Abstract
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 λC 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,λC]. We then show that under some mild conditions there exists a family of quasi-stationary distributions also indexed by λ ∈ [0,λC]. 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 λC-transient. Some deep properties regarding λC-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.
Original language | English |
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Pages (from-to) | 275-311 |
Number of pages | 37 |
Journal | Queueing Systems |
Volume | 66 |
Issue number | 3 |
DOIs | |
Publication status | Published - Nov 2010 |
Externally published | Yes |
Keywords
- Decay parameter
- Invariant measures
- Quasi-stationary distributions
- Stopped Markovian bulk-arrival and bulk-service queues