Hostname: page-component-848d4c4894-p2v8j Total loading time: 0.001 Render date: 2024-05-25T05:00:43.636Z Has data issue: false hasContentIssue false
  • English
  • Français

Ageing properties of certain dependent geometric sums

Part of: Special processes Operations research and management science

Published online by Cambridge University Press:  14 July 2016

Antal Kováts  and
Tamás F. Móri
Antal Kováts
Affiliation:
Eötvös Loránd University
Tamás F. Móri
Affiliation:
Eötvös Loránd University
Get access Rights & Permissions [Opens in a new window]

Abstract

We study some distribution properties of a random sum of i.i.d. non-negative random variables, where the number of terms is geometrically distributed and not independent of the summands. The results are applied to the system failure time of a one-unit system with a single spare and repair facility. In such a system when the operating unit fails it is immediately replaced by the spare and sent to the repair facility. The system continues operating until the first time when the failed unit has not yet been repaired by the failure of the operating unit. Certain ageing properties such as NBU, NWU, NBUE, NWUE, HNBUE, HNWUE, L+ and L– are shown to be inheritable from the working time of the operating unit to the system lifetime.

Keywords

RANDOM SUMGEOMETRIC CONVOLUTIONREPAIRABLE SYSTEMNBUNWUNBUENWUEHNBUEHNWUEVARIABILITY ORDERINGLAPLACE TRANSFORM

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Papers
Information
Journal of Applied Probability , Volume 29 , Issue 3 , September 1992 , pp. 655 - 666
Copyright
Copyright © Applied Probability Trust 1992 

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

∗

Research supported by the Hungarian National Foundation for Scientific Research, Grant N° 1405.

∗∗

Research supported by the Hungarian National Foundation for Scientific Research, Grant N° 1905.

References

[1] Barlow, R. E. and Proschan, F. (1975) Statistical Theory of Reliability and Life Testing. Holt, Rinehart and Winston, New York.Google Scholar
[2] Block, H. W. and Savits, T. H. (1978) Shock models with NBUE survival. J. Appl. Prob. 15, 621–628.Google Scholar
[3] Brown, M. (1990) Error bounds for exponential approximations of geometric convolutions. Ann. Prob. 18, 1388–1402.Google Scholar
[4] Gnedenko, B. V., Belyayev, Yu. K. and Solovyev, A. D. (1969) Mathematical Methods of Reliability Theory. Academic Press, New York.Google Scholar
[5] Klefsjö, B. (1981) HNBUE survival under some shock models. Scand. J. Statist. 8, 9–47.Google Scholar
[6] Ross, S. M. (1983) Stochastic Processes. Wiley, New York.Google Scholar
[7] Shanthikumar, J. G. (1986) First failure time of dependent parallel systems with safety periods. Microelectron. Reliab. 26, 955–972.Google Scholar
[8] Shanthikumar, J. G. (1988) DFR property of first passage times and its preservation under geometric compounding. Ann. Prob. 16, 397–406.Google Scholar
[9] 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