Dynamics of urban traffic congestion: A kinetic Monte Carlo approach to simulating collective vehicular dynamics

Transitions observed in the dynamical patterns of vehicular traffic, for instance, as a result of changes in traffic density, form an important class of phenomena that is sought to be explained by large-scale modeling using many interacting agents. While the dynamics of highway traffic has been the subject of intense investigation over the last few decades, there is as yet comparatively little understanding of the patterns of urban traffic. The macroscopic collective behavior of cars in the network of roads inside a city is marked by relatively high vehicular densities and the presence of signals that coordinate movement of cross-flowing traffic traveling along several directions. In this article, we have presented a novel kinetic Monte Carlo simulation approach for studying the dynamics of urban traffic congestion. This allows us to study continuous-time, continuous-space models of traffic flow in the presence of stochastic fluctuations, which contrast with the dominant paradigm of cellular automata models. We first reproduce well-known results of such discrete models for traffic flow in the absence of any intersections, and then, show the corresponding behavior in the presence of an intersection where cross-flowing traffic is regulated by a signal. The fundamental diagram of traffic flow in the presence of a signal shows a broad plateau indicating that the flow is almost independent of small variations in vehicle density for an intermediate range of densities. This is unlike the case where there are no intersections, where a sharp transition is observed between free flow behavior and jamming on changing vehicle density. The distribution of congestion times shows a power-law scaling regime over an extended range for the stochastic case when exponential-like right skewed probability distributions are used. These results reproduce in a simple setting the empirically observed power-law behavior in congestion time distributions for Indian urban traffic that is validated here with a much larger data-set.