Optimization of the threshold service speed control in the $G/M/1$ queue

Consideration is given to the problem of optimal service speed switching in a $G/M/1$ queuing system with an objective function which takes into account the fee for customer's service, the penalty for customer's delay in the queue, the penalty for customer's rejection, the penalty for server being idle, and the maintenance costs. The case of finite capacity queue is also considered. The nonlinear optimization problem is formulated and solved in which the objective function is the system's revenue and the control variable is the queue length. The author proves that the objective function is unimodal and formulates the conditions for the existence of an optimal threshold service speed switching and necessary and sufficient conditions for the optimal control.