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

Anyue Chen*, Junping Li, Zhenting Hou, Kai Wang Ng

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

10 Citations (Scopus)

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 languageEnglish
Pages (from-to)275-311
Number of pages37
JournalQueueing Systems
Volume66
Issue number3
DOIs
Publication statusPublished - Nov 2010
Externally publishedYes

Keywords

  • Decay parameter
  • Invariant measures
  • Quasi-stationary distributions
  • Stopped Markovian bulk-arrival and bulk-service queues

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