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Dynamic Reliability Modeling of Three-State Networks

Part of: Operations research and management science Special processes

Published online by Cambridge University Press:  30 January 2018

S. Ashrafi  and
M. Asadi
S. Ashrafi*
Affiliation:
University of Isfahan
M. Asadi*
Affiliation:
University of Isfahan and IPM
*
∗ Postal address: Department of Statistics, University of Isfahan, Isfahan, 81744, Iran. Email address: s.ashrafi@sci.ui.ac.ir
∗∗ Postal address: School of Mathematics, Institute of Research in Fundamental Sciences (IPM), PO Box 19395-5764, Tehran, Iran. Email address: m.asadi@sci.ui.ac.ir
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Abstract

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This paper is an investigation into the reliability and stochastic properties of three-state networks. We consider a single-step network consisting of n links and we assume that the links are subject to failure. We assume that the network can be in three states, up (K = 2), partial performance (K = 1), and down (K = 0). Using the concept of the two-dimensional signature, we study the residual lifetimes of the networks under different scenarios on the states and the number of failed links of the network. In the process of doing so, we define variants of the concept of the dynamic signature in a bivariate setting. Then, we obtain signature based mixture representations of the reliability of the residual lifetimes of the network states under the condition that the network is in state K = 2 (or K = 1) and exactly k links in the network have failed. We prove preservation theorems showing that stochastic orderings and dependence between the elements of the dynamic signatures (which relies on the network structure) are preserved by the residual lifetimes of the states of the network (which relies on the network ageing). Various illustrative examples are also provided.

Keywords

Dynamic signature matrixequivalent three-state networkstochastic orderingpositively quadrant dependentassociationNBU

MSC classification

Primary: 60K10: Applications (reliability, demand theory, etc.)
Secondary: 90B25: Reliability, availability, maintenance, inspection
Type
Research Article
Information
Journal of Applied Probability , Volume 51 , Issue 4 , December 2014 , pp. 999 - 1020
Copyright
© Applied Probability Trust 

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