On the unimodality of the income function of a type $G|M|s$ queueing system with controlled queue

The problem of maximizing the average income in a queuing system of type $G|M|s$ on a set of pure stationary threshold strategies with single point switching access restriction mode is considered. The income function depends on the following parameters, measured in value units: the fee received for servicing requests, the cost of maintenance of the device, the deduction of income for the delay applications in the queue, the penalty for unserved applications. It is proved that the income function is unimodal on the set of considered threshold strategies. An algorithm for calculating the optimal threshold value and the corresponding maximum value income is proposed. The results of the computational experiment that illustrate the work of the proposed algorithm are given.