Mean Time to Failure of Weighted k-out-of-n: G Systems

The purpose of this article is to develop a Monte-Carlo simulation algorithm for computing mean time to failure (MTTF) of weighted-k-out-of-n:G and linear consecutive-weighted-k-out-of-n:G systems. Our algorithm is based on the use of appropriately defined stochastic process which represents the total weight of the system at time t. These stochastic processes are explicitly defined and used along with the ordered component lifetimes to simulate MTTF of the systems with weighted components.     

Reliability Research of k-out-of-n: G Supply Chain System Based on Copula

Supply chain management has received considerable attention in the literature and it is meaningful and important to be able to measure the reliability of supply chains. In the article, the suppliers in the supply chain systems are not independent of each other and the dependency relation may be either linear or nonlinear correlation. From the view of the distribution service process, a copula-based method is proposed for analyzing the reliability of supply chains. In this article, by introducing the model of k-out-of-n: G system into the studies of supply chains, an evaluation method is suggested and the reliability indexes are obtained. Finally, a numerical example is presented to illustrate the results obtained in this article.     

Joint Integrated Importance Measure for Multi-State Transition Systems

Joint reliability importance (JRI) evaluates the interaction of two components in contributing to the system reliability in a system. Traditional JRI measures mainly concern the change of the system reliability caused by the interactive change of the reliabilities of the two components and seldom consider the probability distributions, transition rates of the object component states, and system performance. This article extends the JRI concept of two components from multi-state systems to multi-state transition systems and mainly focuses on the joint integrated importance measure (JIIM) which considers the transition rates of component states. Firstly, the concept and physical meaning of JIIM in binary systems are described. Secondly, the JIIM for deterioration process (JIIMDP) and the JIIM for maintenance process (JIIMMP) in multi-state systems are studied respectively. The corresponding characteristics of JIIMDP and JIIMMP for series and parallel systems are also analyzed. Finally, an application to an offshore electrical power generation system is given to demonstrate the proposed JIIM.     

Reliability Evaluation for a Multi-State System Under Stress-Strength Setup

The two most commonly used reliability models in engineering applications are binary k-out-of-n:G and consecutive k-out-of-n:G systems. Multi-state k-out-of-n:G and multi-state consecutive k-out-of-n:G systems have been proposed as an extension of these systems and they have been found to be more flexible tool for modeling engineering systems. In this article, multi-state systems, in particular, multi-state k-out-of-n:G and multi-state consecutive k-out-of-n:G, are considered in a stress-strength setup. The states of the system are classified considering the number of components whose strengths above (below) the multiple stresses available in an environment. The exact state probabilities are provided and the results are illustrated for various stress-strength distributions. Maximum likelihood estimators of state probabilities are also presented.     

Reliability of weighted k-out-of-n: G systems consisting of two types of components and a cold standby component

In this article, the influence of a cold standby component to the reliability of weighted k-out-of-n: G systems consisting of two different types of components is studied. Weighted k-out-of-n: G systems are generalization of k-out-of-n systems that has attracted substantial interest in reliability theory because of their various applications in engineering. A method based on residual lifetimes of mixed components is presented for computing reliability of weighted k-out-of-n: G systems with two types of components and a cold standby component. Reliability and mean time to failure of different structured systems have been computed. Moreover, obtained results are used for defining optimal system configurations that can minimize the overall system costs.     

On the mean residual life of a generalized k-out-of-n system

A generalized k-out-of-n system consists of N modules in which the i th module is composed of n_i components in parallel. The system failswhen at least f components in the whole system or at least k consecutive modules have failed. In this article, we obtain the mean residual life function of such a generalized k-out-of-n system under different conditions, namely, when the number of components in each module is equal or unequal and when the components of the system are independent or exchangeable.     

On the Failure Probability of Used Coherent Systems

In analyzing the lifetime properties of a coherent system, the concept of “signature” is a useful tool. Let T be the lifetime of a coherent system having n iid components. The signature of the system is a probability vector s=(s₁, s₂, …, s_n), such that s_i=P(T=X_i:n), where, X_i:n, i=1, 2, …, n denote the ordered lifetimes of the components. In this note, we assume that the system is working at time t>0. We consider the conditional signature of the system as a vector in which the ith element is defined as p_i(t)=P(T=X_i:n|T>t) and investigate its properties as a function of time.     

