SPDE Limits of Many Server Queues

TL;DR

This paper derives a functional central limit theorem for many-server queues, showing that the scaled total number of customers converges to an Ito diffusion and the age process converges to a Hilbert space-valued SPDE, revealing deep stochastic structure.

Contribution

It introduces a novel SPDE characterization for the age process in many-server queues, extending the understanding of their asymptotic behavior.

Findings

The total number in system converges to an Ito diffusion with constant, service distribution-insensitive diffusion coefficient.

The age process converges to a Hilbert space-valued diffusion characterized by a stochastic PDE.

The limit processes are semimartingales with the strong Markov property.

Abstract

A many-server queueing system is considered in which customers with independent and identically distributed service times enter service in the order of arrival. The state of the system is represented by 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, leading to a Markovian description of the dynamics. Under suitable assumptions, a functional central limit theorem is established for the sequence of (centered and scaled) state processes as the number of servers goes to infinity. The limit process describing the total number in system is shown to be an Ito diffusion with a constant diffusion coefficient that is insensitive to the service distribution. The limit of the sequence of (centered and scaled) age processes is shown to be a Hilbert space valued diffusion that can also be characterized as the unique solution of a stochastic partial differential equation that is coupled with the Ito diffusion. Furthermore, the limit processes are shown to be semimartingales and to possess a strong Markov property.