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Reliability of a k-out-of-n: G System Subjected to Marshall-Olkin Type Shocks Concerning Magnitude |
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PP: 1143-1153 |
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doi:10.18576/jsap/120322
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Author(s) |
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S. M. Bakry,
N. A. Mokhlis,
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Abstract |
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| In this paper the reliability of a k-out-of-n: G system under the effect of shocks having the Marshall-Olkin type shock models, is studied. The magnitudes of the shocks are considered. The system contains n components and only functions when at least k of these components function. The system is subjected to (n + 1) shocks coming from (n + 1) different sources. The shock coming from
the it h source may destroy the it h component, i = 1, . . . , n, while the shock coming from the (n + 1)t h source may destroy all components simultaneously. A shock is fatal, destroys a component (components), whenever its magnitude exceeds an upper threshold. The system reliability is obtained by considering the arrival time and the magnitude of a shock as a bivariate random variable. It is assumed that the bivariate random variables representing the arrival times and the magnitudes of the shocks are independent with non-identical bivariate distributions. Since the computation of the reliability formula obtained is not easy to handle, an algorithm is introduced for calculating the reliability formula. The reliability of a k-out-of-n: G system subjected to independent and identical shocks is obtained as a special case, as well as the reliabilities of the series and the parallel systems. As an application, the bivariate exponential Gumbel distribution is considered. Also, numerical illustrations are performed to highlight the results obtained.
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