An invariant of representations of phase-type distributions and some applications

In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorpti...

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Bibliographic Details
Published in:Journal of applied probability 1996-06, Vol.33 (2), p.368-381
Main Authors: Commault, C., Chemla, J. P.
Format: Article
Language:English
Subjects:
Algebra
Applications
Density distributions
Exact sciences and technology
Input output
Laplace transformation
Laplace transforms
Markov analysis
Markov chains
Markov processes
Mathematical models
Mathematics
Necessary conditions
Polynomials
Probability
Probability and statistics
Probability theory and stochastic processes
Rational functions
Research Papers
Sciences and techniques of general use
Studies
Theory
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Summary:In this paper we consider phase-type distributions, their Laplace transforms which are rational functions and their representations which are finite-state Markov chains with an absorbing state. We first prove that, in any representation, the minimal number of states which are visited before absorption is equal to the difference of degree between denominator and numerator in the Laplace transform of the distribution. As an application, we prove that when the Laplace transform has a denominator with n real poles and a numerator of degree less than or equal to one the distribution has order n. We show that, in general, this result can be extended neither to the case where the numerator has degree two nor to the case of non-real poles.
ISSN:0021-9002
1475-6072
DOI:10.2307/3215060