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

Steady-state analysis of a multi-server queue in the Halfin-Whitt regime