  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.
