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

TL;DR

This paper analyzes the steady-state behavior of large multi-server queues in the Halfin-Whitt regime, providing explicit characterizations of the queue length distribution and critical exponents, revealing connections to single-server heavy-traffic results.

Contribution

It introduces a novel explicit characterization of the limiting stationary queue length distribution and critical exponents for multi-server queues in the Halfin-Whitt regime with lattice-valued service times.

Findings

Explicit stationary distribution in terms of a Markov chain

Closed-form expression for the critical exponent

Matching exponents with single-server heavy-traffic results

Abstract

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.