Journal Article

LAW OF LARGE NUMBERS LIMITS FOR MANY-SERVER QUEUES

Kaspi, Haya, Ramanan, Kavita

2011

Overview

This work considers a many-server queueing system in which customers with independent and identically distributed service times, chosen from a general distribution, enter service in the order of arrival. The dynamics of the system are 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 characterized 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, when the arrival process is time-homogeneous, the measure-valued component of 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.

• Publisher: Institute of Mathematical Statistics,The Institute of Mathematical Statistics
• Subject: 35D99 / 60F17 / 60H99 / 60K25 / 90B22 / Approximation / call centers / Density / Differential equations / fluid limits / GI∕G∕N queue / Integral equations / Law of large numbers / Lebesgue measures / Martingales / mean-field limits / measure-valued processes / Multi-server queues / Network servers / Perceptron convergence procedure / Probability distribution / Queuing theory / Radon / Servers / strong law of large numbers / Uniqueness