A queueing model with bonus service for certain customers (Q1088585)

A queueing model is introduced in which the management has a policy, because of economic reasons, of not operating the service counter unless a certain number, \(R+1\), of customers are available during each busy period. Thus, the first R customers who arrive must wait until the service counter is opened. Such a policy may cause the management to provide or render additional services to the first R customers. Assuming Poisson arrivals and that both regular and additional services follow exponential distributions, explicit expressions are derived for the stationary queue length and busy period distributions and their expected values. In the special case where \(R=1\), an explicit expression is presented for the stationary distribution of the waiting time.