Multiclass processes, dual points and M/M/1 queues

TL;DR

This paper explores the invariant measures of multiclass processes like HAD and TASEP, introduces a new dual points approach for their analysis, and extends Burke's theorem to multiclass queues, providing new insights into their structure.

Contribution

It introduces a novel dual points method for analyzing multiclass processes and extends Burke's theorem to multiclass queueing systems, offering new tools and perspectives.

Findings

Invariant measures are the same for HAD and TASEP multiclass systems.

A new coupling via multi-line processes is established.

Extension of Burke's theorem to multiclass queues.

Abstract

We consider the discrete Hammersley-Aldous-Diaconis process (HAD) and the totally asymmetric simple exclusion process (TASEP) in Z. The basic coupling induces a multiclass process which is useful in discussing shock measures and other important properties of the processes. The invariant measures of the multiclass systems are the same for both processes, and can be constructed as the law of the output process of a system of multiclass queues in tandem; the arrival and service processes of the queueing system are a collection of independent Bernoulli product measures. The proof of invariance involves a new coupling between stationary versions of the processes called a multi-line process; this process has a collection of independent Bernoulli product measures as an invariant measure. Some of these results have appeared elsewhere and this paper is partly a review, with some proofs given only in outline. However we emphasize a new approach via dual points: when the graphical construction is used to construct a trajectory of the TASEP or HAD process as a function of a Poisson process in ZxR, the dual points are those which govern the time-reversal of the trajectory. Each line of the multi-line process is governed by the dual points of the line below. We also mention some other processes whose multiclass versions have the same invariant measures, and we note an extension of Burke's theorem to multiclass queues which follows from the results.