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.