Poissons equation for discrete-time quasi-birth-and-death processes
We consider Poissons equation for quasi-birth-and-death processes (QBDs) and we exploit the special transition structure of QBDs to obtain its solutions in two different forms. One is based on a decomposition through first passage times to lower levels, and the other is based on a recursive expression for the deviation matrix. We revisit the link between a solution of Poissons equation and perturbation analysis and we show that it applies to QBDs. We conclude with the PH/M/1 queue as an illustrative example, and we measure the sensitivity of the expected queue size to the initial value.
poissons equation quasi-birth-and-death process perturbation analysis matrix-analytic method