  In this paper, we are devoted to developing matrix-analytic methods for
solving Poisson's equation for irreducible and positive recurrent discrete-time
Markov chains (DTMCs). Two special solutions, including the deviation matrix D
and the expected additive-type functional matrix K, will be considered. The
results are applied to Markov chains of GI/G/1-type and MAP/G/1 queues with
negative customers. Further extensions to continuous-time Markov chains (CTMCs)
are also investigated.
