Large deviations analysis for the $M/H_2/n + M$ queue in the Halfin-Whitt regime

TL;DR

This paper explicitly computes the large deviations exponent for the steady-state queue length in a non-Markovian multi-server queue with abandonments in the Halfin-Whitt regime, resolving a conjecture and extending stochastic comparison methods.

Contribution

It provides the first explicit large deviations exponent for non-Markovian queues with abandonments, extending stochastic comparison techniques to this setting.

Findings

Derived the true large deviations exponent for the queue length distribution.

Resolved the Dai and He conjecture negatively.

Extended stochastic comparison framework to queues with abandonments.

Abstract

We consider the FCFS $M/H_2/n + M$ queue in the Halfin-Whitt heavy traffic regime. It is known that the normalized sequence of steady-state queue length distributions is tight and converges weakly to a limiting random variable W. However, those works only describe W implicitly as the invariant measure of a complicated diffusion. Although it was proven by Gamarnik and Stolyar that the tail of W is sub-Gaussian, the actual value of $\lim_{x \rightarrow \infty}x^{-2}\log(P(W >x))$ was left open. In subsequent work, Dai and He conjectured an explicit form for this exponent, which was insensitive to the higher moments of the service distribution. We explicitly compute the true large deviations exponent for W when the abandonment rate is less than the minimum service rate, the first such result for non-Markovian queues with abandonments. Interestingly, our results resolve the conjecture of Dai and He in the negative. Our main approach is to extend the stochastic comparison framework of Gamarnik and Goldberg to the setting of abandonments, requiring several novel and non-trivial contributions. Our approach sheds light on several novel ways to think about multi-server queues with abandonments in the Halfin-Whitt regime, which should hold in considerable generality and provide new tools for analyzing these systems.