Abstract. This paper analyses a multiple class single server M/M/1/K queue with finite capacity under weighted fair queuing (WFQ) discipline. The Poisson process has been used to model the multiple classes of arrival streams. The service times have exponential distribution. We assume each class is assigned a virtual queue and incoming jobs enter the virtual queue related to their class and served in FIFO order.We model our system as a two dimensional Markov chain and use the matrix-geometric method to solve its stationary probabilities. This paper presents a matrix geometric solution to the M/M/1/K queue with finite buffer under (WFQ) service. In addition, the paper shows the state transition diagram of the Markov chain and presents the state balance equations, from which the stationary queue length distribution and other measures of interest can be obtained. Numerical experiments corroborating the theoretical results are also offered.

A novel approach is applied to the study of a queue with general correlated traffic, in that the only features of the traffic which are taken into account are the usual measures of its correlation: the traffic is modelled as a batch renewal process. The batch renewal process is a precise tool for investigation into effects of correlation because it is the least biased choice of process which is completely determined by the infinite sets of measures of the traffic's correlation (e.g. indices of dispersion, covariances or correlation functions). The general effect of traffic correlation on waiting time, blocking probability and queue length is well known from simulation studies and numerical analysis of a variety of models. The contribution of this paper is to show that these effects are due to correlation alone (and not to any other features of the traffic or of the models used) and to show explicitly how the magnitudes of blocking, waiting time and queue length distribution are determined by the degree of correlation in the traffic. The study focuses upon a discrete time GlG/D/I/N queue with single server, general batch renewal arrivals process, deterministic service time and finite capacity N. Closed form expressions for basic performance distributions, such as queue length and waiting time distributions and blocking probability, are derived when the batch renewal process is of the least biased form which might be expected to result from actual traffic measurements at the int.erior of a network or of some individual traffic source. The effect of varying degrees of traffic correlation upon basic performance distributions is investigated and illustrative numerical results are presented. Comments on implications of the results on analysis of general discrete time queueing networks with correlated traffic are included. Keywords asynchronous transfer mode (ATM), batch renewal process, correlated traffic, least biased process, discrete time queue, performance distributions S. Fdida et al. (eds.), Data Communications and their Performance

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