A Reliability Bound for Systems of Maintained, Interdependent Components
01 Mar 1970-Journal of the American Statistical Association (Taylor & Francis Group)-Vol. 65, Iss: 329, pp 329-338
TL;DR: In this article, the minimal cut lower bound on the reliability of a coherent system, derived in Esary-Proschan [6] for the case of independent components not subject to maintenance, is shown to hold under a variety of component maintenance policies and in several typical cases of component dependence.
Abstract: In this article the minimal cut lower bound on the reliability of a coherent system, derived in Esary-Proschan [6] for the case of independent components not subject to maintenance, is shown to hold under a variety of component maintenance policies and in several typical cases of component dependence. As an example, the lower bound is obtained for the reliability of a “two out of three” system in which each component has an exponential life length and an exponential repair time. The lower bound is compared numerically with the exact system reliability; for realistic combinations of failure rate, repair rate, and mission time, the discrepancy is quite small.
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TL;DR: In this article, a broad class of latent variable models, namely the monotone unidimensional models, are studied, in which the latent variable is a scalar, the observable variables are conditionally independent given the Latent Variable, and the conditional distribution of the observables given the LSTM is stochastically increasing in the latent Variable.
Abstract: Latent variable models represent the joint distribution of observable variables in terms of a simple structure involving unobserved or latent variables, usually assuming the conditional independence of the observable variables given the latent variables. These models play an important role in educational measurement and psychometrics, in sociology and in population genetics, and are implicit in some work on systems reliability. We study a broad class of latent variable models, namely the monotone unidimensional models, in which the latent variable is a scalar, the observable variables are conditionally independent given the latent variable and the conditional distribution of the observables given the latent variable is stochastically increasing in the latent variable. All models in this class imply a new strong form of positive dependence among the observable variables, namely conditional (positive) association. This positive dependence condition may be used to test whether any model in this class can provide an adequate fit to observed data. Various applications, generalizations and a numerical example are discussed.
321 citations
TL;DR: In this article, a simple method is given for calculating reliability characteristics of repairable and nonrepairable systems, and the importance of the individual system components, assuming independent component failures and constant failure and repair rates for the components.
Abstract: A simple method is given for calculating a) reliability characteristics of repairable and nonrepairable systems, and b) the importance of the individual system components. Assumptions made include independent component failures and constant failure and repair rates for the components. The methods can easily be implemented in a computer program that would be inexpensive to execute and would always overpredict (usually very slightly) the system failure characteristics.
321 citations
TL;DR: The importance measure is a useful guide during the system development phase as to which components should receive more urgent attention in achieving system reliability growth.
281 citations
TL;DR: This paper presents inclusion-exclusion bounds and compares them with disjoint subset bounds and is based on a generalization of Abraham's recursive disjointed products.
Abstract: The reliability literature has recently introduced several multistate models. This paper discusses reliability bounds in the most general of these models. It presents inclusion-exclusion bounds and compares them with disjoint subset bounds. The later bounds are based on a generalization of Abraham's recursive disjoint products.
211 citations
TL;DR: A stochastic matrix is defined to be monotone if its row-vectors are stochastically increasing as discussed by the authors, i.e., the row vectors of the matrix are uniformly increasing.
199 citations
References
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TL;DR: In this paper, the authors present some meaningful derivations of a multivariate exponential distribution that serve to indicate conditions under which the distribution is appropriate, such as the residual life is independent of age.
Abstract: A number of multivariate exponential distributions are known, but they have not been obtained by methods that shed light on their applicability. This paper presents some meaningful derivations of a multivariate exponential distribution that serves to indicate conditions under which the distribution is appropriate. Two of these derivations are based on “shock models,” and one is based on the requirement that residual life is independent of age. It is significant that the derivations all lead to the same distribution. For this distribution, the moment generating function is obtained, comparison is made with the case of independence, the distribution of the minimum is discussed, and various other properties are investigated. A multivariate Weibull distribution is obtained through a change of variables.
1,481 citations
TL;DR: In this paper, it was shown that a random variable can be associated with another random variable if the test functions are either (a) binary or (b) bounded and continuous.
