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On some tractable growth-collapse processes with renewal collapse epochs

Part of: Special processes

Published online by Cambridge University Press:  14 July 2016

Onno Boxma ,
Offer Kella  and
David Perry
Onno Boxma
Affiliation:
EURANDOM and Eindhoven University of Technology, EURANDOM and Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: boxma@win.tue.nl
Offer Kella
Affiliation:
The Hebrew University of Jerusalem, Department of Statistics, The Hebrew University of Jerusalem, Mount Scopus, Jerusalem 91905, Israel. Email address: offer.kella@huji.ac.il
David Perry
Affiliation:
Haifa University, Department of Statistics, Haifa University, Haifa 31905, Israel. Email address: dperry@stat.haifa.ac.il
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Abstract

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In this paper we generalize existing results for the steady-state distribution of growth-collapse processes with independent exponential intercollapse times to the case where they have a general distribution on the positive real line having a finite mean. In order to compute the moments of the stationary distribution, no further assumptions are needed. However, in order to compute the stationary distribution, the price that we are required to pay is the restriction of the collapse ratio distribution from a general distribution concentrated on the unit interval to minus-log-phase-type distributions. A random variable has such a distribution if the negative of its natural logarithm has a phase-type distribution. Thus, this family of distributions is dense in the family of all distributions concentrated on the unit interval. The approach is to first study a certain Markov-modulated shot noise process from which the steady-state distribution for the related growth-collapse model can be inferred via level crossing arguments.

Keywords

Growth collapseshot noise processminus-log-phase-type distributionMarkov modulated

MSC classification

Primary: 60K05: Renewal theory
Secondary: 60K25: Queueing theory
Type
Part 5. Stochastic Growth and Branching
Information
Journal of Applied Probability , Volume 48 , Issue A: New Frontiers in Applied Probability (Journal of Applied Probability Special Volume 48A) , August 2011 , pp. 217 - 234
Copyright
Copyright © Applied Probability Trust 2011 

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