Bounds on Variances of Lifetimes of Coherent and Mixed Systems

Jasiński, Krzysztof; Navarro, Jorge

doi:10.1239/jap/1253279857

Cited by 21 publications

(16 citation statements)

References 24 publications

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“…2 ) and (T (2) 1 , T (2) 2 ) be the joint lifetimes of the first and second paired systems. Our goal is to identify conditions which imply some form of stochastic relationship between (T (…”

Section: T (1)mentioning

confidence: 99%

“…2 ) and (T (2) 1 , T (2) 2 ). Our first result in this direction gives sufficient conditions based on the shift ordering between the signatures of two systems for their respective lifetimes to obey the bivariate lower orthant stochastic ordering (see [10, p. …”

Section: T (1)mentioning

confidence: 99%

“…Moreover, they proved that the distribution of the lifetime of a coherent system with n components can be written as a mixture of the order statistics obtained from the lifetimes of m > n components. The vector of coefficients in that representation was called signature of order m. These representations allow us to compare systems of different orders (see [6]) and to obtain bounds for their variances (see [2]). The following is a simple version of this representation result that relates the signature s of a system of n components with i.i.d.…”

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The Joint Signature of Coherent Systems with Shared Components

Navarro

Samaniego

2010

J. Appl. Probab.

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System signatures are useful tools in the study and comparison of coherent systems. In this paper, we define and study a similar concept, called the joint signature, for two coherent systems which share some components. Under an independent and identically distributed assumption on component lifetimes, a pseudo-mixture representation based on this joint signature is obtained for the joint distribution of the lifetimes of both systems. Sufficient conditions are given based on the respective joint signatures of two pairs of systems, each with shared components, to ensure various forms of bivariate stochastic orderings between the joint lifetimes of the two pairs of systems.

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Section: T (1)mentioning

confidence: 99%

Section: T (1)mentioning

confidence: 99%

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The Joint Signature of Coherent Systems with Shared Components

Navarro

Samaniego

2010

J. Appl. Probab.

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“…or exchangeable dependent components, especially k-outof-n systems. For instance, see [1], [8], [9], [10], [12], [13], [15], [21], [22], [24], [26], [27], and [28].…”

Section: Mixture Representations Of Inactivity Timesmentioning

confidence: 99%

Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

Zhang

2010

J. Appl. Probab.

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In this paper we obtain several mixture representations of the reliability function of the inactivity time of a coherent system under the condition that the system has failed at time t (> 0) in terms of the reliability functions of inactivity times of order statistics. Some ordering properties of the inactivity times of coherent systems with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors between systems.

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“…The bounds were refined by Papadatos (1997) and Jasiński and Rychlik (2011) in the case of symmetrically distributed random variables. Jasiński et al (2009) extended the results of Papadatos (1995) to the case of arbitrarily fixed mixed systems. Rychlik (1994) described methods of calculating sharp lifetime variance bounds for k-out-of-n systems built of exchangeable components with a known marginal lifetime distribution.…”

Section: Introduction and Auxiliary Resultsmentioning

confidence: 90%

Mixed systems with minimal and maximal lifetime variances

Beśka

Jasiński

Spryszyński

2011

Metrika

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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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Bounds on Variances of Lifetimes of Coherent and Mixed Systems

Cited by 21 publications

References 24 publications

The Joint Signature of Coherent Systems with Shared Components

The Joint Signature of Coherent Systems with Shared Components

Mixture Representations of Inactivity Times of Conditional Coherent Systems and their Applications

Mixed systems with minimal and maximal lifetime variances

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