Analysis of the loss probability of the MAP/G/1/K queue. Part I: Asymptotic Theory

The main result in this paper is the characterization of the asymptotic behavior of the loss probability of the MAP/G/1/K queue for large buffer sizes. It is shown that the loss probability tends to 0 at an exponential rate for mean offered loads less than 1. The decay rate is related to the Perron-Frobenius eigenvalue of the matrix generating function describing the arrivals during a service time. The asymptotic constant can be computed at the expense of a computational effort of the same order as that required for the solution of the infinite buffer MAP/G/1 queue A special class of MAP is defined, namely time-reversible MAPs, for which very detailed asymptotic expressions for the loss probability can be found, regardless of the value of the mean offered load. For time-reversible MAPs it is shown that the decay of the loss probability towards its limiting value for K→∞ is exponential, except for a single special case (mean offered load equal to 1), in which it is linear These theoretical results, apart from their inherent interest, yield simple and accurate approximations of the loss probability, which are asymptotically correct and turn out to be accurate for most values of the buffer size. The derivation of such approximations and the discussion of numerical examples are given in a companion paper