Asymptotic bounds for an optimal state-dependent retrial rate of the M/M/1 queue with returning customers

In this paper, we examine the zero capacity M/M/1 queue with returning customers. Asymptotic bounds for the optimal state dependent retrial rate are derived, where the state of the process is determined by queue length and server status. This optimal return rate minimizes the penalty cost to a customer for waiting and checking in to verify that the server is busy. It will be shown that the problem reduces to three cases characterized by a linear relationship between four system parameters: customer arrival rate, service rate, competing retrial rate, and cost. In two of these cases, asymptotic bounds are tight.