A numerical approach to cyclic-service queueing models

Abstract

An iterative numerical technique for the evaluation of queue length distributions is applied to multi-queue systems with one server and cyclic service discipline with Bernoulli schedules. The technique is based on power-series expansions of the state probabilities as functions of the load of the system. The convergence of the series is accelerated by applying a modified form of the epsilon algorithm. Attention is paid to economic use of memory space. The technique is based on power-series expansions of the state probabilities as functions of one parameter (the traffic intensity) of the systems. The coefficients of these power series can be recursively calculated for a large class of multi-queue models. The coefficients of the power-series expansions of the moments of the queue length distributions follow directly from those of the state probabillities. In most instances a bilinear transformation ensures convergence of the power series over the whole range of traffic intensities for which the system is stable. We have introduced in Blanc (2,3) extrapolations of the coefficients of the power series in order to accelerate the convergence of the series. One of these extrapolations will be combined with the epsilon algorithm (cf. Brezinski (6), Wynn (13)) in the present paper. The advantages of the present technique are that quantities are calculated iteratively, that it is relatively easy to compute additional terms of the power series in order to increase accuracy, that algorithms for accelerating the convergence of sequences can be applied, and that, once the coefficients of the