### Abstract

For continuous time birth-death processes on {0,1,2,...}, the first passage time T^{+}_{n} from n to n + 1 is always a mixture of (n + 1) independent exponential random variables. Furthermore, the first passage time T_{0,n+1} from 0 to (n + 1) is always a sum of (n + 1) independent exponential random variables. The discrete time analogue, however, does not necessarily hold in spite of structural similarities. In this paper, some necessary and sufficient conditions are established under which T^{+}_{n} and T_{0,n+1} for discrete time birth-death chains become a mixture and a sum, respectively, of (n + 1) independent geometric random variables on {1,2,...};. The results are further extended to conditional first passage times.

Original language | English |
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Pages (from-to) | 133-147 |

Number of pages | 15 |

Journal | Stochastic Processes and their Applications |

Volume | 20 |

Issue number | 1 |

DOIs | |

Publication status | Published - 1985 Jul |

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### Keywords

- PF
- birth-death processes
- complete monotonicity
- conditional first passage time
- discrete time birth-death chains
- first passage times
- strong unimodality

### ASJC Scopus subject areas

- Statistics and Probability
- Modelling and Simulation
- Applied Mathematics

### Cite this

*Stochastic Processes and their Applications*,

*20*(1), 133-147. https://doi.org/10.1016/0304-4149(85)90021-3