Let Vt be the virtual waiting time at time t in a queue having marked point process input generated by a finite Markov process {Jt}, such that in addition to Markov-modulated Poisson arrivals there may also be arrivals at jump times of {Jt}. In this setting, Poisson’s equation is 𝒜g=¸-f where 𝒜 is the infinitesimal generator of {(Vt,Jt)}. It is shown that the solution g can be expressed as Kf for some suitable kernel K, and the explicit form of K is evaluated. The results are applied to compute limiting variance constants for (normalized) time averages of functions f(Vt,Jt), in particular f(Vt,Jt)=Vt.
