Priorities in M/G/1 queue with server vacations

The M/G/1 queue with single and multiple server vacations is studied under both the preemptive and non-preemptive priority regimes. A unified methodology is developed to derive the Laplace-Stieltjes transform and first two moments of the waiting time W_k of a class-k customer for each of the four models analyzed. The results are given a probabilistic representation involving mean residual lifetimes.