Diffusion limits of limited processor sharing queues

TL;DR

This paper analyzes the diffusion limits of a limited processor sharing queue under heavy traffic, showing that the scaled system converges to a piecewise reflected Brownian motion, providing insights into its long-term behavior.

Contribution

It introduces a measure-valued process approach to model limited processor sharing queues and derives their diffusion limit under heavy traffic conditions.

Findings

System size process converges to a piecewise reflected Brownian motion.

Provides a rigorous diffusion approximation for limited processor sharing queues.

Uses measure-valued processes to model queue dynamics.

Abstract

We consider a processor sharing queue where the number of jobs served at any time is limited to $K$, with the excess jobs waiting in a buffer. We use random counting measures on the positive axis to model this system. The limit of this measure-valued process is obtained under diffusion scaling and heavy traffic conditions. As a consequence, the limit of the system size process is proved to be a piece-wise reflected Brownian motion.