A model of jam formation in congested traffic

Abstract

We study a model of irreversible jam formation in congested vehicular traffic on an open segment of a single-lane road. The vehicles obey a stochastic discrete-time dynamics which is a limiting case of the generalized Totally Asymmetric Simple Exclusion Process. Its characteristic features are: (a) the existing clusters of jammed cars cannot break into parts; (b) when the leading vehicle of a cluster hops to the right, the whole cluster follows it deterministically, and (c) any two clusters of vehicles, occupying consecutive positions on the chain, may become nearest-neighbors and merge irreversibly into a single cluster. The above dynamics was used in a one-dimensional model of irreversible aggregation by Bunzarova and Pesheva [Phys. Rev. E 95, 052105 (2017)]. The model has three stationary non-equilibrium phases, depending on the probabilities of injection (α), ejection (β), and hopping (p) of particles: a many-particle one, MP, a completely jammed phase CF, and a mixed MP+CF phase. An exact expression for the stationary probability P(1) of a completely jammed configuration in the mixed MP+CF phase is obtained. The gap distribution between neighboring clusters of jammed cars at large lengths L of the road is studied. Three regimes of evolution of the width of a single gap are found: (i) growing gaps with length of the order O(L) when β > p; (ii) shrinking gaps with length of the order O(1) when β < p; and (iii) critical gaps at β = p, of the order O(L1/2). These results are supported by extensive Monte Carlo calculations.