Abstract
We consider the mixed systems composed of a fixed number of components whose lifetimes are i.i.d. with a known distribution which has a positive and finite variance. We show that a certain of the k-out-of-n systems has the minimal lifetime variance, and the maximal one is attained by a mixture of series and parallel systems. The number of the k-out-of-n system, and the probability weights of the mixture depend on the first two moments of order statistics of the parent distribution of the component lifetimes. We also show methods of calculating extreme system lifetime variances under various restrictions on the system lifetime expectations, and vice versa.
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Acknowledgments
The authors are grateful to a referee for valuable comments. The third author was supported by the Polish Ministry of Science and Higher Education Grant no. N N201 416739.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Beśka, M., Jasiński, K., Rychlik, T. et al. Mixed systems with minimal and maximal lifetime variances. Metrika 75, 877–894 (2012). https://doi.org/10.1007/s00184-011-0357-5
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DOI: https://doi.org/10.1007/s00184-011-0357-5