Optimal control of a queue-length dependent additional server in $\mathrm{GI}/M/1$ queue

Consideration is given to a $\mathrm{GI}/M/1$ queue in which there is an additional server available for serving customers from the queue. The additional server can be turned on and off depending on the current queue length. The long-run total cost per unit time, equal to the difference between the paid amount for service and the losses due to the waiting of customers and additional server depreciation, is being optimized. The case of finite queue capacity is also considered in which the losses also account for lost customers. It is proved that the cost function considered is unimodal. Necessary and sufficient conditions are given for the existence of the decision point (queue length) at which application of the additional server is optimal. A simple algorithm for controlling the decision point, requiring only observations of the cost function value, is provided.