Poisson's equation for queues driven by a Markovian marked point process

LetV_t be the virtual waiting time at timet in a queue having marked point process input generated by a finite Markov process {J_t}, such that in addition to Markovmodulated Poisson arrivals there may also be arrivals at jump times of {J_t}. In this setting, Poisson's equation isAg=–f whereA is the infinitesimal generator of {(V_t, J_t)}. It is shown that the solutiong can be expressed asKf for some suitable kernelK, and the explicit form ofK is evaluated. The results are applied to compute limiting variance constants for (normalized) time averages of functionsf(V_t, J_t), in particularf(V_t,J_t)=V_t.