Convergence of tandem Brownian queues

TL;DR

This paper proves that passing an arbitrary continuous process through an infinite tandem of Brownian queues results in convergence to a Brownian motion, extending Burke's theorem to a broader class of initial conditions.

Contribution

It demonstrates convergence to the invariant measure for an arbitrary initial process in a tandem Brownian queue network, generalizing Burke's theorem.

Findings

Weak convergence to Brownian motion

Convergence holds under mild initial conditions

Results extend classical Burke's theorem

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.