Fluid limits of many-server queues with reneging

TL;DR

This paper establishes a fluid limit model for many-server queues with customer impatience, providing a deterministic approximation of the system's behavior as the number of servers grows large, extending previous models without reneging.

Contribution

It introduces a novel fluid limit framework for queues with reneging, including explicit integral equations and virtual waiting time analysis, advancing the understanding of such systems.

Findings

Fluid limit characterized by deterministic integral equations

Explicit representation of the limit process

Fluid approximation for virtual waiting time

Abstract

This work considers a many-server queueing system in which impatient customers with i.i.d., generally distributed service times and i.i.d., generally distributed patience times enter service in the order of arrival and abandon the queue if the time before possible entry into service exceeds the patience time. The dynamics of the system is represented in terms of a pair of measure-valued processes, one that keeps track of the waiting times of the customers in queue and the other that keeps track of the amounts of time each customer being served has been in service. Under mild assumptions, essentially only requiring that the service and reneging distributions have densities, as both the arrival rate and the number of servers go to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is shown to be the unique solution of a coupled pair of deterministic integral equations that admits an explicit representation. In addition, a fluid limit for the virtual waiting time process is also established. This paper extends previous work by Kaspi and Ramanan, which analyzed the model in the absence of reneging. A strong motivation for understanding performance in the presence of reneging arises from models of call centers.