Fluid Limit of a PS-queue with Multistage Service

TL;DR

This paper develops fluid limit approximations for a processor sharing queue with multistage service, inspired by freelance job websites, to estimate freelancer success rates in overload conditions.

Contribution

It introduces fluid limit models for PS-queues with multistage service under overload, providing new insights into system behavior and success probability estimation.

Findings

Fluid limit approximations for overload conditions

Estimated freelancer success proportions

Partial solutions for stability analysis

Abstract

The PS-model treated in this paper is motivated by freelance job websites where multiple freelancers compete for a single job. In the context of such websites, multistage service of a job means collection of applications from multiple freelancers. Under Markovian stochastic assumptions, we develop fluid limit approximations for the PS-model in overload. Based on this approximation, we estimate what proportion of freelancers get the jobs they apply for. In addition, the PS-model studied here is an instant of PS with routing and impatience, for which no Lyapunov function is known, and we suggest some partial solutions.