Ergodic risk-sensitive admission control problem for a Markovian multi-server queueing system with abandonment

We consider the ergodic risk-sensitive admission control problem for a Markovian multi-server queueing system with abandonment, where costs are incurred for server idleness, customer abandonment, and rejecting incoming arrivals. We first derive the Bellman optimality equation for this problem and show that a threshold policy—one that rejects incoming arrivals whenever the system-size exceeds a threshold—is optimal among all admissible control policies. We then propose a policy iteration algorithm to identify the optimal threshold, where we prove, under certain conditions on the problem’s parameters, that the algorithm will terminate at the optimal threshold level. We also characterize the effect of risk sensitivity on the optimal threshold, proving that this threshold monotonically decreases with respect to the sensitivity parameter and converges to the average cost optimal threshold from below as the sensitivity parameter tends to zero.