About the optimal threshold of queue length in a particular problem of profit maximization in the $M/G/1$ queuing system

The paper considers the problem of maximizing the average profit per time in the $M/G/1$ system on the set of access restriction stationary threshold strategies with one “switch point”. Profit in the described model is defined as the following measures: service fee, hardware maintenance fee, fine for service delay, fine for unhandled requests, and fine for system idle. The conditions of existence of optimal and finite threshold values are obtained. The method and the algorithm for calculating the lower bound for the optimal threshold and corresponding value of maximal profit per time are proposed. The auxiliary problem of maximizing the system profit, averaged by number of handled requests on the set of the considered threshold strategies, is solved. The necessary and sufficient conditions of existence of solution of the auxiliary problem are found. The method and algorithm for its solution are proposed.