Abstract: It is customary to consider that two random variables $S$ and $T$ are associated if $\operatorname{Cov}\lbrack S, T\rbrack = EST - ES\cdot ET$ is nonnegative. If $\operatorname{Cov}\lbrack f(S), g(T)\rbrack \geqq 0$ for all pairs of nondecreasing functions $f, g$, then $S$ and $T$ may be considered more strongly associated. Finally, if $\operatorname{Cov}\lbrack f(S, T), g(S, T)\rbrack \geqq 0$ for all pairs of functions $f, g$ which are nondecreasing in each argument, then $S$ and $T$ may be considered still more strongly associated. The strongest of these three criteria has a natural multivariate generalization which serves as a useful definition of association: DEFINITION 1.1. We say random variables $T_1,\cdots, T_n$ are associated if \begin{equation*}\tag{1.1}\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack \geqq 0\end{equation*} for all nondecreasing functions $f$ and $g$ for which $Ef(\mathbf{T}), Eg(\mathbf{T}), Ef(\mathbf{T})g(\mathbf{T})$ exist. (Throughout, we use $\mathbf{T}$ for $(T_1,\cdots, T_n)$; also, without further explicit mention we consider only test functions $f, g$ for which $\operatorname{Cov}\lbrack f(\mathbf{T}), g(\mathbf{T})\rbrack$ exists.) In Section 2 we develop the fundamental properties of association: Association of random variables is preserved under (a) taking subsets, (b) forming unions of independent sets, (c) forming sets of nondecreasing functions, (d) taking limits in distribution. In Section 3 we develop some simpler criteria for association. We show that to establish association it suffices to take in (1.1) nondecreasing test functions $f$ and $g$ which are either (a) binary or (b) bounded and continuous. In Section 4 we develop the special properties of association that hold in the case of binary random variables, i.e., random variables that take only the values 0 or 1. These properties turn out to be quite useful in applications. We also discuss association in the bivariate case. We relate our concept of association in this case to several discussed by Lehmann (1966). Finally, in Section 5 applications in probability and statistics are presented yielding results by Robbins (1954), Marshall-Olkin (1966), and Kimball (1951). An application in reliability which motivated our original interest in association will be presented in a forthcoming paper.
1,246 citations
"A Reliability Bound for Systems of ..." refers background in this paper
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...The notion of association is discussed in [7]....
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TL;DR: The present study deals with general classes of systems which contain two-terminal networks and most other kinds of systems considered previously as special cases, and investigates their combinatorial properties and their reliability.
Abstract: A number of recent publications have dealt with problems of analyzing the performance of multi-component systems and evaluating their reliability. For example, a comprehensive theory of two-terminal networks was presented in [I] by Moore and Shannon who, among other results, have developed methods for obtaining highly reliable systems using components of low reliability; some of their procedures are credited to earlier work by von Neumann [2]. Several of the concepts and results of the present paper are generalizations of the corresponding concepts and results of the Moore-Shannon paper. A discussion of complex systems interpreted as Boolean functions may be found in the paper [3] by Mine. The present study deals with general classes of systems which contain two-terminal networks and most other kinds of systems considered previously as special cases, and investigates their combinatorial properties and their reliability. These classes consist, with several variants, of systems such that the more components...
297 citations
TL;DR: The Moore-Shannon inequality is extended to the case of unequal component reliabilities, permitting a simple demonstration of the S-shapedness properties of system reliability functions.
Abstract: Some general aspects of the reliability of coherent systems whose components are independent, but not necessarily of the same reliability are explored. Upper and lower bounds, which can be computed directly from the minimal paths and minimal cuts of a system, are found for system reliability. The Moore-Shannon inequality is extended to the case of unequal component reliabilities, permitting a simple demonstration of the S-shapedness properties of system reliability functions.
143 citations
"A Reliability Bound for Systems of ..." refers background in this paper
...In this article the minimal cut lower bound on the reliability of a coherent system, derived in [6] for the case of independent components not subject to maintenance, is shown to hold under a variety of component maintenance policies and in several typical cases of component dependence....
[...]