Decay properties and quasi-stationary distributions for stopped Markovian bulk-arrival and bulk-service queues |
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Authors: | Anyue Chen Junping Li Zhenting Hou Kai Wang Ng |
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Institution: | (1) Department of Mathematics, University of Queensland, Brisbane, Qld 4072, Australia |
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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 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. |
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Keywords: | |
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