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Decay property of stopped Markovian bulk-arriving queues

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

Junping Li  and
Anyue Chen
Junping Li*
Affiliation:
Central South University
Anyue Chen*
Affiliation:
University of Liverpool and University of Hong Kong
*
∗ Postal address: School of Mathematical Science and Computing Technology, Central South University, Changsha, 410075, P. R. China. Email address: [email protected]
∗∗ Postal address: Division of Statistics and Probability, Department of Mathematical Sciences, The University of Liverpool, Liverpool, L69 7ZL, UK. Email address: [email protected]
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Abstract

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We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions of a Markovian bulk-arriving queue that stops immediately after hitting the zero state. Investigating such behavior is crucial in realizing the busy period and some other related properties of Markovian bulk-arriving queues. The exact value of the decay parameter λC is obtained and expressed explicitly. The invariant measures, invariant vectors, and quasistationary distributions are then presented. We show that there exists a family of invariant measures indexed by λ ∈ [0, λC]. We then show that, under some conditions, there exists a family of quasistationary distributions, also indexed by λ ∈ [0, λC]. The generating functions of these invariant measures and quasistationary distributions are presented. We further show that a stopped Markovian bulk-arriving queue is always λC-transient and some deep properties are 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.

Keywords

Markovian bulk-arriving queuedecay parameterinvariant measureinvariant vectorquasistationary distribution

MSC classification

Primary: 60J27: Continuous-time Markov processes on discrete state spaces
Secondary: 60J35: Transition functions, generators and resolvents
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 40 , Issue 1 , March 2008 , pp. 95 - 121
Copyright
Copyright © Applied Probability Trust 2008 

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