Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues

We consider decay properties including the decay parameter, invariant measures and quasi-stationary distributions for a Markovian bulk-arrival and bulk-service queue which stops when the waiting line is empty. Investigating such a model is crucial for understanding the busy period and other related properties of the Markovian bulk-arrival and bulk-service queuing processes. The exact value of the decay parameter λ_C is first obtained. We show that the decay parameter can be easily expressed explicitly. The invariant measures and quasi-distributions are then revealed. We show that there exists a family of invariant measures indexed by λ ∈ [0,λ_C]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ ∈ [0,λ_C]. The generating functions of these invariant measures and quasi-stationary distributions are presented. We further show that this stopped Markovian bulk-arrival and bulk-service queueing model is always λ_C-transient. Some deep properties regarding λ_C-transience are examined and revealed. The clear geometric interpretation of the decay parameter is explained. A few examples are then provided to illustrate the results obtained in this paper.