Heavy-tailed queues in the Halfin-Whitt regime

TL;DR

This paper investigates heavy-tailed queueing systems in the Halfin-Whitt regime, establishing new tightness results under minimal assumptions and analyzing large deviations, revealing fundamentally different tail behaviors from light-tailed cases.

Contribution

It extends tightness results for stationary queue lengths to heavy-tailed distributions with minimal moment assumptions and characterizes large deviations with sub-exponential decay.

Findings

Stationary queue length normalized by n^{1/2} is tight under minimal assumptions.

Large deviations exhibit sub-exponential decay, contrasting with exponential tails in light-tailed cases.

Results are tight and robust even for deterministic processing times.

Abstract

We consider the FCFS G/G/n queue in the Halfin-Whitt regime, in the presence of heavy-tailed distributions (i.e. infinite variance). We prove that under minimal assumptions, i.e. only that processing times have finite 1 + epsilon moment and inter-arrival times have finite second moment, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{2}}$, is tight. All previous tightness results for the stationary queue length required that processing times have finite 2 + epsilon moment. Furthermore, we develop simple and explicit bounds on the stationary queue length in that setting. When processing times have an asymptotically Pareto tail with index alpha in (1,2), we bound the large deviations behavior of the limiting process, and derive a matching lower bound when inter-arrival times are Markovian. Interestingly, we find that the large deviations behavior of the limit has a sub-exponential decay, differing fundamentally from the exponentially decaying tails known to hold in the light-tailed setting, and answering an open question of Gamarnik and Goldberg. For the setting where instead the inter-arrival times have an asymptotically Pareto tail with index alpha in (1,2), we extend recent results of Hurvich and Reed (who analyzed the case of deterministic processing times) by proving that for general processing time distributions, the sequence of stationary queue length distributions, normalized by $n^{\frac{1}{\alpha}}$, is tight (here we use the scaling of Hurvich and Reed, i.e. Halfin-Whitt-Reed regime). We are again able to bound the large-deviations behavior of the limit, and find that our derived bounds do not depend on the particular processing time distribution, and are in fact tight even for the case of deterministic processing times.