  We consider a multi-server queue in the Halfin-Whitt regime: as the number of
servers $n$ grows without a bound, the utilization approaches 1 from below at
the rate $\Theta(1/\sqrt{n})$. Assuming that the service time distribution is
lattice-valued with a finite support, we characterize the limiting stationary
queue length distribution in terms of the stationary distribution of an
explicitly constructed Markov chain. Furthermore, we obtain an explicit
expression for the critical exponent for the moment generating function of a
limiting (scaled) steady-state queue length. This exponent has a compact
representation in terms of three parameters: the amount of spare capacity and
the coefficients of variation of interarrival and service times. Interestingly,
it matches an analogous exponent corresponding to a single-server queue in the
conventional heavy-traffic regime.
