### Abstract

We consider birth-death processes taking values in ${\cal N} \equiv \{0,1,\ldots\}$, but allow the death rate in state $0$ to be positive, so that escape from ${\cal N}$ is possible. Two such processes with transition functions $\{p_{ij}(t)\}$ and $\{{\tilde p}_{ij}(t)\}$ are said to be {\it similar} if, for all $i,j \in {\cal N},$ there are constants $c_{ij}$ such that ${\tilde p}_{ij}(t) = c_{ij}p_{ij}(t)$ for all $t \geq 0.$ We determine conditions on the birth and death rates of a birth-death process for the process to be a member of a family of similar processes, and we identify the members of such a family.

Original language | Undefined |
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Place of Publication | Enschede |

Publisher | University of Twente, Department of Applied Mathematics |

Number of pages | 14 |

ISBN (Print) | 0169-2690 |

Publication status | Published - 1999 |

### Publication series

Name | Memorandum / Department of Applied Mathematics |
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Publisher | Department of Applied Mathematics, University of Twente |

No. | 1486 |

ISSN (Print) | 0169-2690 |

### Keywords

- EWI-3306
- IR-65675
- METIS-141298
- MSC-60J80

## Cite this

Lenin, R. B., Parthasarathy, P. R., Scheinhardt, W. R. W., & van Doorn, E. A. (1999).

*Families of birth-death processes with similar time-dependent behaviour*. (Memorandum / Department of Applied Mathematics; No. 1486). Enschede: University of Twente, Department of Applied Mathematics.