The M/M/1 queue is Bernoulli

TL;DR

This paper demonstrates that the departure process of a stationary M/M/1 queue is metrically isomorphic to a two-sided Bernoulli shift, extending Burke's classical theorem and exploring related open problems.

Contribution

It establishes a new measure-theoretic isomorphism for the M/M/1 queue departure process, linking queue theory with Bernoulli shifts.

Findings

The departure process is metrically isomorphic to a two-sided Bernoulli shift.

Extensions of Burke's theorem are discussed with open problems identified.

Provides a measure-theoretic perspective on classical queue results.

Abstract

The classical output theorem for the M/M/1 queue, due to Burke (1956), states that the departure process from a stationary M/M/1 queue, in equilibrium, has the same law as the arrivals process, that is, it is a Poisson process. In this paper we show that the associated measure-preserving transformation is metrically isomorphic to a two-sided Bernoulli shift. We also discuss some extensions of Burke's theorem where it remains an open problem to determine if, or under what conditions, the analogue of this result holds.