Maximization of average stationary profit in $M$/$G$/1 queuing system

The problem of optimization of the queue length threshold in a $M/G/1$ system is considered in terms of maximizing the marginal return received by the system per unit of time. The profit value consists of the following measures: service fee; hardware maintenance fee; cost of service delay; fine for unhandled requests; and fine for system idle. The author formulates the necessary conditions of existence of a finite threshold in an $M/G/1$ system and prove the statements of necessary and sufficient conditions for threshold optimality and existence of the finite optimal threshold. The author proposes an algorithm for calculating the optimal threshold value and the corresponding maximal profit. The author presents the results of computational experiments that illustrate the work of the proposed algorithm.