  We consider the problem of risk-sensitive control of a stochastic network. In
controlling such a network, an escape time criterion can be useful if one
wishes to regulate the occurrence of large buffers and buffer overflow. In this
paper a risk-sensitive escape time criterion is formulated, which in comparison
to the ordinary escape time criteria penalizes exits which occur on short time
intervals more heavily. The properties of the risk-sensitive problem are
studied in the large buffer limit, and related to the value of a deterministic
differential game with constrained dynamics. We prove that the game has value,
and that the value is the (viscosity) solution of a PDE. For a simple network,
the value is computed, demonstrating the applicability of the approach.
