Optimal control of a type of M/G/1 queueing system

This paper discusses a type of queueing system with finite capacity. Service discipline is first come first served. The server turns on when a certain number of customers are present and turns off in idle period. By use of supplemental variables method, we get the generating function of equilibrium distribution. We then study the optimal control policy of this queueing system. A method of optimal control is obtained. The objective is to dynamically determine the optimal number of customers based on the state of system, so as to maximize the average profit. In order to determine the optimal number of customers when the server turns on, we present the objective function.