On uniform exponential ergodicity of Markovian multiclass many-server queues in the Halfin-Whitt regime

TL;DR

This paper establishes uniform exponential ergodicity for Markovian multiclass many-server queues in the Halfin-Whitt regime, using Lyapunov functions to analyze diffusion limits and queueing processes.

Contribution

It provides a unified Lyapunov-based approach to prove ergodicity and tail properties for controlled diffusion and queueing systems, extending previous results to multiclass models with renewal arrivals.

Findings

Exponential ergodicity when spare capacity is positive

Sub-Gaussian tails for invariant measures with all positive abandonment rates

Uniform exponential convergence for Poisson arrivals under all work-conserving policies

Abstract

We study ergodic properties of Markovian multiclass many-server queues which are uniform over scheduling policies, as well as the size n 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 queueing processes simultaneously. We first study the limiting controlled diffusion, and we show that if the spare capacity (safety staffing) parameter is positive, then the diffusion is exponentially ergodic uniformly over all stationary Markov controls, and the invariant probability measures have uniform exponential tails. This result is sharp, since when there is no abandonment and the spare capacity parameter is negative, then 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 the above results, we proceed to establish the corresponding ergodic properties for the diffusion-scaled queueing processes. In addition to providing a simpler proof of the results in Gamarnik and Stolyar [Queueing Syst (2012) 71:25-51], we extend these results to the 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.