# On the reliability modeling of weighted *k*-out-of-*n* systems with randomly chosen components

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## Abstract

The weighted *k*-out-of-*n* (briefly denoted as weighted *k* / *n*) systems are among the most important kind of redundancy structures. We consider a weighted *k* / *n* system with dependent components where the system is built up from two classes \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\) of components that are categorized according to their weights and reliability functions. It is assumed that a random number *M*, \({M=0,1,\dots ,m}\), of the components are chosen from set \({\mathfrak {C}}_X\) whose components are distributed as \(F_X\) and the remaining \(n-M\) components selected from the set \({\mathfrak {C}}_Y\) whose components have distribution function \(F_Y\). We further assume that the structure of dependency of the components can be modeled by a copula function. The reliability of the system, at any time *t*, is expressed as a mixture of reliability of weighted *k* / *n* systems with fixed number of the components of types \({\mathfrak {C}}_X\) and \({\mathfrak {C}}_Y\) in terms of the probability mass function *M*. Some stochastic orderings are made between two different weighted *k* / *n* systems. It is shown that when the random mechanism of the chosen components for two systems are ordered in usual stochastic (*st*) order then, under some conditions, the lifetimes of the two systems are also ordered in *st* order. We also compare the lifetimes of two different systems in the sense of stochastic precedence concept. The results are examined by several illustrative examples under different conditions.

## Keywords

Reliability Weighted*k*-out-of-

*n*system Copulas Stochastic order Stochastic precedence

## Mathematics Subject Classification

90B25 62H20## Notes

### Acknowledgements

The authors express their sincere thanks to the Editor, and associate editor, and two referees for providing constructive comments which led to improvement of the paper. Asadi’s research work was carried out in IPM Isfahan branch and was in part supported by a Grant from IPM (No. 96620411).

