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. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-1-9,
author = {Michael Keane and Neil O'Connell},
title = {The M/M/1 queue is Bernoulli},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {205-210},
zbl = {1177.60072},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-1-9}
}
Michael Keane; Neil O'Connell. The M/M/1 queue is Bernoulli. Colloquium Mathematicae, Tome 111 (2008) pp. 205-210. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm110-1-9/