Ergodicity of an SPDE Associated with a Many-Server Queue

TL;DR

This paper models a many-server queueing network using a coupled stochastic PDE and SDE system, proving ergodicity and uniqueness of the invariant distribution, thus solving a long-standing open problem in queueing theory.

Contribution

It introduces a novel infinite-dimensional diffusion model for large-scale queues and proves its ergodicity, addressing an open problem from 1981.

Findings

Established the unique invariant distribution of the model.

Connected the invariant distribution to the scaled stationary distributions of the queue.

Developed methods applicable to broader network analysis.

Abstract

We consider the so-called GI/GI/N queueing network in which a stream of jobs with independent and identically distributed service times arrive according to a renewal process to a common queue served by $N$ identical servers in a First-Come-First-Serve manner. We introduce a two-component infinite-dimensional Markov process that serves as a diffusion model for this network, in the regime where the number of servers goes to infinity and the load on the network scales as $1 - \beta N^{-1/2}+ o(N^{-1/2})$ for some $\beta > 0$. Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued It\^{o} equation and a stochastic partial differential equation on the positive half line, which are coupled together by a nonlinear boundary condition. We construct an asymptotic (equivalent) coupling to show that this Markov process has a unique invariant distribution. This invariant distribution is shown in a companion paper [1] to be the limit of the sequence of suitably scaled and centered stationary distributions of the GI/GI/N network, thus resolving (for a large class service distributions) an open problem raised by Halfin and Whitt in 1981. The methods introduced here are more generally applicable for the analysis of a broader class of networks.