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Phase-type distributions and invariant polytopes

Published online by Cambridge University Press:  01 July 2016

Colm Art O'Cinneide
Colm Art O'Cinneide*
Affiliation:
Louisiana State University
*
Present address: Department of Mathematical Sciences, SE 301, University of Arkansas, Fayetteville AR 72701, USA.
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Abstract

The notion of an invariant polytope played a central role in the proof of the characterization of phase-type distributions. The purpose of this paper is to develop invariant polytope techniques further. We derive lower bounds on the number of states needed to represent a phase-type distribution based on poles of its Laplace–Stieltjes transform. We prove that every phase-type distribution whose transform has only real poles has a bidiagonal representation. We close with three short applications of the invariant polytope idea. Taken together, the results of this paper show that invariant polytopes provide a natural approach to many questions about phase-type distributions.

Keywords

REPRESENTATIONS OF PHASE-TYPE DISTRIBUTIONSTRIANGULAR REPRESENTATIONSMARKOV CHAINSLAPLACE TRANSFORMSGENERATORSBOUNDS ON EIGENVALUES
Type
Research Article
Information
Advances in Applied Probability , Volume 23 , Issue 3 , September 1991 , pp. 515 - 535
Copyright
Copyright © Applied Probability Trust 1991 

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Footnotes

Work partially supported by the National Science Foundation under grant no. DMS-8907630, and carried out at the University of Arizona and Louisiana State University.

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