Reliability modeling of coherent systems with shared components based on sequential order statistics

2018 ◽  
Vol 55 (3) ◽  
pp. 845-861
Author(s):  
S. Ashrafi ◽  
S. Zarezadeh ◽  
M. Asadi
Keyword(s):  
Order Statistics ◽  
Reliability Function ◽  
Failure Model ◽  
Coherent Systems ◽  
Sequential Order ◽  
Birth Process ◽  
Reliability Modeling ◽  
Joint Reliability ◽  
Sequential Order Statistics ◽  
Signature Matrix

Abstract In this paper we are concerned with the reliability properties of two coherent systems having shared components. We assume that the components of the systems are two overlapping subsets of a set of n components with lifetimes X1,...,Xn. Further, we assume that the components of the systems fail according to the model of sequential order statistics (which is equivalent, under some mild conditions, to the failure model corresponding to a nonhomogeneous pure-birth process). The joint reliability function of the system lifetimes is expressed as a mixture of the joint reliability functions of the sequential order statistics, where the mixing probabilities are the bivariate signature matrix associated to the structures of systems. We investigate some stochastic orderings and dependency properties of the system lifetimes. We also study conditions under which the joint reliability function of systems with shared components of order m can be equivalently written as the joint reliability function of systems of order n (n>m). In order to illustrate the results, we provide several examples.

scholarly journals Systems with Failure-Dependent Lifetimes of Components

2009 ◽  
Vol 46 (04) ◽  
pp. 1052-1072 ◽  
Author(s):  
M. Burkschat
Keyword(s):  
Order Statistics ◽  
Joint Distribution ◽  
Coherent System ◽  
Coherent Systems ◽  
Sequential Order ◽  
Failure Times ◽  
Sequential Order Statistics ◽  
Definition Of ◽  
General Component

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequentialk-out-of-nsystem, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

Representations of component importance for coherent systems with exchangeable components

2020 ◽  
Vol 57 (2) ◽  
pp. 385-406
Author(s):  
S. Pitzen ◽  
M. Burkschat
Keyword(s):  
Order Statistics ◽  
Distribution Functions ◽  
Importance Measure ◽  
Coherent Systems ◽  
Sequential Order ◽  
Limiting Behavior ◽  
Sequential Order Statistics ◽  
Component Importance ◽  
Dependent Component ◽  
Two Measures

AbstractTwo definitions of Birnbaum’s importance measure for coherent systems are studied in the case of exchangeable components. Representations of these measures in terms of distribution functions of the ordered component lifetimes are given. As an example, coherent systems with failure-dependent component lifetimes based on the notion of sequential order statistics are considered. Furthermore, it is shown that the two measures are ordered in the case of associated component lifetimes. Finally, the limiting behavior of the measures with respect to time is examined.

scholarly journals Dynamic Signatures of Coherent Systems Based on Sequential Order Statistics

2013 ◽  
Vol 50 (1) ◽  
pp. 272-287 ◽  
Author(s):  
M. Burkschat ◽  
J. Navarro
Keyword(s):  
Order Statistics ◽  
Failure Time ◽  
Residual Lifetime ◽  
Coherent Systems ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Additional Information ◽  
Structured Systems ◽  
Sequential Order Statistics ◽  
Inactivity Time

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

scholarly journals Systems with Failure-Dependent Lifetimes of Components

2009 ◽  
Vol 46 (4) ◽  
pp. 1052-1072 ◽  
Author(s):  
M. Burkschat
Keyword(s):  
Order Statistics ◽  
Joint Distribution ◽  
Coherent System ◽  
Coherent Systems ◽  
Sequential Order ◽  
Failure Times ◽  
Sequential Order Statistics ◽  
Definition Of ◽  
General Component

A model for describing the lifetimes of coherent systems, where the failures of components may have an impact on the lifetimes of the remaining components, is proposed. The model is motivated by the definition of sequential order statistics (cf. Kamps (1995)). Sequential order statistics describe the successive failure times in a sequential k-out-of-n system, where the distribution of the remaining components' lifetimes is allowed to change after every failure of a component. In the present paper, general component lifetimes which can be influenced by failures are considered. The ordered failure times of these components can be used to extend the concept of sequential order statistics. In particular, a definition of sequential order statistics based on exchangeable components is proposed. By utilizing the system signature (cf. Samaniego (2007)), the distribution of the lifetime of a coherent system with failure-dependent exchangeable component lifetimes is shown to be given by a mixture of the distributions of sequential order statistics. Furthermore, some results on the joint distribution of sequential order statistics based on exchangeable components are given.

Dynamic Signatures of Coherent Systems Based on Sequential Order Statistics

2013 ◽  
Vol 50 (01) ◽  
pp. 272-287 ◽  
Author(s):  
M. Burkschat ◽  
J. Navarro
Keyword(s):  
Order Statistics ◽  
Failure Time ◽  
Residual Lifetime ◽  
Coherent Systems ◽  
Sequential Order ◽  
Stochastic Comparisons ◽  
Additional Information ◽  
Structured Systems ◽  
Sequential Order Statistics ◽  
Inactivity Time

Sequential order statistics can be used to describe the ordered lifetimes of components in a system, where the failure of a component may affect the performance of remaining components. In this paper mixture representations of the residual lifetime and the inactivity time of systems with such failure-dependent components are considered. Stochastic comparisons of differently structured systems are obtained and properties of the weights in the mixture representations are examined. Furthermore, corresponding representations of the residual lifetime and the inactivity time of a system given the additional information about a previous failure time are derived.

Univariate and multivariate stochastic orderings of residual lifetimes of live components in sequential (𝑛-𝑟+ 1)-out-of-𝑛 systems

2018 ◽  
Vol 55 (3) ◽  
pp. 834-844
Author(s):  
Ghobad Barmalzan ◽  
Abedin Haidari ◽  
Narayanaswamy Balakrishnan
Keyword(s):  
Order Statistics ◽  
Exponential Distribution ◽  
Sufficient Conditions ◽  
Sequential Order ◽  
Stochastic Orderings ◽  
Sequential Order Statistics ◽  
Residual Lifetimes

Abstract Sequential order statistics can be used to describe the ordered lifetimes of components of a system when the failure of a component may affect the reliability of the remaining components. After a reliability system consisting of n components fails, some of its components may still be alive. In this paper we first establish some univariate stochastic orderings and ageing properties of the residual lifetimes of the live components in a sequential (n-r+1)-out-of-n system. We also obtain a characterizing result for the exponential distribution based on uncorrelated residual lifetimes of live components. Finally, we provide some sufficient conditions for comparing vectors of residual lifetimes of the live components from two sequential (n-r+1)-out-of-n systems. The results established here extend some well-known results in the literature.

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