A Differential Game for a Multiclass Queueing Model in the Moderate-Deviation Heavy-Traffic Regime

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.