A differential game for a multiclass queueing model in the moderate-deviation heavy-traffic regime

TL;DR

This paper develops a differential game model for multiclass queueing systems under moderate deviations in heavy traffic, deriving stationary policies and connecting to classical free boundary problems.

Contribution

It introduces a novel differential game framework for multiclass queueing control in the moderate-deviation heavy-traffic regime, with explicit solutions and saddle point characterization.

Findings

Characterization via a one-dimensional free boundary problem

Explicit semi-closed form of an optimal strategy

Identification of a saddle point in the game

Abstract

We study a differential game that governs the moderate-deviation heavy-traffic asymptotics of a multiclass single-server queueing control problem with a risk-sensitive cost. We consider a cost set on a finite but sufficiently large time horizon, and show that this formulation leads to stationary feedback policies for the game. Several aspects of the game are explored, including its characterization via a (one-dimensional) free boundary problem, the semi-explicit solution of an optimal strategy, and the specification of a saddle point. We emphasize the analogy to the well-known Harrison-Taksar free boundary problem which plays a similar role in the diffusion-scale heavy-traffic literature.