Many-Server Heavy-Traffic Limits for Queueing Systems with Perfectly Correlated Service and Patience Times

TL;DR

This paper analyzes heavy-traffic limits for many-server queueing systems with perfect correlation between service and patience times, revealing that the system behaves like a no-abandonment model under diffusion scaling.

Contribution

It provides a novel characterization of the process and steady-state limits for correlated service and patience times, including a refined fluid limit for transient cases.

Findings

Diffusion limit matches the Halfin-Whitt diffusion for $M/M/n$ queues.

Queue scales like $n^{3/4}$ when the diffusion limit is transient.

Stationary distributions converge to the process limit behavior.

Abstract

We characterize heavy-traffic process and steady-state limits for systems staffed according to the square-root safety rule, when the service requirements of the customers are perfectly correlated with their individual patience for waiting in queue. Under the usual many-server diffusion scaling, we show that the system is asymptotically equivalent to a system with no abandonment. In particular, the limit is the Halfin-Whitt diffusion for the $M/M/n$ queue when the traffic intensity approaches its critical value $1$ from below, and is otherwise a transient diffusion, despite the fact that the prelimit is positive recurrent. To obtain a refined measure of the congestion due to the correlation, we characterize a lower-order fluid (LOF) limit for the case in which the diffusion limit is transient, demonstrating that the queue in this case scales like $n^{3/4}$. Under both the diffusion and LOF scalings, we show that the stationary distributions converge weakly to the time-limiting behavior of the corresponding process limit.