Validity of heavy-traffic steady-state approximations in many-server queues with abandonment

TL;DR

This paper proves that in large-scale multi-server queues with abandonment, the steady-state distribution converges under heavy traffic conditions, validating the use of approximations in such systems.

Contribution

It establishes the interchange of heavy traffic and steady-state limits for GI/Ph/n+M queues, a significant theoretical validation for these models.

Findings

Stationary distributions are tight in the Halfin-Whitt regime.

Heavy-traffic and steady-state limits can be interchanged.

Results apply to systems with phase-type service and exponential patience times.

Abstract

We consider GI/Ph/n+M parallel-server systems with a renewal arrival process, a phase-type service time distribution, n homogenous servers, and an exponential patience time distribution with positive rate. We show that in the Halfin-Whitt regime, the sequence of stationary distributions corresponding to the normalized state processes is tight. As a consequence, we establish an interchange of heavy traffic and steady state limits for GI/Ph/n+M queues.