The complete analysis of the discrete time finite DBMAP/G/1/N queue

A comprehensive analysis of the DBMAP/G/1/N queueing system for both the “arrival first” and “departure first” policies is given including detailed proofs and latest results on an algorithmic recipe to compute the z-transform of the waiting time probability function as well as the probability function itself. Additionally, expressions for the probability of simultaneous arrivals and departures and an algorithm to compute them are presented for the first time. The framework presented allows dealing with a superposition DMAP+DBMAP as an input process. For both streams in the superposition, per stream waiting time probability functions are given for the case of a deterministic service time, which is relevant for the modeling of ATM networking (both wired and wireless). Together with findings for the continuous time case, this paper completes the insights into finite queueing systems of the M/G/1-type. In comparison with the method of the unfinished work, which can be used to analyze finite queueing systems with deterministic service-time only and, besides other semi-Markovian input processes, with the DBMAP as an input, the M/G/1-paradigm provides much faster algorithms to compute loss probabilities and waiting time moments, due to the smaller system matrix. Note that the D[B]MAP has proved a versatile stochastic process, which can also be tuned to represent periodic correlation functions and not only geometrically decaying ones.