Optimization of a queue-length dependent additional server in the multiserver queue

The problem of optimal control of an additional server in a stationary $G/M/s$ queue is considered. The additional server can be turned on and off at instants when the queue length is changed. It is formulated as the nonlinear optimization problem, in which the objective function accounts for amounts for service, losses due to the waiting of customers, maintenance, and downtime of the additional server. The functioning of the system is described as a controlled Markov chain. Only stationary control policies are considered. For Poisson arrivals, necessary and sufficient conditions are given for the existence of the optimal decision point (threshold) and it is proved that the objective function is unimodal. A simple algorithm for the computation of the threshold is provided.