The $M/M/1$ queue is Bernoulli
Volume 110 / 2008
Colloquium Mathematicum 110 (2008), 205-210
MSC: Primary 60K25, 37A50; Secondary 60J25, 60J65, 37H99.
DOI: 10.4064/cm110-1-9
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. 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.
