Attractiveness of Brownian queues in tandem

TL;DR

This paper proves that in a sequence of Brownian queues in tandem, the departure process converges to a Brownian motion, establishing it as an attractive invariant measure, using coupling techniques related to Brownian percolation.

Contribution

It demonstrates the convergence of departure processes to Brownian motion in tandem queues and introduces a coupling approach via Brownian percolation models.

Findings

Departure process converges to Brownian motion as number of queues increases

Brownian motion is an attractive invariant measure for the queueing operator

Coupling technique links tandem queues with Brownian percolation models

Abstract

Consider a sequence of n bi-infinite and stationary Brownian queues in tandem. Assume that the arrival process entering in the first queue is a zero mean ergodic process. We prove that the departure process from the n-th queue converges in distribution to a Brownian motion as n goes to infinity. In particular this implies that the Brownian motion is an attractive invariant measure for the Brownian queueing operator. Our proof exploits the relationship between the Brownian queues in tandem and the last-passage Brownian percolation model, developing a coupling technique in the second setting. The result is also interpreted in the related context of Brownian particles acting under one sided reflection.