Proof of the unimodality of the objective function in $M/M/N$ queue with threshold-based congestion control

The problem of limiting the load in the system $M/M/N/\infty$ is considered using a simple threshold strategy. In addition to the service time, each task is characterized by a deadline. Depending on the quality of service, the system receives either a fixed income or a penalty. The quality of control is determined by the marginal average income and the threshold value that maximizes this value is considered as optimal. Usually, it is much easier to find the optimal threshold if the objective function has a single maximum. The experimental results show the unimodality of the objective function for a wide class of arrival flows. However, there is no rigorous proof of this fact and in the paper, this gap is filled up for the Poisson arrivals. The proof is based on the results of the Markov chain theory and queueing theory.