Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-19T21:10:36.798Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

∗

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

Supported by NSF Grant MCS 82–00098.

References

Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing: Probability Models. Holt, Rinehart and Winston, New York.Google Scholar
Brown, M. and Chaganty, N. R. (1983) On the first passage time distribution for a class of Markov chains. Ann. Prob. 11, 1000–1008.Google Scholar
Derman, C., Ross, S. M. and Schechner, Z. (1983) A note on first passage times in birth and death and nonnegative diffusion processes. Naval Res. Logist. Quart. 30, 283–285.Google Scholar
Esary, J. D., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Ghosh, M. and Ebrahimi, N. (1982) Shock models leading to increasing failure rate and decreasing mean residual life survival. J. Appl. Prob. 19, 158–166.Google Scholar
Karlin, S. (1964) Total positivity, absorption probabilities and applications. Trans. Amer. Math. Soc. 111, 33–107.CrossRefGoogle Scholar
Karlin, S. (1968) Total Positivity. Stanford University Press, Stanford, CA.Google Scholar
Karlin, S. and Mcgregor, J. (1959) A characterization of birth and death processes. Proc. Nat. Acad. Sci. USA 45, 375–379.CrossRefGoogle ScholarPubMed
Karlin, S. and Proschan, F. (1960) Polya type distributions of convolutions. Ann. Math. Statist. 31, 721–736.Google Scholar
Keilson, J. (1971) Log-concavity and log-convexity in passage time densities of diffusion and birth-death processes. J. Appl. Prob. 8, 391–398.Google Scholar
Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality. Springer-Verlag, New York.Google Scholar
Keilson, J. and Kester, A. (1977) Monotone matrices and monotone Markov processes. Stoch. Proc. Appl. 5, 231–241.Google Scholar
Schechner, Z. (1981) A note on two IFR systems. Naval Res. Logist. Quart. 28, 685–692.CrossRefGoogle Scholar
Widder, D. V. (1941) The Laplace Transform. Princeton University Press, Princeton, NJ.Google Scholar