A queueing theory approach for a multi-speed exclusion process

TL;DR

This paper models multi-speed traffic flow using a generalized exclusion process, mapping it to queueing networks to analyze vehicle dynamics and congestion phenomena.

Contribution

It introduces a novel reaction-diffusion model with variable vehicle speeds and maps it to queueing networks to study traffic flow and condensation effects.

Findings

Derived the fundamental diagram of traffic flow.

Identified conditions for vehicle condensation.

Showed how speed changes impact traffic congestion.

Abstract

We consider a one-dimensional stochastic reaction-diffusion generalizing the totally asymmetric simple exclusion process, and aiming at describing single lane roads with vehicles that can change speed. To each particle is associated a jump rate, and the particular dynamics that we choose (based on 3-sites patterns) ensures that clusters of occupied sites are of uniform jump rate. When this model is set on a circle or an infinite line, classical arguments allow to map it to a linear network of queues (a zero-range process in theoretical physics parlance) with exponential service times, but with a twist: the service rate remains constant during a busy period, but can change at renewal events. We use the tools of queueing theory to compute the fundamental diagram of the traffic, and show the effects of a condensation mechanism.