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A Markovian growth-collapse model

  • Onno Boxma (a1), David Perry (a2), Wolfgang Stadje (a3) and Shelemyahu Zacks (a4)

Abstract

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, (X t ) t≥0, and the distributions of the hitting times T a = inf{t ≥ 0 : X t = 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[T a , 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.

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Copyright

Corresponding author

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

References

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