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First-passage times with PFr densities

Published online by Cambridge University Press:  14 July 2016

David Assaf ,
Moshe Shared  and
J. George shanthikumar
David Assaf
Affiliation:
University of Arizona
Moshe Shared*
Affiliation:
University of Arizona
J. George shanthikumar*
Affiliation:
University of Arizona
*
∗∗Postal address: Department of Mathematics, Building 89, The University of Arizona, Tucson, AZ 85721, USA.
∗∗∗Postal address: Department of Systems and Industrial Engineering, The University of Arizona, Tucson, AZ 85721, USA.
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Abstract

It is shown that if a finite-state continuous-time Markov process can be uniformized such that the embedded Markov chain has a TPr (totally positive of order r) transition matrix, then the first-passage time from state 0 to any other state has a PFr (Polya frequency of order r) density. As a consequence, results of Keilson (1971), Esary, Marshall and Proschan (1973), Ghosh and Ebrahimi (1982) and Derman, Ross and Schechner (1983) are strengthened. It is also shown that some cumulative damage shock models, with an underlying compound Poisson process and ‘damages' which are not necessarily non-negative, are associated with wear processes having PFr first-passage times to any threshold. First-passage times with completely monotone densities are also discussed.

Keywords

TOTAL POSITIVITYCOMPLETE MONOTONICITYCUMULATIVE DAMAGE SHOCK MODELSPFr PROCESSESCOMPOUND POISSON PROCESSIFR
Type
Research Papers
Information
Journal of Applied Probability , Volume 22 , Issue 1 , March 1985 , pp. 185 - 196
Copyright
Copyright © Applied Probability Trust 1985 

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Footnotes

∗

Permanent address: Department of Statistics, Hebrew University, Jerusalem, Israel.

Supported by NSF Grant MCS 82–00098.

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