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Mixture Representations for the Joint Distribution of Lifetimes of two Coherent Systems with Shared Components

Part of: Distribution theory - Probability Special processes

Published online by Cambridge University Press:  04 January 2016

Jorge Navarro ,
Francisco J. Samaniego  and
N. Balakrishnan
Jorge Navarro*
Affiliation:
Universidad de Murcia
Francisco J. Samaniego*
Affiliation:
University of California, Davis
N. Balakrishnan*
Affiliation:
McMaster University and King Abdulaziz University
*
∗ Postal address: Facultad de Matemáticas, Universidad de Murcia, 30100 Murcia, Spain. Email address: jorgenav@um.es
∗∗ Postal address: Department of Statistics, University of California, Davis, 1 Shields Avenue, Davis, CA 95616, USA. Email address: fjsamaniego@ucdavis.edu
∗∗∗ Postal address: Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario L8S 4K1, Canada. Email address: bala@mcmaster.ca
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Abstract

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The signature of a system is defined as the vector whose ith element is the probability that the system fails concurrently with the ith component failure. The signature vector is known to be a distribution-free measure and a representation of the system's survival function has been developed in terms of the system's signature. The present work is devoted to the study of the joint distribution of lifetimes of pairs of systems with shared components. Here, a new distribution-free measure, the ‘joint bivariate signature’, of a pair of systems with shared components is defined, and a new representation theorem for the joint survival function of the system lifetimes is established. The theorem is shown to facilitate the study of the dependence between systems and the comparative performance of two pairs of such systems.

Keywords

Coherent systemk-out-of-n systemorder statisticssignaturemixturestochastic order

MSC classification

Primary: 60E15: Inequalities; stochastic orderings
Secondary: 60K10: Applications (reliability, demand theory, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 45 , Issue 4 , December 2013 , pp. 1011 - 1027
Copyright
© Applied Probability Trust 

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