On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime

Abstract

We study ergodic properties of Markovian multiclass many-server queues that are uniform over scheduling policies and the size of the system. The system is heavily loaded in the Halfin–Whitt regime, and the scheduling policies are work conserving and preemptive. We provide a unified approach via a Lyapunov function method that establishes Foster–Lyapunov equations for both the limiting diffusion and the prelimit diffusion-scaled queuing processes simultaneously. We first study the limiting controlled diffusion and show that if the spare capacity (safety staffing) parameter is positive, the diffusion is exponentially ergodic uniformly over all stationary Markov controls, and the invariant probability measures have uniform exponential tails. This result is sharp because when there is no abandonment and the spare capacity parameter is negative, the controlled diffusion is transient under any Markov control. In addition, we show that if all the abandonment rates are positive, the invariant probability measures have sub-Gaussian tails regardless whether the spare capacity parameter is positive or negative. Using these results, we proceed to establish the corresponding ergodic properties for the diffusion-scaled queuing processes. In addition to providing a simpler proof of previous results in Gamarnik and Stolyar [Gamarnik D, Stolyar AL (2012) Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime: asymptotics of the stationary distribution. Queueing Systems 71(1–2):25–51], we extend these results to multiclass models with renewal arrival processes, albeit under the assumption that the mean residual life functions are bounded. For the Markovian model with Poisson arrivals, we obtain stronger results and show that the convergence to the stationary distribution is at an exponential rate uniformly over all work-conserving stationary Markov scheduling policies.