Nagel-Schreckenberg Model Research Articles

Comment on “Analytical investigation of the open boundary conditions in the Nagel-Schreckenberg model”

In their recent paper [Phys. Rev. E 79, 031115 (2009)], Ning Jia and Shoufeng Ma used some Markov chain arguments for the analytical description of inflow in the deterministic Nagel-Schreckenberg model with open boundaries. In this context, they considered two different mechanisms of injecting vehicles: the standard injection rule and a popular expanded injection rule. While the results for the first one seem to be correct, simulations show that the inflow formula in case of the expanded injection rule yields only approximate results. Therefore, this comment provides the exact formula also in this case and explains the shortcoming in the derivation of Jia and Ma.

Analytical investigation of the open boundary conditions in the Nagel-Schreckenberg model

We theoretically investigate the mechanism of the open boundary conditions in the deterministic Nagel-Schreckenberg model, which was studied mainly by numerical simulations before. Our studies concentrate on the open boundaries and have found an effective approach for deducing the analytical expression of inflow. We also raise a removal rule which is analyzable. These findings provide a theoretical explanation of the behaviors of the open boundaries and allow the exact prediction of the traffic state by using the injection rate and the removal rate.

Intersection effects of arterial road for traffic flow

Based on the Nagel-Schreckenberg model, a traffic model for an arterial road with equal-distance-distributed intersections is established. The traffic flow is numerically simulated under the open boundary condition and the control of red and green lights. It is found from the results that the capacity of the arterial road is determined by the capacity of the intersection. For small densities, the flow has no relation with the number of intersections. Moreover, the rule of variation of the maximum point of average velocity with the green-light period is obtained with the mean-field method. On the contrary, the number of intersections plays a crucial role in the flow at larger densities. The arterial road can achieve optimal traffic state via regulating the period of traffic lights.

A cellular automaton model of traffic considering the dynamic evolution of velocity randomization probability

Few cellular automaton models of traffic have taken the relationship between the randomization probability of velocity and the specific value of current velocity into account. On the basis of Nagel-Schreckenberg model, this paper presents a modified cellular automaton model in which the randomization probability evolves with the driver’s remembering to historic experience and with the traffic condition around. Numerical simulation results show that the proposed model can reveal the influence of driver’s remembering on the traffic and generate higher average velocity and flux.

TASEP related models with traffic light boundary

Cellular automata models play an important role in traffic modeling. For some variants of the Nagel-Schreckenberg model, the effects of traffic light boundary conditions are considered. Based on previous results, the exact density profiles can be derived easily for deterministic dynamics. Additionally, the exact average outflow per traffic light cycle is presented not only in the deterministic case, but also for an important semi-stochastic variant with slow-to-start behaviour. Thereby, the models are strongly related to the well-known totally asymmetric simple exclusion process (TASEP) which can be regarded as a generic model for many driven particle systems.

Deceleration in advance in the Nagel–Schreckenberg traffic flow model

Based on the Nagel–Schreckenberg model, we study the impact of deceleration in advance on the dynamics of traffic flow. In the process of deceleration in advance, the effect of reaction delay and the effect of expectation are considered respectively. The traffic flow properties are studied by analyzing the fundamental diagram, spatio-temporal patterns, distance headway distribution and car accidents. The simulation results show that reaction delay brings complex traffic flow patterns and expectation makes the serious car accidents rarely happen.

Energy dissipation in the deterministic and nondeterministic Nagel–Schreckenberg models

In this paper, we numerically study energy dissipation per unit time per car E d in the Nagel–Schreckenberg (NS) model. Numerical results show that there is a critical density over which energy dissipation occurs, but below which no energy loss happens in the deterministic NS model. Energy dissipation depends on both the car density and initial configuration in the deterministic case. The energy dissipation rate, E d , calculated starting from a completely jammed state whose value is minimum, decreases monotonously with the increase of car density above the critical density. In the nondeterministic case, however, there is no critical density and energy dissipation happens in the whole density region, and initial configuration has no effects on energy dissipation. The dissipated energy has two different contributions: one coming from the interactions and another from the braking noise in the stochastic case. The relative contributions of the two dissipation mechanisms are presented. In the free-flow state, E d is proportional to p ( 1 − p ) where p is the stochastic braking probability. In the case of v max = 1 , E d is proportional to the mean density of “go and stop” car per time step ρ g s , which is equal to n 0 ( 1 − n 0 ) where n 0 is the fraction of stopped car. Theoretical analyses give an excellent agreement with numerical results.

