Hostname: page-component-586b7cd67f-rcrh6 Total loading time: 0 Render date: 2024-11-26T06:04:45.105Z Has data issue: false hasContentIssue false
  • English
  • Français

A class of correlated cumulative shock models

Published online by Cambridge University Press:  01 July 2016

Ushio Sumita  and
J. George Shanthikumar
Ushio Sumita*
Affiliation:
University of Rochester
J. George Shanthikumar*
Affiliation:
University of Arizona
*
∗Postal address: Graduate School of Management, University of Rochester, Rochester, NY 14627, USA.
†Postal address: Systems and Industrial Engineering Department, University of Arizona, Tucson, AZ 85721, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we define and analyze a class of cumulative shock models associated with a bivariate sequence {Xn, Yn}∞n=0 of correlated random variables. The {Xn} denote the sizes of the shocks and the {Yn} denote the times between successive shocks. The system fails when the cumulative magnitude of the shocks exceeds a prespecified level z. Two models, depending on whether the size of the nth shock is correlated with the length of the interval since the last shock or with the length of the succeeding interval until the next shock, are considered. Various transform results and asymptotic properties of the system failure time are obtained. Further, sufficient conditions are established under which system failure time is new better than used, new better than used in expectation, and harmonic new better than used in expectation.

Keywords

FIRST-PASSAGE TIMESNEW BETTER THAN USEDNEW BETTER THAN USED IN EXPECTATIONHARMONIC NEW BETTER THAN USED IN EXPECTATION: CUMULATIVE DAMAGE
Type
Research Article
Information
Advances in Applied Probability , Volume 17 , Issue 2 , June 1985 , pp. 347 - 366
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.)

References

1. Anscombe, F. J. (1952) Large-sample theory of sequential estimation. Proc. Camb. Phil. Soc. 48, 600–607.CrossRefGoogle Scholar
2. Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing–Probability Models. Holt, Rinehart, and Winston, New York.Google Scholar
3. Block, H. W. and Savits, T. H. (1978) Shock models with NBUE survival. J. Appl. Prob. 15, 621–628.CrossRefGoogle Scholar
4. Klefsjö, B. (1981) HNBUE survival under some shock models. Scand. J. Statist. 8, 39–47.Google Scholar
5. Klefsjö, B. (1982) The HNBUE and HNWUE classes of life distributions. Naval Res. Logist. Quart. 29, 331–344.Google Scholar
6. Marshall, A. W. and Shaked, M. (1983) New better than used processes. Adv. Appl. Prob. 15, 601–615.CrossRefGoogle Scholar
7. Rolski, R. (1975) Mean residual life. Bull. Internat. Statist. Inst. 46, 266–270.Google Scholar
8. Ross, S. (1983) Stochastic Processes. Wiley, New York.Google Scholar
9. Shanthikumar, J. G. (1984) Processes with new better than used first-passage times. Adv. Appl. Prob. 16, 667–686.CrossRefGoogle Scholar
10. Shanthikumar, J. G. and Sumita, U. (1983) General shock models associated with correlated renewal sequences. J. Appl. Prob. 20, 600–614.CrossRefGoogle Scholar
11. Shanthikumar, J. G. and Sumita, U. (1984) Distribution properties of the system failure time in a general shock model. Adv. Appl. Prob. 16, 363–377.CrossRefGoogle Scholar
12. Shanthikumar, J. G. and Sumita, U. (1984) A central limit theorem for random sums of random variables. O.R. Letters 3, 153–155.CrossRefGoogle Scholar
13. Smith, W. L. (1954) Asymptotic renewal theorems. Proc. R. Soc. Edinburgh A 64, 9–48.Google Scholar
14. Wilson, J. R. (1983) The inspection paradox in renewal-reward processes. O.R. Letters 2, 27–30.CrossRefGoogle Scholar