Stationary distributions of the multi-type ASEP

TL;DR

This paper presents a recursive construction for the stationary distribution of multi-type ASEP on finite rings or infinite lines, using queueing systems and matrix product methods, with applications in sampling and analyzing convoy phenomena.

Contribution

It introduces a novel queueing-based recursive method for multi-type ASEP stationary distributions, extending previous models to include probabilistic unused services.

Findings

Provides an exact sampling method for the stationary distribution on rings.

Derives rational function expressions for stationary probabilities.

Describes convoy formation phenomena in large systems.

Abstract

We give a recursive construction of the stationary distribution of multi-type asymmetric simple exclusion processes on a finite ring or on the infinite line $Z$. The construction can be interpreted in terms of "multi-line diagrams" or systems of queues in tandem. Let $q$ be the asymmetry parameter of the system. The queueing construction generalises the one previously known for the totally asymmetric ($q=0$) case, by introducing queues in which each potential service is unused with probability $q^k$ when the queue-length is $k$. The analysis is based on the matrix product representation of Prolhac, Evans and Mallick. Consequences of the construction include: a simple method for sampling exactly from the stationary distribution for the system on a ring; results on common denominators of the stationary probabilities, expressed as rational functions of $q$ with non-negative integer coefficients; and probabilistic descriptions of "convoy formation" phenomena in large systems.