## Abstract

Let {Y_{t}, t ≥ 0} be a compound Poisson process, i.e., Y_{t} = ∑_{i=0}^{Nt}X_{i}, X_{0} = 0, X_{1}, X_{2}, . . . i.i.d. positive random variables and {N_{t}, t ≥ 0} a time homogeneous Poisson jump process. We consider two linear boundaries, b_{L} (t) = -β_{1} + 1 and b_{U} (t) = β_{2} + t, β_{1}, β_{2} > 0 and the stopping times T_{L} = inf{t : Y_{t} = b_{L}(t)}, T_{U} = inf{t : Y_{t} ≥ b_{U}(t)} and T = min{T_{L},T_{U}}. Laplace Stieltjes transforms are developed for the distributions of T_{U} and T for general distribution of X_{i} (i ≥ 1). These transforms are obtained by analyzing the sample path behavior of the process {Y_{t}}. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more.

Original language | English |
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Pages (from-to) | 89-101 |

Number of pages | 13 |

Journal | Communications in Statistics. Part C: Stochastic Models |

Volume | 15 |

Issue number | 1 |

DOIs | |

State | Published - 1999 |

## Keywords

- Laplace transforms
- Linear boundaries
- Martingales
- Queuing theory
- Stopping times

## ASJC Scopus subject areas

- Modeling and Simulation