Critical fluid limit of a gated processor sharing queue

TL;DR

This paper analyzes a queueing system with a gated processor sharing policy, showing that under critical conditions and scaling, the system's behavior converges to a predictable, periodic limit.

Contribution

It introduces a measure-valued process model for gated processor sharing queues and proves convergence to a periodic limit under critical scaling.

Findings

Convergence of the queueing process to a periodic limit

Description of the limiting measure-valued process

Insights into the system's behavior under critical conditions

Abstract

We consider a sequence of single-server queueing models operating under a service policy that incorporates batches into processor sharing: arriving jobs build up behind a gate while waiting to begin service, while jobs in front of the gate are served according to processor sharing. When they have been completed, the waiting jobs move in front of the gate and the cycle repeats. We model this system with a pair of measure valued processes describing the jobs in front of and behind the gate. Under mild asymptotically critical conditions and a law-of-large-numbers scaling, we prove that the pair of measure-valued processes converges in distribution to an easily described limit, which has an interesting periodic dynamics.