Nondeterministic Nagel-Schreckenberg traffic model with open boundary conditions

Renormalization-group study of the Nagel-Schreckenberg model

We study the phase transition from free flow to congested phases in the Nagel-Schreckenberg (NS) model by using the dynamically driven renormalization group (DDRG). The breaking probability p that governs the driving strategy is investigated. For the deterministic case p=0, the dynamics remain invariant in each renormalization-group (RG) transformation. Two fully attractive fixed points, ρ_{c}^{*}=0 and 1, and one unstable fixed point, ρ_{c}^{*}=1/(v_{max}+1), are obtained. The critical exponent ν which is related to the correlation length is calculated for various v_{max}. The critical exponent appears to decrease weakly with v_{max} from ν=1.62 to the asymptotical value of 1.00. For the random case p>0, the transition rules in the coarse-grained scale are found to be different from the NS specification. To have a qualitative understanding of the effect of stochasticity, the case p→0 is studied with simulation, and the RG flow in the ρ-p plane is obtained. The fixed points p=0 and 1 that govern the driving strategy of the NS model are found. A short discussion on the extension of the DDRG method to the NS model with the open-boundary condition is outlined.

Transport dynamics of molecular motors that switch between an active and inactive state

Molecular motors are involved in key transport processes in the cell. Many of these motors can switch from an active to a nonactive state, either spontaneously or depending on their interaction with other molecules. When active, the motors move processively along the filaments, while when inactive they are stationary. We treat here the simple case of spontaneously switching motors, between the active and inactive states, along an open linear track. We use our recent analogy with vehicular traffic, where we go beyond the mean-field description. We map the phase diagram of this system, and find that it clearly breaks the symmetry between the different phases, as compared to the standard total asymmetric exclusion process. We make several predictions that may be testable using molecular motors in vitro and in living cells.

Traffic-light boundary in the deterministic Nagel-Schreckenberg model

The characteristics of the deterministic Nagel-Schreckenberg model with traffic-light boundary conditions are investigated and elucidated in a mostly theoretically way. First, precise analytical results of the outflow are obtained for cases in which the duration of the red phase is longer than one step. Then, some results are found and studied for cases in which the red phase equals one step. The main findings include the following. The maximum outflow is "road-length related" if the inflow is saturated; otherwise, if the inbound cars are generated stochastically, multiple theoretical outflow volumes may exist. The findings indicate that although the traffic-light boundary can be implemented in a simple and deterministic manner, the deterministic Nagel-Schreckenberg model with such a boundary has some unique and interesting behaviors.

Autocorrelations in the totally asymmetric simple exclusion process and Nagel-Schreckenberg model

We study via Monte Carlo simulation the dynamics of the Nagel-Schreckenberg model on a finite system of length L with open boundary conditions and parallel updates. We find numerically that in both the high and low density regimes the autocorrelation function of the system density behaves like 1-|t|/τ with a finite support [-τ,τ] . This is in contrast to the usual exponential decay typical of equilibrium systems. Furthermore, our results suggest that in fact τ=L/c , and in the special case of maximum velocity v{max}=1 (corresponding to the totally asymmetric simple exclusion process) we can identify the exact dependence of c on the input, output and hopping rates. We also emphasize that the parameter τ corresponds to the integrated autocorrelation time, which plays a fundamental role in quantifying the statistical errors in Monte Carlo simulations of these models.

Analytical results of the Nagel-Schreckenberg model with stochastic open boundary conditions

The deterministic Nagel-Schreckenberg model with stochastic open boundary conditions is investigated in a mostly analytical way. By means of the Markov Chain, we model the working process of the stochastic open boundaries. First, the analytical expression of the free-flow density profiles is derived. Then, we discuss theoretically how the right boundary determines the traffic capacity, global density, and density profiles. For these features, the analytical and numerical results agree well. This paper implies that the deterministic Nagel-Schreckenberg model with stochastic open boundaries is almost totally analyzable.

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.

Totally asymmetric exclusion process with long-range hopping

Generalization of the one-dimensional totally asymmetric exclusion process (TASEP) with open boundary conditions in which particles are allowed to jump l sites ahead with the probability pl approximately 1l/sigma+1 is studied by Monte Carlo simulations and the domain-wall approach. For sigma>1 the standard TASEP phase diagram is recovered, but the density profiles display additional features when 1<sigma<2. At the first-order transition line, the domain wall is localized and phase separation is observed. In the maximum-current phase the profile has an algebraic decay with a sigma -dependent exponent. Within the sigma<or=1 regime, where the transitions are found to be absent, analytical results in the continuum mean-field approximation are derived in the limit sigma=-1.

Open boundaries in a cellular automata model for synchronized flow: effects of nonmonotonicity

In this paper, we have discussed the traffic situations arising from the open boundary conditions (OBC) of a cellular automata model that can reproduce the synchronized flow. The model is different from the slow-to-start (STS) model in that the upper branch of the fundamental diagram in the periodic boundary conditions (PBC) is not monotonous but has an extremum. The phase diagram and the fundamental diagram of the model in the OBC are investigated. The results are compared with those of the STS model and those in the PBC. The current in the OBC as well as the density profiles in the different phases is also investigated.

Anisotropy effect on two-dimensional cellular-automaton traffic flow with periodic and open boundaries

Using computer simulations we investigate, in a version of the Biham-Middleton-Levine model with random sequential update on a square lattice, the anisotropy effect of the probabilities of the change of the motion directions of cars, from up to right (p(ur)) and from right to up (p(ru)), on the dynamical jamming transition and velocities under periodic boundaries on one hand and the phase diagram under open boundaries on the other hand. However, in the former case, the sharp jamming transition appears only for p(ur)=0=p(ru)=0 (i.e., when the cars alter their motion directions). In the open boundary conditions, it is found that the first-order line transition between jamming and moving phases is curved. Hence, by increasing the anisotropy, the moving phase region expands as well as the contraction of the jamming and maximal current phases takes place. Moreover, in the anisotropic case, the transition between the jamming phase (or moving phase) and the maximal current phase is of second order while in the isotropic case, and when each car changes its direction of motion at every time step (p(ru)=p(ur)=1), the transition is of first order. Furthermore, in the maximal current phase, the density profile decays with an exponent gamma approximately 1/4.

Open boundaries in a cellular automaton model for traffic flow with metastable states

The effects of open boundaries in the velocity-dependent randomization (VDR) model, a modified version of the well-known Nagel-Schreckenberg (NaSch) cellular automaton model for traffic flow, are investigated. In contrast to the NaSch model, the VDR model exhibits metastable states and phase separation in a certain density regime. A proper insertion strategy allows us to investigate the whole spectrum of possible system states and the structure of the phase diagram by Monte Carlo simulations. We observe an interesting microscopic structure of the jammed phases, which is different from the one of the NaSch model. For finite systems, the existence of high flow states in a certain parameter regime leads to a special structure of the fundamental diagram measured in the open system. Apart from that, the results are in agreement with an extremal principle for the flow, which has been introduced for models with a unique flow-density relation. Finally, we discuss the application of our findings for a systematic flow optimization. Here some surprising results are obtained, e.g., a restriction of the inflow can lead to an improvement of the total flow through a bottleneck.

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.