## References

- Arcones MA, Kvam PH, Samaniego FJ (2002) Nonparametric estimation of a distribution subject to a stochastic precedence constraint. J Am Stat Assoc 97:170–182MathSciNetCrossRefGoogle Scholar
- Ball MO, Hagstrom JN, Provan JS (1995) Threshold reliability of networks with small failure sets. Networks 25:101–115CrossRefGoogle Scholar
- Barlow R.E, Proschan F (1981) Statistical theory of reliability and life testing: Probability models. Silver Spring, MDGoogle Scholar
- Berdichevsky V, Gitterman M (1998) Stochastic resonance and ratchets new manifestations. Physica A 249:88–95CrossRefGoogle Scholar
- Bier M (1997) Brownian ratchets in physics and biology. Contemp Phys 38:371–379CrossRefGoogle Scholar
- Chen Y, Yang Q (2005) Reliability of two-stage weighted k-out-of-n systems with components in common. IEEE Trans Reliab 54:431–440CrossRefGoogle Scholar
- Coolen FPA, Coolen-Maturi T (2012) Generalizing the signature to systems with multiple types of components. Complex Syst Dependability 115–130Google Scholar
- Cui L, Xie M (2005) On a generalized k-out-of-n system and its reliability. Int J Syst Sci 36:267–274MathSciNetCrossRefGoogle Scholar
- De Santis E, Fantozzi F, Spizzichino F (2015) Relations between stochastic orderings and generalized stochastic precedence. Probab Eng Inf Sci 29:329–343MathSciNetCrossRefGoogle Scholar
- Di Crescenzo A (2007) A Parrondo paradox in reliability theory. Math Sci 32:17–22MathSciNetzbMATHGoogle Scholar
- Di Crescenzo A, Pellerey F (2011) Stochastic comparisons of series and parallel systems with randomized independent components. Oper Res Lett 39:380–384MathSciNetCrossRefGoogle Scholar
- Eryilmaz S (2011) Dynamic behavior of k-out-of-n: G systems. Oper Res Lett 39:155–159MathSciNetCrossRefGoogle Scholar
- Eryilmaz S (2013) Mean instantaneous performance of a system with weighted components that have arbitrarily distributed lifetimes. Reliab Eng Syst Saf 119:290–293CrossRefGoogle Scholar
- Eryilmaz S (2014) Multivariate copula based dynamic reliability modeling with application to weighted-k-out-of-n systems of dependent components. Struct Saf 51:23–28CrossRefGoogle Scholar
- Eryilmaz S (2015) Capacity loss and residual capacity in weighted k-out-of-n: G systems. Reliab Eng Syst Saf 136:140–144CrossRefGoogle Scholar
- Eryilmaz S, Coolen FPA, Coolen-Maturi T (2018) Marginal and joint reliability importance based on survival signature. Reliab Eng Syst Saf 172:118–128CrossRefGoogle Scholar
- Eryilmaz S, Sarikaya K (2013) Modeling and analysis of weighted-k-out-of-n: G system consisting of two different types of components. Proc Inst Mech Eng Part O J Risk Reliab 201:1–7Google Scholar
- Gammaitoni L, Hanggi P, Jung P, Marchesoni F (1998) Stochastic resonance. Rev Mod Phys 70:223–287CrossRefGoogle Scholar
- Genest C, Neslehova J, Ghorbal NB (2011) Estimators based on Kendall’s tau in multivariate copula models. Australian New Zealand J Stat 53:157–177MathSciNetCrossRefGoogle Scholar
- Hazra NK, Finkelstein M, Cha JH (2017) On optimal grouping and stochastic comparisons for heterogenous items. J Multivar Anal 160:146–156CrossRefGoogle Scholar
- Hazra NK, Nanda AK (2014) Some results on series and parallel systems of randomized components. Oper Res Lett 42:132–136MathSciNetCrossRefGoogle Scholar
- Higashiyama Y (2001) A factored reliability formula for weighted-k-out-of-n system. Asia-Pacific J Oper Res 18:61–66Google Scholar
- Jia X, Cui L (2012) Reliability research of k-out-of-n: G supply chain system based on copula. Commun Stat Theory Methods 41:4023–4033MathSciNetCrossRefGoogle Scholar
- Kochar S, Xu M (2010) On residual lifetimes of k-out-of-n systems with nonidentical components. Probab Eng Inf Sci 24:109–127MathSciNetCrossRefGoogle Scholar
- Kuo W, Zuo MJ (2003) Optimal reliability modeling: principles and applications. Wiley, New YorkGoogle Scholar
- Li H, Li X (2013) Stochastic orders in reliability and risk. Springer, New YorkCrossRefGoogle Scholar
- Li X, You Y, Fang R (2016) On weighted k-out-of-n systems with statistically dependent component lifetimes. Probab Eng Inf Sci 30:533–546MathSciNetCrossRefGoogle Scholar
- Li X, Zhao P (2008) Stochastic comparison on general inactivity time and general residual life of k-out-of-n systems. Commun Stat Simul Comput 37:1005–1019MathSciNetCrossRefGoogle Scholar
- Navarro J, Del Aguila Y (2017) Stochastic comparisons of distorted distributions, coherent systems and mixtures with ordered components. Metrika 80:627–648MathSciNetCrossRefGoogle Scholar
- Navarro J, Pellerey F, Di Crescenzo A (2015) Orderings of coherent systems with randomized dependent components. Eur J Oper Res 240:127–139MathSciNetCrossRefGoogle Scholar
- Navarro J, Spizzichino F (2010) Comparisons of series and parallel systems with components sharing the same copula. Appl Stoch Models Bus Ind 26:775–791MathSciNetCrossRefGoogle Scholar
- Nelsen RB (2006) An introduction to copulas. Springer, New YorkzbMATHGoogle Scholar
- Rushdi AM (1990) Threshold systems and their reliability. Microelectron Reliab 30:299–312CrossRefGoogle Scholar
- Samaniego FJ, Navarro J (2016) On comparing coherent systems with heterogeneous components. Adv Appl Probab 48(1):88–111MathSciNetCrossRefGoogle Scholar
- Samaniego FJ, Shaked M (2008) Systems with weighted components. Stat Probab Lett 78:815–823MathSciNetCrossRefGoogle Scholar
- Shaked M, Shanthikumar JG (2007) Stochastic orders. Springer series in statistics. Springer, New YorkCrossRefGoogle Scholar
- Tang XS, Li DQ, Zhou CB, Phoon KK, Zhang LM (2013a) Impact of copulas for modeling bivariate distributions on system reliability. Struct Saf 44:80–90CrossRefGoogle Scholar
- Tang XS, Li DQ, Zhou CB, Zhang LM (2013b) Bivariate distribution models using copulas for reliability analysis. Proc Inst Mech Eng Part O J Risk Reliab 227:499–512Google Scholar
- Wang Y, Pham H (2012) Modeling the dependent competing risks with multiple degradation processes and random shock using time-varying copulas. IEEE Trans Reliab 61:13–22CrossRefGoogle Scholar
- Wu JS, Chen RJ (1994a) An algorithm for computing the reliability of a weighted-k-out-of-n system. IEEE Trans Reliab 43:327–328CrossRefGoogle Scholar
- Wu JS, Chen RJ (1994b) Efficient algorithms for k-out-of-n and consecutive weighted k-out-of-n : F system. IEEE Trans Reliab 43:650–655CrossRefGoogle Scholar