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Mathematics 2018, 6(10), 197; https://doi.org/10.3390/math6100197

Network Reliability Modeling Based on a Geometric Counting Process

1
Department of Statistics, Shiraz University, Shiraz 71454, Iran
2
Department of Statistics, University of Isfahan, Isfahan 81744, Iran
3
School of Mathematics, Institute of Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran
*
Author to whom correspondence should be addressed.
Received: 4 August 2018 / Revised: 28 September 2018 / Accepted: 8 October 2018 / Published: 11 October 2018
(This article belongs to the Special Issue Stochastic Processes with Applications)
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Abstract

In this paper, we investigate the reliability and stochastic properties of an n-component network under the assumption that the components of the network fail according to a counting process called a geometric counting process (GCP). The paper has two parts. In the first part, we consider a two-state network (with states up and down) and we assume that its components are subjected to failure based on a GCP. Some mixture representations for the network reliability are obtained in terms of signature of the network and the reliability function of the arrival times of the GCP. Several aging and stochastic properties of the network are investigated. The reliabilities of two different networks subjected to the same or different GCPs are compared based on the stochastic order between their signature vectors. The residual lifetime of the network is also assessed where the components fail based on a GCP. The second part of the paper is concerned with three-state networks. We consider a network made up of n components which starts operating at time t = 0 . It is assumed that, at any time t > 0 , the network can be in one of three states up, partial performance or down. The components of the network are subjected to failure on the basis of a GCP, which leads to change of network states. Under these scenarios, we obtain several stochastic and dependency characteristics of the network lifetime. Some illustrative examples and plots are also provided throughout the article. View Full-Text
Keywords: two-dimensional signature; multi-state network; totally positive of order 2; stochastic order; stochastic process two-dimensional signature; multi-state network; totally positive of order 2; stochastic order; stochastic process
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Zarezadeh, S.; Ashrafi, S.; Asadi, M. Network Reliability Modeling Based on a Geometric Counting Process. Mathematics 2018, 6, 197.

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