Time to congestion in Homogeneous Quasi-Birth-and-Death Processes

Quasi-Birth-and-Death Processes are useful tools for studying queuing systems. Of particular interest is the distribution of the first passage time to a certain level of congestion, a given number of customers in the queue, for instance. We present an algorithmic method based on a decomposition of the set of levels preceding the level of congestion. It aims to compute the two first moments of the time till congestion and it is well-adapted to the particular configuration of QBD process transition matrix.