Fixed points for multi-class queues

TL;DR

This paper extends Burke's fixed-point theorem to multi-class queues with priority levels, analyzing their properties across various queue models and revealing unique clustering behaviors of lower-priority customers.

Contribution

It generalizes fixed-point results to multi-type queues with priorities, linking queue interchangeability with particle system frameworks.

Findings

Fixed points exhibit clustering of lower-priority customers.

Lower-priority work can occur only at measure-zero times in Brownian queues.

Interchangeability properties are key to proving fixed points in multi-type queues.

Abstract

Burke's theorem can be seen as a fixed-point result for an exponential single-server queue; when the arrival process is Poisson, the departure process has the same distribution as the arrival process. We consider extensions of this result to multi-type queues, in which different types of customer have different levels of priority. We work with a model of a queueing server which includes discrete-time and continuous-time M/M/1 queues as well as queues with exponential or geometric service batches occurring in discrete time or at points of a Poisson process. The fixed-point results are proved using interchangeability properties for queues in tandem, which have previously been established for one-type M/M/1 systems. Some of the fixed-point results have previously been derived as a consequence of the construction of stationary distributions for multi-type interacting particle systems, and we explain the links between the two frameworks. The fixed points have interesting "clustering" properties for lower-priority customers. An extreme case is an example of a Brownian queue, in which lower-priority work only occurs at a set of times of measure 0 (and corresponds to a local time process for the queue-length process of higher priority work).