Let \(\{J_ t\}_{t>0}\) be a finite Markov process, generating a marked point process input, and denote by \(V_ t\) the virtual waiting time at time \(t\) in the queue. This paper dals with Poisson's equation which in the setting \(\{(V_ t, J_ t)\}\) has the form \((*)\) \({\mathcal A}_ g = - f\), where \({\mathcal A}\) is the infinitesimal generator of \(\{(V_ t,J_ t)\}\). The explicit form of a suitable kernel \(K\) is calculated, so that the solution of \((*)\) has the form \(Kf\). Further, these results are applied to the service times having phase-type distributions and for evaluating of variance constants. Though of complicated matrix-analytical form, the results are explicit and computational tractable.