The study of a cellular automaton traffic flow model with public transit，harbor-shaped bus stop and mixed different-maximum-speed vehicles on single lane

Based on the Nagel-Schreckenberg models of traffic flow， a cellular automaton traffic model with different-maximum-speed vehicles mixed on single lane is proposed， which considers public transit and harbor-shaped bus stop. The influence of deceleration probability， the density of vehicles， the average stop time of public transit vehicles， the proportion of public transit vehicles occupied in the mixed vehicles and the bus stop spacing on mixed traffic flow are analyzed by computer simulation. The models traffic flow characters are presented and analyzed.

Abnormal hysteresis effect and phase transitions in a velocity-difference dependent randomization CA model

Additional to the basic Nagel–Schreckenberg model for single-lane traffic flow, we consider the effect of the velocity-difference between two successive vehicles on the randomization of the latter one. This modification improves the transit capacity of the model, which results in a much higher simulated flux. Moreover, in a circular road, this model simulates an abnormal hysteresis effect. The characteristic shape of hysteresis curves may shed some light on the distinguish of synchronized flow and jams.

A modified NaSch model with density-dependent randomization for traffic flow

Based on the Nagel-Schreckenberg (NaSch) model of traffic flow, a modified cellular automaton (CA) traffic model with the density-dependent randomization (abbreviated as the DDR model) is proposed to simulate traffic flow. The fundamental diagram obtained by simulation shows the ability of this modified NaSch model to capture the essential features of traffic flow, e.g., synchronized flow, metastable state, hysteresis and phase separation at higher densities. Comparisons are made between this DDR model and the NaSch model, also between this DDR model and the VDR model. And the underlying mechanism is analyzed. All these results indicate that the presented model is reasonable and more realistic.

Cellular automaton model for mixed traffic flow with motorcycles

A single-lane cellular automaton model is proposed to simulate mixed traffic with motorcycles. By performing numerical simulations under the periodic boundary condition, some density-flow relations and the “lane-changing” behavior of motorcycles are investigated in detail. It is found that the maximum car flow decreases because of the “lane-changing” behavior of motorcycles. The maximum total flow increases first and then decreases with increasing motorcycle density. Moreover, the transition of the total flow from the free flow to the congested flow is smooth in this model. The “lane-changing” rate of motorcycles will decrease to zero finally with the increase of the car density. But its evolutionary trend is considerably complex. Another interesting fact is that, with the increase of the motorcycle density, the “lane-changing” rate increases first and decreases later. This phenomenon is very similar to the findings in other papers on multi-lane car flows. The“lane-changing” is almost of no use in increasing the flow of motorcycles as the motorcycle density is small. But it distinctly causes the increase in the flow of motorcycles when the motorcycle density is sufficiently large, and in this density regime, the flow of motorcycles gradually decreases to the one given with the Nagel–Schreckenberg model for motorcycles with the increase of the car density. The simulation results indicate that it is necessary to set a barrier or a lane for separating the motorcycle flow from the car flow except in some special density regime.

The effects of reaction delay in the Nagel-Schreckenberg traffic flow model

This paper studies a new model, which considers the effects of drivers reaction delay in the Nagel-Schreckenberg model. We studied the traffic flow properties in the new model under both periodic and open boundary conditions. The fundamental diagram, spatio-temporal patterns, density-density correlation functions, relaxation time, and distance headway distribution are investigated. Several interesting results are reported, for example, (i) the jam becomes less condensed when the delay effect is considered; (ii) the distance headway of the new model exhibits a multi-peak distribution when randomization p is small; (iii) for large p, the distribution of distance headway follows a power law in the new model; (iv) under open boundary conditions, the existence of a stationary jam near the left boundary will lower the flow rate.

Effects of a type of quenched randomness on car accidents in a cellular automaton model

In this paper we numerically study the probability Pac of the occurrence of car accidents in the Nagel-Schreckenberg (NS) model with a defect. In the deterministic NS model, numerical results show that there exists a critical value of car density below which no car accident happens. The critical density Pc1 is not related only to the maximum speed of cars, but also to the braking probability at the defect. The braking probability at a defect can enhance, not suppress, the occurrence of car accidents when its value is small. Only the braking probability at the defect is very large, car accidents can be reduced by the bottleneck. In the nondeterministic NS model, the probability Pac exhibits the same behaviors with that in the deterministic model except the case of vmax=1 under which the probability Pac is only reduced by the defect. The defect also induces the inhomogeneous distribution of car accidents over the whole road. Theoretical analyses give an agreement with numerical results in the deterministic NS model and in the nondeterministic NS model with vmax=1 in the case of large defect braking probability.

