Ergodicity of an SPDE Associated with a Many-Server Queue

Reza Aghajani, Kavita Ramanan

Published 2015-12-09Version 1

We introduce a two-component infinite-dimensional Markov process that arises as the diffusion limit of a sequence of parallel server or GI/GI/N queues, in the so-called Halfin-Whitt asymptotic regime. We characterize this process as the unique solution of 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 result is used in a companion paper to establish convergence of the sequence of scaled stationary queue lengths for a large class of service distributions, thus resolving an open problem raised by Halfin and Whitt in 1981. The methods introduced here are likely to be applicable for studying a broader class of many-server network models.