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A Markovian growth-collapse model

Part of: Special processes Markov processes Limit theorems

Published online by Cambridge University Press:  01 July 2016

Onno Boxma ,
David Perry ,
Wolfgang Stadje  and
Shelemyahu Zacks
Onno Boxma*
Affiliation:
EURANDOM and Eindhoven University of Technology
David Perry*
Affiliation:
University of Haifa
Wolfgang Stadje*
Affiliation:
University of Osnabrück
Shelemyahu Zacks*
Affiliation:
Binghamton University
*
∗ Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: boxma@win.tue.nl
∗∗ Postal address: Department of Statistics, University of Haifa, Haifa, 31909, Israel. Email address: dperry@haifa.ac.il
∗∗∗ Postal address: Department of Mathematics and Computer Science, University of Osnabrück, 49069 Osnabrück, Germany. Email address: wolfgang@mathematik.uni-osnabrueck.de
∗∗∗∗ Postal address: Department of Mathematical Sciences, Binghamton University, Binghamton, NY 13902-6000, USA. Email address: shelly@math.binghamton.edu
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Abstract

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We consider growth-collapse processes (GCPs) that grow linearly between random partial collapse times, at which they jump down according to some distribution depending on their current level. The jump occurrences are governed by a state-dependent rate function r(x). We deal with the stationary distribution of such a GCP, (Xt)t≥0, and the distributions of the hitting times Ta = inf{t ≥ 0 : Xt = a}, a > 0. After presenting the general theory of these GCPs, several important special cases are studied. We also take a brief look at the Markov-modulated case. In particular, we present a method of computing the distribution of min[Ta, σ] in this case (where σ is the time of the first jump), and apply it to determine the long-run average cost of running a certain Markov-modulated disaster-ridden system.

Keywords

Growth-collapse processpiecewise-deterministic Markov processstationary distributionhitting timeuniform cut-offdualityMarkov modulation

MSC classification

Primary: 60K30: Applications (congestion, allocation, storage, traffic, etc.)
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces 60J75: Jump processes 60F05: Central limit and other weak theorems
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 1 , March 2006 , pp. 221 - 243
Copyright
Copyright © Applied Probability Trust 2006 

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