fast vehicles. The effect of evacuation of damaged vehicles from the road with probabilities Pevf and Pevs of fast and slow vehicles respectively on the traffic flow behavior is also computed.