POISSON EQUATION FOR QUEUES DRIVEN BY A MARKOVIAN MARKED POINT PROCESS

Let V(t) be the virtual waiting time at time t in a queue having marked point process input generated by a finite Markov process {J(t)}, such that in addition to Markov-modulated Poisson arrivals there may also be arrivals at jump times of {J(t)}. In this setting, Poisson's equation is Ag = -f where A is the infinitesimal generator of {(V(t), J(t))}. 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(V(t), J(t)), in particular f(V(t), J(t)) = V(t).