Transient analysis of quasi-birth-death processes

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Abstract
The concept of Complete Level Crossing Information is shown to lead to a matrix geometric solution for the transform of the distribution of the number in queue at each time t in queues modeled as Quasi-Birth-Death (QBD) processes. Specifically, if (t) is the vector of probabilities of state occupancy at time t for the states on level n of a QBD-process given the process starts from level 1 at , and if is the Laplace transform of , then there exists a square matrix W(s) such that: We also show how to determine the boundary vectors and , for this recursion To illustrate this method, a QBD process which models a CSMA/CD network is analyzed in detail. The mean number of busy users in the network and the tail distribution, both of which are functions of time, are computed from the closed-form expressions for the transform domain recursion. These results are significant since they should greatly expand the class of queues for which transient analysis is tractable.

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