Convergence of tandem Brownian queues

Abstract

It is known that in a stationary Brownian queue with both arrival and service processes equal in law to Brownian motion, the departure process is a Brownian motion, that is, Burke's theorem in this context. In this short note we prove convergence to this invariant measure: if we have an arbitrary continuous process satisfying some mild conditions as initial arrival process and pass it through an infinite tandem network of queues, the resulting process weakly converges to a Brownian motion. We assume independent and exponential initial workloads for all queues.

Attractiveness of Brownian queues in tandem

“…Our method of proof differs substantially from the methods developed for discrete valued queueing systems: In [21] a coupling between the departure times in every step of the tandem queue of each user is accomplished, while in [24] the waiting times of each user in every node of the tandem queue system are considered for the coupling. A rather simplified version of this result was presented in [15] where, using a path coupling of the departures processes, a non-stationary and one-sided (in time) system is studied with some particular initial conditions. Those techniques are non applicable to the current bi-infinite stationary setting.…”

“…A rather simplified version of this result was presented in [15] where, using a path coupling of the departures processes, a non-stationary and one-sided (in time) system is studied with some particular initial conditions. Those techniques are non applicable to the current bi-infinite stationary setting.…”