## 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.

Original language | English |
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Pages (from-to) | 221-243 |

Number of pages | 23 |

Journal | Advances in Applied Probability |

Volume | 38 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2006 |

## Keywords

- Duality
- Growth-collapse process
- Hitting time
- Markov modulation
- Piecewise-deterministic Markov process
- Stationary distribution
- Uniform cut-off

## ASJC Scopus subject areas

- Statistics and Probability
- Applied Mathematics