A simple analysis for an M/GI/1/N queue with vacation time and exhaustive service discipline

We consider an M/GI/1/N queue with vacation time and exhaustive service discipline, the same queue as presented by Lee (1984). Focusing only on the service completion epochs (as opposed to both service completion epochs and vacation completion epochs by Lee), we present a simple analysis for the queue length distribution at an arbitrary time as well as the waiting time distribution. We consider the limiting behaviours of these distributions and provide some rigorous proofs for a more general setting (including a G/GI/1/N queue with a general vacation policy) by means of the rate conservation law.