Hostname: page-component-74656f9884-j7cmm Total loading time: 0 Render date: 2026-11-10T17:54:14.457Z Has data issue: false hasContentIssue false
  • English
  • Français

Pure jump shock models in reliability

Published online by Cambridge University Press:  01 July 2016

James W. Drosen
James W. Drosen*
Affiliation:
The Pennsylvania State University
*
∗Postal address: Division of Management Science, College of Business Administration, The Pennsylvania State University, 310 Business Administration Building, University Park, PA 16802, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

There are many examples of a device suffering damage from random environmental shocks. We model the damage level of such a device as a pure jump Markov process, where the incremental damage caused by a shock depends both on the magnitude of the shock and on the damage level just before the shock. We also look at the time until failure of the device, which occurs when the damage level exceeds a random threshold. The distribution of the failure time and the failure rate are examined, and conditions for the failure rate to be increasing or to have an increasing average are found.

Keywords

STOCHASTIC INTEGRAL EQUATIONSJUMP PROCESSES

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 18 , Issue 2 , June 1986 , pp. 423 - 440
Copyright
Copyright © Applied Probability Trust 1986 

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.)

Article purchase

Temporarily unavailable

References

Abdel-Hameed, M. S. (1984a) Life distribution properties of devices subject to a Lévy wear process. Math. Operat. Res. 9, 606–614.CrossRefGoogle Scholar
Abdel-Hameed, M. S. (1984b) Life distribution properties of devices subject to a pure jump damage process. J. Appl. Prob. 21, 816–825.CrossRefGoogle Scholar
Abdel-Hameed, M. S. and Proschan, F. (1973) Non-stationary shock models. Stoch. Proc. Appl. 1, 383–404.Google Scholar
Blumenthal, R. M. and Getoor, R. K. (1968) Markov Processes and Potential Theory. Academic Press, New York.Google Scholar
Çinlar, E. (1975). Introduction to Stochastic Processes and their Applications. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Çinlar, E. and Jacod, J. (1981) Representation of semimartingale Markov processes in terms of Wiener processes and Poisson random measures. In Seminar on Stochastic Processes 1981, ed. Çinlar, E., Chung, K. L., and Getoor, R. K., Birkhauser, Boston, 159–242.CrossRefGoogle Scholar
Çinlar, E., Jacod, J., Potter, P. and Sharpe, M. J. (1980) Semimartingales and Markov processes. Z. Wahrscheinlichkeitsth. 54, 161–219.CrossRefGoogle Scholar
Çinlar, E. and Pinsky, M. (1972) On dams with additive inputs and general release rule. J. Appl. Prob. 9, 422–429.Google Scholar
Drosen, J. W. (1983) Failure times and optimal stopping rules of generalized shock models. Doctoral Dissertation, Northwestern University, Evanston, Illinois.Google Scholar
Esary, J., Marshall, A. W. and Proschan, F. (1973) Shock models and wear processes. Ann. Prob. 1, 627–649.Google Scholar
Gihman, I. L. and Skorohod, A. V. (1972) Stochastic Differential Equations. Springer-Verlag, New York.Google Scholar
Karlin, S. (1968) Total Positivity, Vol. 1. Stanford University Press, Stanford, Ca.Google Scholar
Komatsu, T. (1973) Markov processes associated with certain integro-differential operators. Osaka J. Math. 36, 1–26.Google Scholar
Stroock, D. W. (1975) Diffusion processes associated with Lévy generators. Z. Wahrscheinlichkeitsth. 32, 209–244.Google Scholar