# General standby allocation in series and parallel systems

20 Jun 2017-Communications in Statistics-theory and Methods (Taylor & Francis)-Vol. 46, Iss: 19, pp 9842-9858

TL;DR: In this paper, three different models of one or more standby components are discussed and the results related to the cold and hot standby models are obtained as particular cases of the results discussed in this article because the model considered here is a general one.

Abstract: Stochastic orders are very useful tools to compare the lifetimes of two systems. Optimum lifetime of a series (resp. parallel) system with general standby component(s) depends on the allocation strategy of standby component(s) into the system. Here, we discuss three different models of one or more standby components. In each model, we compare different series (resp. parallel) systems (which are formed through different allocation strategies of standby component(s)) with respect to the usual stochastic order and the stochastic precedence order. The results related to the cold as well as the hot standby models are obtained as particular cases of the results discussed in this article because the model considered here is a general one.

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TL;DR: This paper studies the optimal component allocation policy of a k-out-of-n system composed of n independent subsystems and gets the optimal allocation policy under the constraint of the maximum number of components allowed for each subsystem and the entire system.

Abstract: This paper studies the optimal component allocation policy of a k-out-of-n system composed of n independent subsystems. Each subsystem includes one or several allocative components arranged in parallel [series], and these allocative components are drawn from a randomly selected batch of manufactured components. We compare two allocation policies by means of the majorization order. As a consequence, we get the optimal allocation policy under the constraint of the maximum number of components allowed for each subsystem and the entire system. We also analyze the influence of the batches heterogeneity on the k-out-of-n system reliability. Finally, several illustrative examples are presented.

27 citations

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TL;DR: In this article, the authors considered an open problem of obtaining the optimal operational sequence for the 1-out-of-n system with warm standby and showed that the components should be activated in accordance with the increasing sequence of their lifetimes.

Abstract: We consider an open problem of obtaining the optimal operational sequence for the 1-out-of-n system with warm standby. Using the virtual age concept and the cumulative exposure model, we show that the components should be activated in accordance with the increasing sequence of their lifetimes. Lifetimes of the components and the system are compared with respect to the stochastic precedence order and its generalization. Only specific cases of this optimal problem were considered in the literature previously.

17 citations

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TL;DR: In this article, the authors considered an open problem of optimal operational sequence for the $1$-out-of-n$ system with warm standby, and showed that the components should be activated in accordance with the increasing sequence of their lifetimes.

Abstract: We consider an open problem of optimal operational sequence for the $1$-out-of-$n$ system with warm standby. Using the virtual age concept and the cumulative exposure model, we show that the components should be activated in accordance with the increasing sequence of their lifetimes. Lifetimes of the components and the system are compared with respect to the stochastic precedence order. Only specific cases of this optimal problem were considered in the literature previously.

10 citations

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TL;DR: In this paper, the reliability of a k-out-of-n:G system in the stress-strength setup with three different types (cold, warm or hot) of standby components was investigated and the optimal time to activate the standby components to the working state was investigated.

Abstract: This paper deals with the reliability of a k-out-of-n:G system in the stress–strength setup with three different types (cold, warm or hot) of standby components. The switching time of the standby component to the k-out-of-n:G stress–strength system has been studied and how its effect on the stress–strength reliability and costs have been assessed. By taking into account the switching time of the standby component, some expressions for the stress–strength reliability and the mean remaining strength functions are obtained. The results for exponential and Weibull distributions are given in detail and the optimal time to activate the standby components to the working state is investigated.

4 citations

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TL;DR: The allocation of one active redundancy when it differs depending on the component with which it is to be allocated is investigated, and the allocation of two active redundancies in two different ways are compared.

Abstract: An effective way of improving the reliability of a system is the allocation of active redundancy. Let $X_{1}$, $X_{2}$ be independent lifetimes of the components $C_{1}$ and $C_{2}$, respectively, which form a series system. Let denote $U_{1} = \min ( \max (X_{1},X),X_{2})$ and $U_{2} = \min (X_{1},\max (X_{2},X))$, where X is the lifetime of a redundancy (say S) independent of $X_{1}$ and $X_{2}$. That is $U_{1}(U_{2})$ denote the lifetime of a system obtained by allocating S to $C_{1}(C_{2})$ as an active redundancy. Singh and Misra (1994) considered the criterion where $C_{1}$ is preferred to $C_{2}$ for redundancy allocation if $P(U_{1} > U_{2})\geq P(U_{2} > U_{1})$. In this paper we use the same criterion of Singh and Misra (1994) and we investigate the allocation of one active redundancy when it differs depending on the component with which it is to be allocated. We find sufficient conditions for the optimization which depend on the components and redundancies probability distributions. We also compare the allocation of two active redundancies (say $S_{1}$ and $S_{2}$) in two different ways, that is $S_{1}$ with $C_{1}$ and $S_{2}$ with $C_{2}$ and viceversa. For this case the hazard rate order plays an important role. We obtain results for the allocation of more than two active redundancies to a k-out-of-n systems.

