Ergodicity of an SPDE associated with a many-server queue

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..N.1/2+o(N.1/2)for some .>0.Under suitable assumptions, we characterize this process as the unique solution to a pair of stochastic evolution equations comprised of a real-valued Itô 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 [Aghajani and Ramanan (2016)] 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 [Oper. Res. 29 (1981) 567.588].The methods introduced here are more generally applicable for the analysis of a broader class of networks.