Heavy-traffic limits for waiting times in many-server queues with abandonment

TL;DR

This paper derives heavy-traffic limits for waiting times in large-server queues with customer abandonment, showing different behaviors in critically loaded and overloaded regimes, using advanced stochastic process techniques.

Contribution

It introduces a novel approach to analyze waiting times in many-server queues with abandonment under different load conditions, including new stochastic process limits and invariance principles.

Findings

Abandonment does not affect waiting times in critically loaded (QED) regime.

Waiting times in overloaded (ED) regime are characterized by queue emptying times.

New two-parameter invariance principles are established for process convergence.

Abstract

We establish heavy-traffic stochastic-process limits for waiting times in many-server queues with customer abandonment. If the system is asymptotically critically loaded, as in the quality-and-efficiency-driven (QED) regime, then a bounding argument shows that the abandonment does not affect waiting-time processes. If instead the system is overloaded, as in the efficiency-driven (ED) regime, following Mandelbaum et al. [Proceedings of the Thirty-Seventh Annual Allerton Conference on Communication, Control and Computing (1999) 1095--1104], we treat customer abandonment by studying the limiting behavior of the queueing models with arrivals turned off at some time $t$. Then, the waiting time of an infinitely patient customer arriving at time $t$ is the additional time it takes for the queue to empty. To prove stochastic-process limits for virtual waiting times, we establish a two-parameter version of Puhalskii's invariance principle for first passage times. That, in turn, involves proving that two-parameter versions of the composition and inverse mappings appropriately preserve convergence.