An approximation for performance evaluation of stationary single server queues

This paper provides a method for approximating the probability distributions of stationary statistics in FIFO single server queues. The method is based on the Wiener-Hopf factorization technique, and is applied to semi-Markov queues where the underlying state space is of unlimited size. It has been established that the tail of the distribution of the waiting time or unfinished work in such queues is negative exponential, and in this paper we estimate the parameters of that exponential term. A particularly important case, which is treated here, is a model for a statistical multiplexer where the net input process forms a stationary ergodic Gaussian discrete-time stochastic process. In this case, it is possible to derive analytically a simple formula in a closed form for the approximation. The formula is in terms of three parameters of the net input process: the mean, the variance, and the autocovariance sum. This provides a solid theoretical basis for traffic characterization by these parameters. Comparison with simulation results show that the method is accurate. Also presented is a result for a special case where the arrival process is autoregressive