### Abstract

A probability distribution of discrete phase-type must have a rational generating function. We give an algorithm that constructs, from a given rational function G(z), a Markov chain whose absorption-time distribution has G(z) as generating function. The algorithm, which is based on an automata-theoretic algorithm of Soittola, may be applied to any G{z) that satisfies the conditions on discrete phase-type generating functions discovered by O'Cinneide. So it provides an alternative, algebraic proof of O'Cinneide's characterisation of discrete phase-type distributions. We also clarify the relation between the classes of continuous and discrete phase-type distributions, and show that O'Cinneide's characterisation of continuous phase-type distributions is a corollary of his discrete characterisation. In conjunction with our discrete-time algorithm, this engenders an algorithm for constructing a Markov process representation for any distribution of continuous phase-type.

Original language | English (US) |
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Pages (from-to) | 573-602 |

Number of pages | 30 |

Journal | Communications in Statistics. Stochastic Models |

Volume | 7 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1991 |

### Keywords

- Laplace transforms
- Markov chains
- first passage times
- generating functions
- phase-type distributions
- representations of phase-type distributions

### ASJC Scopus subject areas

- Modeling and Simulation