On single-threshold queue management in a queuing system with impatient customers

The results of a theoretical study of a managed queuing system of $M/M/k$ type with impatient customers and single-threshold queue management are presented. The task of optimizing single-threshold queue management is set, the essence of which is to calculate for the queue length a certain threshold value that maximizes a given objective function. In the system under study, a customer leaves the system unattended if the waiting time in the queue (or the service time on the device) exceeds a certain time interval of random length distributed according to an exponential law with a given parameter. A cost function is used as an indicator of the effectiveness of queue management (objective function) which takes into account the losses per unit of time due to system technical maintenance, rejection of customers at the entrance of the system, and leaving of customers until the end of the service. A method for solving the problem of maximizing the cost objective function on a set of single-threshold queue controls and an algorithm for guaranteed calculation of the optimal threshold are proposed.