Queues Served in Cyclic Order: Waiting Times

In a recent paper1 we studied two models of a system of queues served in cyclic order: In each model, the t'th queue is characterized by general service time distribution function #,(·) and Poisson input with parameter X,- . In the exhaustive service model, the server continues to serve a particular queue until for the first time there are no units in service or waiting in that queue; at this time the server advances to and immediately starts service on the next nonempty queue in the cyclic order. The gating model differs from the exhaustive service model in that when the server advances to a nonempty queue, a gate closes behind the waiting units. Only those units waiting in front of the gate are served during this cycle, with the service of subsequent arrivals deferred to the next cycle. In Ref. 1 we found, for the exhaustive service model, expressions for the mean number of units in a queue at the instant it starts service, the mean cycle time, and the Laplace-Stieltjes transform of the cycle time distribution function. In the present paper, we extend the analysis to obtain, for each 399