We consider Poisson's 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, the other is based on a recursive expression for the
deviation matrix.
We revisit the link between a solution of Poisson's 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
