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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
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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 

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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.

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