Analytical studies on a modified Nagel–Schreckenberg model with the Fukui–Ishibashi acceleration rule

We propose and study a one-dimensional traffic flow cellular automaton model of high-speed vehicles with the Fukui-Ishibashi-type (FI) acceleration rule for all cars, and the Nagel-Schreckenberg-type (NS) stochastic delay mechanism. By using the car-oriented mean field theory, we obtain analytically the fundamental diagrams of the average speed and vehicle flux depending on the vehicle density and stochastic delay probability. Our theoretical results, which may contribute to the exact analytical theory of the NS model, are in excellent agreement with numerical simulations.

Toward an improvement over Kerner-Klenov-Wolf three-phase cellular automaton model

The Kerner-Klenov-Wolf (KKW) three-phase cellular automaton model has a nonrealistic velocity of the upstream front in widening synchronized flow pattern which separates synchronized flow downstream and free flow upstream. This paper presents an improved model, which is a combination of the initial KKW model and a modified Nagel-Schreckenberg (MNS) model. In the improved KKW model, a parameter is introduced to determine the vehicle moves according to the MNS model or the initial KKW model. The improved KKW model can not only simulate the empirical observations as the initial KKW model, but also overcome the nonrealistic velocity problem. The mechanism of the improvement is discussed.

Optimizing on-ramp entries to exploit the capacity of a road

In this paper, we perform simulations of an extended Nagel-Schreckenberg model for one-lane and two-lane roads. We consider the presence of many entry ramps, placed in different locations, and we determine how the flux of cars of each entry ramp must be controlled in order to better exploit the capacity of the road. Our results are of relevance for the optimization of daily traffic, and set rules for the design of evacuation plans of urban conglomerations exposed to natural hazards, such as volcanoes.

Phase transition at an on-ramp in the Nagel–Schreckenberg traffic flow model

Recent empirical observations show that traffic flow is related to various phase transitions, especially at bottlenecks. This paper studies traffic flow induced by an on-ramp in the Nagel–Schreckenberg model. It is shown that when considering the situation that vehicles on main road and on-ramp interact in a relatively long road section, first-order phase transition from free flow to congested flow occurs. In contrast, it is second-order phase transition from free flow to congested flow if the interaction road section is short. We have investigated the reason why first-order phase transition occurs. The interaction of vehicles may be viewed as a kind of cooperation behavior of drivers. Therefore, our results indicate that in future traffic flow research, investigation on cooperation behavior should be carried out.

A stopped time dependent randomization cellular automata model for traffic flow controlled by traffic light

Modelling road traffic behavior using cellular automata has become a well-established method to analyze, understand, and even forecast the behavior of real road traffic, because the automata's evolution rules are simple, computationally efficient. In this paper, we presented a new model. In this model, the randomization probability is defined to be function of the stopped time of the vehicle: the longer the vehicle stops, the larger the randomization probability is. This means that the sensitivity of the drivers depends on the stopped time. The simulations show that although the fundamental diagram of the new model is similar to that of Nagel–Schreckenberg model, the saturated current depends on the cycle time of traffic light. We have explained the dependence of saturated current on cycle time and made the outlook of the future work. Our results indicate that we can adjust the cycle time of the traffic lights to enhance the road capacity.

Hysteresis Phenomenon in Deterministic Traffic Flows

We study phase transitions of a system of particles on the one-dimensional integer lattice moving with constant acceleration, with a collision law respecting slower particles. This simple deterministic ``particle-hopping'' traffic flow model being a straightforward generalization to the well known Nagel-Schreckenberg model covers also a more recent slow-to-start model as a special case. The model has two distinct ergodic (unmixed) phases with two critical values. When traffic density is below the lowest critical value, the steady state of the model corresponds to the ``free-flowing'' (or ``gaseous'') phase. When the density exceeds the second critical value the model produces large, persistent, well-defined traffic jams, which correspond to the ``jammed'' (or ``liquid'') phase. Between the two critical values each of these phases may take place, which can be interpreted as an ``overcooled gas'' phase when a small perturbation can change drastically gas into liquid. Mathematical analysis is accomplished in part by the exact derivation of the life-time of individual traffic jams for a given configuration of particles.

Effects of changing orders in the update rules on traffic flow

Based on the Nagel-Schreckenberg (NaSch) model of traffic flow, we study the effects of the orders of the evolutive rule on traffic flow. It has been found from simulation that the cellular automaton (CA) traffic model is very sensitively dependent on the orders of the evolutive rule. Changing the evolutive steps of the NaSch model will result in two modified models, called the SDNaSch model and the noise-first model, with different fundamental diagrams and jamming states. We analyze the mechanism of these two different traffic models and corresponding traffic behaviors in detail and compare the two modified model with the NaSch model. It is concluded that the order arrangement of the stochastic delay and deterministic deceleration indeed has remarkable effects on traffic flow.