Some Waiting-Time Distributions for Queues with Multiple Feedback and Priorities

This paper considers a single-server queue where incoming calls require exactly N services before leaving the system. Each service has a certain priority in relation to all other services, the server always accepting for service the call at the head of the nonempty queue with highest priority whenever he completes a service. Originating calls arrive according to a Poisson process and the service times for each type of service are independent random variables with the same arbitrary distribution. We show that for the N ! possible permutations of the priorities, exactly 2 ( N −1) different waiting-time distributions will arise. The four waiting-time distributions obtained for N = 3 are discussed and an inequality established for the mean waiting times. For arbitrary N only the distributions giving rise to the minimum and maximum mean waiting times are discussed. It is possible, however, to calculate the waiting-time distribution for arbitrary N and any priority relation. The procedure is outlined by considering a specific example.