  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.
