Generalizations of the stochastic decomposition results for single server queues with vacations

In this paper we show that the sample path comparison used in Doshi [l] to prove the stochastic decomposition of the steady state waiting time for GI/G/1 queue with exhaustive service and multiple vacations can also be used to prove that this decomposition result is valid for more general (stationary) arrival and service processes. In addition, we use the same approach to show a similar decomposition result for the steady state virtual waiting time. The special case of semi-Markov arrival process and i.i.d. service times is analysed further and stronger results are obtained for that case: we show that there is a unique invariant waiting time (virtual waiting time) distribution which is the limiting distribution for any initial conditions. We also show that conditional distributions of the steady state waiting time and virtual waiting time (given the supplementary variables describing the state of the arrival process) have the stochastic decomposition property. Finally, the steady state queue length and th...