Risk-Sensitive Optimal Control of Queues

We consider the problem of designing risk-sensitive optimal control policies for scheduling packet transmissions in a stochastic wireless network. A single client is connected to an access point (AP) through a wireless channel. Packet transmission incurs a cost $C$, while packet delivery yields a reward of $R$ units. The client maintains a finite buffer of size $B$, and a penalty of $L$ units is imposed upon packet loss which occurs due to finite queueing buffer. We show that the risk-sensitive optimal control policy for such a simple set-up is of threshold type, i.e., it is optimal to carry out packet transmissions only when $Q(t)$, i.e., the queue length at time $t$ exceeds a certain threshold $\tau$. It is also shown that the value of threshold $\tau$ increases upon increasing the cost per unit packet transmission $C$. Furthermore, it is also shown that a threshold policy with threshold equal to $\tau$ is optimal for a set of problems in which cost $C$ lies within an interval $[C_l,C_u]$. Equations that need to be solved in order to obtain $C_l,C_u$ are also provided.