Hostname: page-component-cd9895bd7-7cvxr Total loading time: 0 Render date: 2024-12-26T11:53:19.231Z Has data issue: false hasContentIssue false
  • English
  • Français

Shock models with underlying birth process

Published online by Cambridge University Press:  14 July 2016

M. S. A-Hameed  and
F. Proschan
M. S. A-Hameed
Affiliation:
Florida State University
F. Proschan
Affiliation:
Florida State University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper extends results of Esary, Marshall and Proschan (1973) and A-Hameed and Proschan (1973). We consider the life distribution of a device subject to a sequence of shocks occurring randomly in time according to a nonstationary pure birth process: given k shocks have occurred in [0, t], the probability of a shock occurring in (t, t + Δ] is λ kλ (t)Δ + o (Δ). We show that various fundamental classes of life distributions (such as those with increasing failure rate, or those with the ‘new better than used' property, etc.) are obtained under appropriate assumptions on λ k, λ (t), and on the probability of surviving a given number of shocks.

Keywords

SHOCK MODELSBIRTH PROCESSTOTAL POSITIVITYLIFE DISTRIBUTIONSINCREASING FAILURE RATENEW BETTER THAN USEDVARIATION DIMINISHING PROPERTYRELIABILITY
Type
Research Papers
Information
Journal of Applied Probability , Volume 12 , Issue 1 , March 1975 , pp. 18 - 28
Copyright
Copyright © Applied Probability Trust 1975 

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.)

References

A-Hameed, M. S. and Proschan, F. (1973) Nonstationary shock models. Stoch. Proc, Applic. 1, 383–404.Google Scholar
Barlow, R. E. and Proschan, F. (1965) Mathematical Theory of Reliability . John Wiley, New York.Google Scholar
Esary, J. D., Marshall, A. W., and Proschan, F. (1973) Shock models and wear processes. Ann. Probab. 1, 627–649.Google Scholar
Karlin, S. (1966) A First Course in Stochastic Processes. Academic Press, New York.Google Scholar
Karlin, S. (1968) Total Positivity. Vol. 1. Stanford University Press, Stanford.Google Scholar
Karlin, S. and Proschan, F. (1960) Pólya type distributions of convolutions. Ann. Math. Statist. 31, 721–736.Google Scholar
Marshall, A. W. and Proschan, F. (1972) Classes of distributions applicable in replacement, with renewal theory implications. Proc. 6th Berkeley Symp. Math. Statist. Prob. 1, 395–415. University of California Press, Berkeley.Google Scholar