Law of Large Numbers Limits for Many Server Queues

TL;DR

This paper establishes a law of large numbers for many-server queueing systems, describing their behavior as the number of servers grows large, with applications to computer data systems and call centers.

Contribution

It introduces a fluid limit framework for many-server queues with general service times, characterizing the limit via integral equations and analyzing convergence to equilibrium.

Findings

Fluid limit characterized by integral equations

Convergence to equilibrium in time-homogeneous systems

Explicit representation of the limit process

Abstract

This work considers a many-server queueing system in which customers with i.i.d., generally distributed service times enter service in the order of arrival. The dynamics of the system is represented in terms of a process that describes the total number of customers in the system, as well as a measure-valued process that keeps track of the ages of customers in service. Under mild assumptions on the service time distribution, as the number of servers goes to infinity, a law of large numbers (or fluid) limit is established for this pair of processes. The limit is characterised as the unique solution to a coupled pair of integral equations, which admits a fairly explicit representation. As a corollary, the fluid limits of several other functionals of interest, such as the waiting time, are also obtained. Furthermore, in the time-homogeneous setting, the fluid limit is shown to converge to its equilibrium. Along the way, some results of independent interest are obtained, including a continuous mapping result and a maximality property of the fluid limit. A motivation for studying these systems is that they arise as models of computer data systems and call centers.