Effect of overpasses in the Biham-Middleton-Levine traffic flow model with random and parallel update rule

The effect of overpasses in the Biham-Middleton-Levine traffic flow model with random and parallel update rules has been studied. An overpass is a site that can be occupied simultaneously by an eastbound car and a northbound one. Under periodic boundary conditions, both self-organized and random patterns are observed in the free-flowing phase of the parallel update model, while only the random pattern is observed in the random update model. We have developed mean-field analysis for the moving phase of the random update model, which agrees with the simulation results well. An intermediate phase is observed in which some cars could pass through the jamming cluster due to the existence of free paths in the random update model. Two intermediate states are observed in the parallel update model, which have been ignored in previous studies. The intermediate phases in which the jamming skeleton is only oriented along the diagonal line in both models have been analyzed, with the analyses agreeing well with the simulation results. With the increase of overpass ratio, the jamming phase and the intermediate phases disappear in succession for both models. Under open boundary conditions, the system exhibits only two phases when the ratio of overpasses is below a threshold in the random update model. When the ratio of the overpass is close to 1, three phases could be observed, similar to the totally asymmetric simple exclusion process model. The dependence of the average velocity, the density, and the flow rate on the injection probability in the moving phase has also been obtained through mean-field analysis. The results of the parallel model under open boundary conditions are similar to that of the random update model.