On dynamical analysis of a horizontal trafic queue

We consider a horizontal traffic queue on a circular road, where vehicles arrive according to a spatio-temporal Poisson process, and, upon arrival, move clockwise until they reach a common destination point, from where they depart the queue. When inside the queue, the speed of a vehicle is proportional to the distance to the vehicle in front. We provide a tight characterization of the throughput of such a queueing system. The dynamics of inter-vehicular spacings in between jumps is reminiscent of linear consensus on directed graphs. The finer properties of the departure process of the proposed queue depend on the dynamics of the minimum inter-vehicular distance. Motivated by the need to study such finer dynamics, we study the evolution of the Kullback-Leibler divergence of the inter-vehicular spacings in between and during jumps. Simulations that illustrate phase transitions in the throughput under nonlinear car following models are also provided.