  We consider the so-called GI/GI/N queue, in which a stream of jobs with
independent and identically distributed service times arrive as a renewal
process to a common queue that is served by $N$ identical parallel servers in a
first-come-first-serve manner. We introduce a new representation for the state
of the system and, under suitable conditions on the service and interarrival
distributions, establish convergence of the corresponding sequence of centered
and scaled stationary distributions in the so-called Halfin-Whitt asymptotic
regime. In particular, this resolves an open question posed by Halfin and Whitt
in 1981. We also characterize the limit as the stationary distribution of an
infinite-dimensional two-component Markov process that is the unique solution
to a certain stochastic partial differential equation. Previous results were
essentially restricted to exponential service distributions or service
distributions with finite support, for which the corresponding limit process
admits a reduced finite-dimensional Markovian representation. We develop a
different approach to deal with the general case when the Markovian
representation of the limit is truly infinite-dimensional. This approach is
more broadly applicable to a larger class of networks.
