Waiting time distributions for token-passing systems with limited-one service via discrete Fourier transforms

An interactive numerical solution to the waiting time distributions for asymmetric token-passing systems (of more than two queues) with a limited-one service policy is proposed. Customer service times and changeover times (incurred by the server to switch from one queue to another) have general distributions. A set of four embedded Markov chains is obtained by observing the system state at the instants of (customer) service beginning, service completion (server) visit beginning, and visit completion. Using results of Eisenberg (1972) the probability generating function (PGF) for the marginal queue-length distribution for each queue at a service completion is obtained. This PGF involves an unknown PGF for the system state probabilities at visit-completion epochs, where the latter are solved by a numerical technique based on discrete Fourier transforms. Thus, the waiting time distribution for each queue is found. Several numerical examples are presented to validate the proposed approach. Areas for improvements and extensions of this numerical technique are also discussed