On the Percentile Residual Lifetime of Parallel Systems

In this paper, we consider a parallel system consisting of n components. Then, the percentile residual lifetime of the system given survival of at least n ? r + 1, r = 1, 2, …, n component(s) has been introduced, and some properties of this measure have been investigated. We show that the system accommodates decreasing percentile residual lifetime function, provided the components have increasing hazard rate functions. Different parallel systems have been compared with each other in terms of the introduced measure. Furthermore, behavior of the percentile residual lifetime of the system and the components have been compared in terms of some reliability notions. Also, a characterization result has been presented.     

Reliability Analysis of Unrepairable Systems with k-out-of-m:G Subsystems Subject to Suspended Animation

Two unrepairable series structure systems with k-out-of-m:G subsystems and spares are investigated. The first one consists of a k-out-of-m:G subsystem and a series subsystem while the other consists of two k-out-of-m:G subsystems. The systems have identical components with identical lifetime distributions and the working components are suspended as soon as the systems are down. Two Markov models are proposed for the reliability analysis of such systems and closed form results on the reliability and the mean time to failure (MTTF) are presented. Numerical examples are given to illustrate the impact of several parameters on the reliability of the systems.     

On the Mean Remaining Strength of the k-Out-of-n:F System with Exchangeable Components

In this article, we consider the mean remaining strength of a k-out-of-n:F system in the stress–strength setup for the exchangeable components. We provide some results for parallel and series systems under this setup, where X₁, X₂, …, X_n are the strengths of the components designed under the common stress. An illustrative example is given for the k-out-of- n:F system using the multivariate FGM distribution.     

On moments of the supremum of normed weighted averages

Let {X, X_n; n ≥ 1} be a sequence of real-valued iid random variables, 0 < r < 2 and p > 0. Let D = { A = (a_nk; 1 ≤ k ≤ n, n ≥ 1); a_nk, ? R and sup_{n, k} |a_n,k| < ∞}. Set S_n( A ) = ∑ⁿ_k=1a_{n, k}X_k for A ? D and n ≥ 1. This paper is devoted to determining conditions whereby E{sup_{n ≥ 1}, |S_n( A )|/n^1/r}^p < ∞ or E{sup_{n ≥ 2} |S_n( A )|/2n log n)^1/2}^p < ∞ for every A ? D. This generalizes some earlier results, including those of Burkholder (1962), Choi and Sung (1987), Davis (1971), Gut (1979), Klass (1974), Siegmund (1969) and Teicher (1971).     

On nonparametric Bayesian inference for the distribution of a random sample

The nonparametric Bayesian approach for inference regarding the unknown distribution of a random sample customarily assumes that this distribution is random and arises through Dirichlet-process mixing. Previous work within this setting has focused on the mean of the posterior distribution of this random distribution, which is the predictive distribution of a future observation given the sample. Our interest here is in learning about other features of this posterior distribution as well as about posteriors associated with functionals of the distribution of the data. We indicate how to do this in the case of linear functionals. An illustration, with a sample from a Gamma distribution, utilizes Dirichlet-process mixtures of normals to recover this distribution and its features.     

On the comparison of the pre-test and shrinkage estimators for the univariate normal mean

The estimation of the mean of an univariate normal population with unknown variance is considered when uncertain non-sample prior information is available. Alternative estimators are defined to incorporate both the sample as well as the non-sample information in the estimation process. Some of the important statistical properties of the restricted, preliminary test, and shrinkage estimators are investigated. The performances of the estimators are compared based on the criteria of unbiasedness and mean square error in order to search for a ‘best’ estimator. Both analytical and graphical methods are explored. There is no superior estimator that uniformly dominates the others. However, if the non-sample information regarding the value of the mean is close to its true value, the shrinkage estimator over performs the rest of the estimators. Received: June 19, 1999; revised version: March 23, 2000     

Testing for lack of dependence in the functional linear model

The authors consider the linear model Y_n = ψX_n + ?_n relating a functional response with explanatory variables. They propose a simple test of the nullity of ψ based on the principal component decomposition. The limiting distribution of their test statistic is chi‐squared, but this distribution is also an excellent approximation in finite samples. The authors illustrate their method using data from terrestrial magnetic observatories.