3 citations

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4,258 citations

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TL;DR: A number of new classes of life distributions arising naturally in reliability models are treated systematically and each provides a realistic probabilistic description of a physical property occurring in the reliability context, thus permitting more realistic modeling of commonly occurring reliability situations.

Abstract: : This is the first of two books on the statistical theory of reliability and life testing. The present book concentrates on probabilistic aspects of reliability theory, while the forthcoming book will focus on inferential aspects of reliability and life testing, applying the probabilistic tools developed in this volume. This book emphasizes the newer, research aspects of reliability theory. The concept of a coherent system serves as a unifying theme for much of the book. A number of new classes of life distributions arising naturally in reliability models are treated systematically: the increasing failure rate average, new better than used, decreasing mean residual life, and other classes of distributions. As the names would seem to indicate, each such class of life distributions provides a realistic probabilistic description of a physical property occurring in the reliability context. Also various types of positive dependence among random variables are considered, thus permitting more realistic modeling of commonly occurring reliability situations.

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TL;DR: Accelerated Testing: Statistical Models, Test Plans, and Data Analyses, by W. Nelson.

Abstract: Preface.1. Introduction and Background.2. Models for Life Tests with Constant Stress.3. Graphical Data Analysis.4. Complete Data and Least Squares Analyses.5. Censored Data and Maximum Likelihood Methods.6. Test Plans.7. Competing Failure Modes and Size Effect.8. Least-Squares Comparisons for Complete Data.9. Maximum Likelihood Comparisons for Censored and Other Data.10. Models and Data Analyses for Step and Varying Stress.11. Accelerated Degradation.Appendix A. Statistical Tables.References.Index.

1,816 citations

### "General standby allocation in serie..." refers methods in this paper

...Cha et al. (2008), to the best of our knowledge, are the first to develop a new technique to handle the general standby based on the concept of accelerated life model (see, Nelson, 1990) and virtual age model (see, Kijima, 1989, Finkelstein (1999, 2007, 2008), and Finkelstein and Cha (2013)....

[...]

...Then, for all t ≥ 0, FY∗ (t ) = FY (ω(t )), or equivalently, γ (t ) = ω(t ) (cf. Nelson, 1990; Yun and Cha, 2010)....

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TL;DR: In this article, the authors developed general repair models for a repairable system by using the idea of the virtual age process of the system and derived an upper bound for E[Sn] when a general repair is used.

Abstract: In this paper, we develop general repair models for a repairable system by using the idea of the virtual age process of the system. If the system has the virtual age V, _ = y immediately after the (n - I)th repair, the nth failure-time X, is assumed to have the survival function F(x + y)IF(y) where F(x) is the survival function of the failure-time of a new system. A general repair is represented as a sequence of random variables An taking a value between 0 and 1, where An denotes the degree of the nth repair. For the extremal values 0 and 1, An = 1 means a minimal repair and An = 0 a perfect repair. Two models are constructed depending on how the repair affects the virtual age process: Vn= V,_ + A,Xn as Model 1 and Vn = An(V,_n + Xn) as Model II. Various monotonicity properties of the process Sn = 2k_ I Xk with respect to stochastic orderings of general repairs are obtained. Using a result, an upper bound for E[Sn] when a general repair is used is derived.

786 citations

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23 Feb 2009

TL;DR: In this paper, the Failure Rate and Mean Remaining Lifetime of Software Demographic and Biological Applications (FRE) were constructed for software failure rate in Demographic, Biological and Artificial Intelligence applications.

Abstract: Introduction Failure Rate and Mean Remaining Lifetime More on Exponential Representation Point Processes and Minimal Repair Virtual Age and Imperfect Repair Mixture Failure Rate Modelling Limiting Behaviour of Mixture Failure Rates 'Constructing' the Failure Rate Failure Rate of Software Demographic and Biological Applications

284 citations