Dynamic Reliability Evaluation of Consecutive-k-Within-m-Out-of-n:F System

A consecutive k-within-m-out-of-n:F system consists of n linearly ordered components and fails if and only if there are m consecutive components which include among them at least k failed components. This system model generalizes both consecutive k-out-of-n:F and k-out-of-n:F systems. In this article, we study the dynamic reliability properties of consecutive k-within-m-out-of-n:F system consisting of exchangeable dependent components. We also obtain some stochastic ordering results and use them to get simple approximation formulae for the survival function and mean time to failure of this system.     

A Sensitivity Analysis for System Reliability

In terms of the relationship between the lower confidence limits of reliabilities for system and component, using the finite Markov chain imbedding approach, we analyze the sensitivity of reliability for a linear consecutive-k-out-of-n:F system, a circular consecutive-k-out-of-n:F system, and a k-within-consecutive-r-out-of-n:F system. The analysis is based on two parameters: the number of experiments and the number of failed components.     

Failure Data Analysis with Extended Weibull Distribution

A consecutive k-out-of-n: G system consists of n linearly ordered components functions if and only if at least k consecutive components function. In this article we investigate the consecutive k-out-of-n: G system in a setup of multicomponent stress-strength model. Under this setup, a system consists of n components functions if and only if there are at least k consecutive components survive a common random stress. We consider reliability and its estimation of such a system whenever there is a change and no change in strength. We provide minimum variance unbiased estimation of system reliability when the stress and strength distributions are exponential with unknown scale parameters. A nonparametric minimum variance unbiased estimator is also provided.     

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.     

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.     

A Note on the Regression of Order Statistics

This paper examines the relationships between the mean residual life functions of parallel and k-out-of-n systems with the regression of order statistics. Using these relationships, the results and properties about the mean residual life function of those systems can be used for the regression of order statistics and vice versa. Finally, the paper proposes a definition for the mean residual life function of a k-out-of-n system when the number of failed components of the system is known.     

On residual lifetimes in sequential (n ? k?+?1)-out-of-n systems

Sequential order statistics is an extension of ordinary order statistics. They model the successive failure times in sequential k-out-of-n systems, where the failures of components possibly affect the residual lifetimes of the remaining ones. In this paper, we consider the residual lifetime of the components after the kth failure in the sequential (n − k + 1)-out-of-n system. We extend some results on the joint distribution of the residual lifetimes of the remaining components in an ordinary (n − k + 1)-out-of-n system presented in Bairamov and Arnold (Stat Probab Lett 78(8):945–952, 2008) to the case of the sequential (n − k + 1)-out-of-n system.     

The mean general residual lifetime of (n − k + 1)-out-of-n systems with exchangeable components

The study of the reliability properties of (n ? k + 1)-out-of-n systems has gained a great deal of attention, from both theoretical and practical perspectives. In this article, we consider (n ? k + 1)-out-of-n systems with exchangeable components and study the stochastic properties of two forms of residual lifetimes of such systems under the following conditions: n ? r + 1 (r ? k) components of the system are operating at time t > 0, and/or the rth (r < k) component has failed, but the system is working at time t. In addition, some results relating to the functions of the mean general residual lifetimes (MGRL) are derived for these systems. Finally, in accordance with the generalized Farlie–Gumbel–Morgenstern model, we present the reliability properties of the general residual lifetime of (n ? k + 1)-out-of-n systems and investigate the asymptotic behavior of the proposed MGRL functions with exponential marginals.     

Reliability analysis of consecutive k-out-of-n systems with non-identical components lifetimes

In recent years, the study of reliability properties of consecutive k-out-of-n systems has attracted a great deal of attention from both theoretical and practical perspectives. In this paper we consider linear and circular consecutive k-out-of-n systems. It is assumed that lifetimes of components of the systems are independent but their probability distributions are non-identical. We study the reliability properties of the residual lifetimes of such systems under the condition that at least (n−r+1), r≤n, components of the system are operating. We also investigate the probability that a specific number of components of the above-mentioned system operate at time t, t>0, under the condition that the system is alive at time t.     

