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

摘要

LetVtbe the virtual waiting time at timetin a queue having marked point process input generated by a finite Markov process {Jt}, such that in addition to Markovmodulated Poisson arrivals there may also be arrivals at jump times of {Jt}. In this setting, Poisson's equation isAg=−fwhereAis the infinitesimal generator of {(Vt, Jt)}. It is shown that the solutiongcan be expressed asKffor some suitable kernelK, and the explicit form ofKis evaluated. The results are applied to compute limiting variance constants for (normalized) time averages of functionsf(Vt, Jt), in particularf(Vt,Jt)=V