Sharp bounds for the reliability of systems and mixtures with ordered components

Miziuła, Patryk; Navarro, Jorge

doi:10.1002/nav.21735

Cited by 10 publications

(6 citation statements)

References 15 publications

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“…The survival copula is an n ‐dimensional distribution function with uniform marginals over the interval (0, 1) which can be used to represent the joint reliability function of ( X 1 , …, X n ) as

\Pr (X_{1} > x_{1}, \dots, X_{n} > x_{n}) = K ({\overset{F}{true}}_{1} (x_{1}), \dots, {\overset{F}{true}}_{n} (x_{n})),

where

{\overset{F}{true}}_{i} (x_{i}) = \Pr (X_{i} > x_{i})

is the reliability function of the i th component, for i = 1,…, n . Then, it is well known that the system reliability function

{\overset{F}{true}}_{T}

can be written as

{\overset{F}{true}}_{T} (t) = \overset{true‾}{Q} ({\overset{F}{true}}_{1} (t), \dots, {\overset{F}{true}}_{n} (t)),

where

\overset{true‾}{Q}

depends on the structure of the system and on the copula K (see, eg, Miziula & Navarro, ; Navarro, ; Navarro, del Aguila, Sordo, & Suárez‐Llorens, ). Analogously, its distribution function

F_{T} = 1 - {\overset{F}{true}}_{T}

can be written as

F_{T} (t) = Q (F_{1} (t), \dots, F_{n} (t))

…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

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Stochastic comparisons of replacement policies in coherent systems under minimal repair

Arriaza

Navarro

Suárez‐Llorens

2018

Naval Research Logistics

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In this paper, we study stochastic comparisons of the coherent systems obtained under different repair policies. We consider minimal repairs of the components, that is, a failed component is replaced by another component which is similar (ie, it has the same distribution) to the replaced one and with the same age. Under these assumptions, we study how to determine the best replacement policies. Two general methods are proposed, one for systems with identically distributed components and another one for systems with ordered (in different stochastic senses) components. Both methods can be applied to systems with independent or dependent components. We use these methods to determine the optimal repair policies in some particular cases. We pay special attention to series and parallel systems with independent or dependent components. An algorithm to find the optimal replacement policy (when it exists) is provided.

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\Pr (X_{1} > x_{1}, \dots, X_{n} > x_{n}) = K ({\overset{F}{true}}_{1} (x_{1}), \dots, {\overset{F}{true}}_{n} (x_{n})),

where

{\overset{F}{true}}_{i} (x_{i}) = \Pr (X_{i} > x_{i})

is the reliability function of the i th component, for i = 1,…, n . Then, it is well known that the system reliability function

{\overset{F}{true}}_{T}

can be written as

{\overset{F}{true}}_{T} (t) = \overset{true‾}{Q} ({\overset{F}{true}}_{1} (t), \dots, {\overset{F}{true}}_{n} (t)),

where

\overset{true‾}{Q}

depends on the structure of the system and on the copula K (see, eg, Miziula & Navarro, ; Navarro, ; Navarro, del Aguila, Sordo, & Suárez‐Llorens, ). Analogously, its distribution function

F_{T} = 1 - {\overset{F}{true}}_{T}

can be written as

F_{T} (t) = Q (F_{1} (t), \dots, F_{n} (t))

…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

“…where Q depends on the structure of the system and on the copula K (see, eg, Miziula & Navarro, 2017;Navarro, 2018;Navarro, del Aguila, Sordo, & Suárez-Llorens, 2016). Analogously, its distribution function F T = 1 − F T can be written as…”

Section: Notation and Preliminary Resultsmentioning

confidence: 99%

Stochastic comparisons of replacement policies in coherent systems under minimal repair

Arriaza

Navarro

Suárez‐Llorens

2018

Naval Research Logistics

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“…If the components are ID, the joint reliability function of the random vector ( X 1 , … , X n ) can be expressed as

\Pr (X_{1} > x_{1}, \dots, X_{n} > x_{n}) = C (\overline{F} (x_{1}), \dots, \overline{F} (x_{n})),

where

\overline{F} (x_{i}) = \Pr (X_{i} > x_{i})

is the common reliability function of the components and C is the survival copula associated with the random vector ( X 1 , … , X n ). Then, the system reliability function

{\overline{F}}_{T}

can be written as

{\overline{F}}_{T} (t) = h (\overline{F} (t)),

where h (·) is a distortion function which depends on the structure of the system and on the survival copula C (see Navarro et al, 12 Mizula and Navarro, 13 Navarro, 14 and Navarro and Rychlik 15 ). Analogously, the joint distribution function of ( X 1 , … , X n ) can be expressed as

\Pr (X_{1} \leq x_{1}, \dots, X_{n} \leq x_{n}) = Ĉ (F (x_{1}), \dots, F (x_{n}

…”

Section: Introductionmentioning

confidence: 99%

Preservation of some stochastic orders by distortion functions with application to coherent systems with exchangeable components

Arriaza

Sordo

2020

Appl Stoch Models Bus & Ind

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The preservation of stochastic orders by distortion functions has become a topic of increasing interest in the reliability analysis of coherent systems. The reason of this interest is that the reliability function of a coherent system with identically distributed components can be represented as a distortion function of the common reliability function of the components. In this framework, we study the preservation of the excess wealth order, the total time on test transform order, the decreasing mean residual live order, and the quantile mean inactivity time order by distortion functions. The results are applied to study the preservation of these stochastic orders under the formation of coherent systems with exchangeable components.

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“…, 1) = 1. The explicit expression forQ can be seen in, for example, formula (2.3) of Miziula and Navarro [15]. The distortion functionQ depends on both the structure of the system and the dependence structure among the components, that is, the copula function associated to the component lifetimes.…”

Section: Resultsmentioning

confidence: 99%

Relationships Between Importance Measures and Redundancy in Systems With Dependent Components

Navarro¹,

Fernández-Martínez²,

Sánchez

et al. 2019

Prob. Eng. Inf. Sci.

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The paper shows the connections between some importance indices for the components in an engineering coherent system and the performance of the system obtained when a redundancy mechanism is applied to a specific component. A copula approach is used to model the dependency among the components. This approach includes the popular case of independent components. Under some assumptions, it is proved that if component i is more important than component j, then the system obtained by applying a redundancy procedure to the ith component is better, under different stochastic criteria, than that obtained with the jth component. These results can be applied to several redundancy mechanisms. A new importance index is defined to study active redundancies. Some illustrative examples are provided.

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Sharp bounds for the reliability of systems and mixtures with ordered components

Cited by 10 publications

References 15 publications

Stochastic comparisons of replacement policies in coherent systems under minimal repair

Stochastic comparisons of replacement policies in coherent systems under minimal repair

Preservation of some stochastic orders by distortion functions with application to coherent systems with exchangeable components

Relationships Between Importance Measures and Redundancy in Systems With Dependent Components

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