Heavy traffic limit with discontinuous coefficients via a non-standard semimartingale decomposition

TL;DR

This paper analyzes a single server queue in heavy traffic with discontinuous rates, showing the limit process solves a stochastic differential equation with discontinuous coefficients, using a novel semimartingale decomposition.

Contribution

It introduces a new semimartingale decomposition approach for point processes to study heavy traffic limits with discontinuous coefficients in queueing models.

Findings

Limit process characterized by a stochastic differential equation with discontinuous coefficients.

Semimartingale decomposition proves useful for analyzing complex queueing models.

Potential for broader application in queueing theory and stochastic process analysis.

Abstract

This paper studies a single server queue in heavy traffic, with general inter-arrival and service time distributions, where arrival and service rates vary discontinuously as a function of the (diffusively scaled) queue length. It is proved that the weak limit is given by the unique-in-law solution to a stochastic differential equation in $[0,\infty)$ with discontinuous drift and diffusion coefficients. The main tool is a semimartingale decomposition for point processes introduced in \cite{dal-miy}, which is distinct from the Doob-Meyer decomposition of a counting process. Whereas the use of this tool is demonstrated here for a particular model, we believe it may be useful for investigating the scaling limits of queueing models very broadly.