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.     

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.     

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.     

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.     

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.     

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.     

Optimization problems for consecutive-2-out-of-n:G system

Abstract

This paper considers the optimization problems for a consecutive-2-out-of-n:G system where n is considered to be fixed or random. When the number of components is constant, the optimal number of components and the optimal replacement time are discussed by minimizing the expected cost rates. Furthermore, we focus on the above discussions again when n is a random variable. We give an approximate value of MTTF and propose the preventive replacement policy, respectively.     

Reliability and Some Characteristics of Circular Consecutive-k-Within-m-Out-of-n:F System

A system can be classified with respect to the physical arrangement of its components and the functioning principle. A circular consecutive k-within-m-out-of-n:F system consists of n circularly ordered components and fails if and only if there are m consecutive components that include among them at least k failed components. A circular consecutive k-within-m-out-of-n:F system turns into circular consecutive k-out-of-n:F for m = k and k-out-of-n:F system for m = n. In this study, signature-based analysis of circular consecutive k-within-m-out-of-n:F system is performed. A new approximation to this system is provided based on maximum number of failed components and an illustrative example is given for different values of n, m, k to compare the approximate results with simulated and exact results.     

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.     

Ordering New Conditional Residual Lifetimes of k-out-of-n Systems

This article introduces a new residual life of (n ? k + 1)-out-of-n systems consisting of n independent and identically distributed components, given that the total number of failures of components is not less than l at time t, and the system is still working. Some stochastic comparisons of residual lifetimes are conducted, and behaviors of DLR and ILR are given.     

Extremum sieve estimation in k-out-of-n systems

This article considers the non parametric estimation of absolutely continuous distribution functions of independent lifetimes of non identical components in k-out-of-n systems, 2 ? k ? n, from the observed “autopsy” data. In economics, ascending “button” or “clock” auctions with n heterogeneous bidders with independent private values present 2-out-of-n systems. Classical competing risks models are examples of n-out-of-n systems. Under weak conditions on the underlying distributions, the estimation problem is shown to be well-posed and the suggested extremum sieve estimator is proven to be consistent. This article considers the sieve spaces of Bernstein polynomials which allow to easily implement constraints on the monotonicity of estimated distribution functions.     

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.     

An Integral Bhattacharyya Type Bound for the Bayes Risk

In this article, some results are derived on stochastic comparisons of the residual and past lifetimes of an (n ? k + 1)-out-of-n system with dependent components. These findings generalize some recent results obtained on systems with independent components and provide some interesting results for a system with dependent components following an Archimedean copula. An illustrative example is provided.     

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.     

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.     

Ordering k-out-of-n systems with interdependent components and one active redundancy

This paper studies the optimal allocation of one active redundancy to a k-out-of-n system in the context of statistically dependent component lifetimes. It is proved that assigning the redundancy to the weakest component is more favorable so as to get a larger system lifetime in terms of both the usual stochastic order and the stochastic preference order.     

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.     

A note on the mean residual life of a coherent system with a cold standby component

In this paper, we investigate the effect of a cold standby component on the mean residual life (MRL) of a system. When the system fails, a cold standby component is immediately put in operation. We particularly focus on the coherent systems in which, after putting the standby component into operation, the failure of the system is due to the next component failure. For these systems, we define MRL functions and obtain their explicit expressions. Also some stochastic ordering results are provided. Such systems include k-out-of-n systems. Hence, our results extend some results in literature.     

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.     

Some Properties of Order Statistics When the Sample Size Is Random

ABSTRACT

The study of r-out-of-n systems is of utmost importance in reliability theory. In this note, we study closure of different partial orders under the formation of r-out-of-N and (N ? s)-out-of-N systems when the number of components N, forming the system, is a random variable having support {k, k + 1,…}, where k is a fixed positive integer, r ∈ {1,…, k} and s ∈ {0, 1,…, k ? 1}. This generalizes quite a few results already known in the literature. We also study the closure of different partial orders when two systems are formed out of different random number of components.