Circular consecutive k-out-of-n systems with exchangeable dependent components

In this paper we study circular consecutive k-out-of-n systems consisting of exchangeable components. We derive explicit expressions for both unconditional and conditional survival functions for 2k+1≥n, while signature based mixture representations for general k are obtained. The applications and computational results concerned with mean residual life function and stochastic ordering are presented.     

On capacity evaluation for multi-state weighted k-out-of-n system

ABSTRACT

Weighted k-out-of-n system has been widely used in various engineering areas. Performance of such system is characterized by the total capacity of the components. Therefore, capacity evaluation is of great importance for research on the behavior of the system over time. Capacity evaluation for binary weighted k-out-of-n system has been reported in the literature. In this paper, to shorten computational time, we first develop a multiplication method for capacity evaluation of binary weighted k-out-of-n system. We then generalize capacity evaluation to multi-state weighted k-out-of-n system. Recursive algorithm and multiplication algorithm are developed for capacity evaluation for such system. Comparison is made of the two methods in different aspects. An illustrative example of an oil transmission system is presented to demonstrate the implementation of the proposed methods.     

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.     

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.     

Nonparametric model checking for k-out-of-n systems

It is an important problem in reliability analysis to decide whether for a given k-out-of-n system the static or the sequential k-out-of-n model is appropriate. Often components are redundantly added to a system to protect against failure of the system. If the failure of any component of the system induces a higher rate of failure of the remaining components due to increased load, the sequential k-out-of-n model is appropriate. The increase of the failure rate of the remaining components after a failure of some component implies that the effects of the component redundancy are diminished. On the other hand, if all the components have the same failure distribution and whenever a failure occurs, the remaining components are not affected, the static k-out-of-n model is adequate. In this paper, we consider nonparametric hypothesis tests to make a decision between these two models. We analyze test statistics based on the profile score process as well as test statistics based on a multivariate intensity ratio and derive their asymptotic distribution. Finally, we compare the different test statistics.     

Allocating Redundancies to k-out-of-n Systems with Independent and Heterogeneous Components

This paper studies the allocation of independent redundancies with a common life distribution to k-out-of-n systems of independent components with non identical life distributions. A sufficient condition is found for allocating more active redundancies to the weaker component to gain a larger lifetime for k-out-of-n systems, and assigning more standby redundancies to the weaker (stronger) components is proved to yield larger lifetime for series (parallel) systems in the sense of the increasing concave (convex) order. Also, the optimal policy is proved to be majorized by all other policies when the system’s components are stochastically ordered.     

Local asymptotic normality of intermediate and central order statistics

Consider n independent random variables Z_i,…, Z_n on R with common distribution function F, whose upper tail belongs to a parametric family F(t) = F_θ(t),t ≥ x₀, where θ ∈ ? ? R ^d. A necessary and sufficient condition for the family F_θ, θ ∈ ?, is established such that the k-th largest order statistic Z_n?k+1:n alone constitutes the central sequence yielding local asymptotic normality ( LAN ) of the loglikelihood ratio of the vector (Z_n?i+1:n)¹ _i=kof the k largest order statistics. This is achieved for k = k(n)→_n→∞∞ with k/n→_n→∞ 0.

In the case of vectors of central order statistics ( Z_r:n, Z_r+1:n,…, Z_s:n ), with r/n and s/n both converging to q ∈ ( 0,1 ), it turns out that under fairly general conditions any order statistic Z_m:n with r ≤ m ≤s builds the central sequence in a pertaining LAN expansion.These results lead to asymptotically optimal tests and estimators of the underlying parameter, which depend on single order statistics only     

Number of Working Components in Consecutive k-out-of-n System While It Is Working

In this article we study the distribution and expected value of the number of working components at time t in a consecutive k-out-of-n system under the condition that it is working at time t. We provide the exact distribution of the corresponding conditional random variable and compute its expected value for the system consisting of exchangeable dependent components. The results are also extended to any coherent system by the aid of system signature. Finally, we present illustrative and computational results for some systems having Lomax components.