Heavy traffic limit for a processor sharing queue with soft deadlines

TL;DR

This paper derives a diffusion limit for a GI/GI/1 processor sharing queue with soft deadlines under heavy traffic, providing new approximations for lead times and sojourn times.

Contribution

It introduces a measure-valued process model and characterizes its diffusion limit, offering novel insights into queue dynamics with soft deadlines.

Findings

Diffusion approximation for residual service times and deadlines

Explicit limit characterization as a function of queue length

New insights into lead time and sojourn time profiles

Abstract

This paper considers a GI/GI/1 processor sharing queue in which jobs have soft deadlines. At each point in time, the collection of residual service times and deadlines is modeled using a random counting measure on the right half-plane. The limit of this measure valued process is obtained under diffusion scaling and heavy traffic conditions and is characterized as a deterministic function of the limiting queue length process. As special cases, one obtains diffusion approximations for the lead time profile and the profile of times in queue. One also obtains a snapshot principle for sojourn times.