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.     

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.     

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 Use of the Importance Measure for Multi-State Repairable k-out-of-n: G Systems

Importance measures in reliability systems are used to identify weak components in contributing to proper functioning of the system. Traditional importance measures mainly concern the change of the system reliability as the change of the reliability of one component and seldom consider the expected number of repairs of the objective component in unit time. This paper proposes an improvement potential rate importance (IPR) to verify the effectiveness of the improvement in system reliability for multi-state repairable k-out-of-n: G systems. Then the comparisons between IPR and Birnbaum importance are discussed. Finally, a case study is given to demonstrate the proposed IPR.     

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.     

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.     

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.     

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.     

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.     

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.     

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.     

Profust reliability of linear and circular type F and G systems having two failure criteria under Markov dependency

In this study, the profust reliabilities of (n, f, k): F(G) and < n, f, k > : F(G) systems for Markov dependent components are investigated. Having two failure criteria are the common features of these four systems. The usage of both fuzzy approach and two failure criteria in the same system provides us more realistic approach to evaluate the reliability of more complex systems. The component configurations are examined for both linear and circular sequences and the working principle of systems are studied for both F and G systems. Under all these assumptions, the profust reliabilities of (n, f, k): F(G) and < n, f, k > : F(G) systems are obtained using the distribution of run statistics. Also a new membership function different from the linear membership function which is generally used in the literature is proposed. Some numerical results which allow the comparison of the systems from various perspectives and various figures for both linear and circular type systems are presented. Some special cases (between Markov – iid assumption, conventional – profust reliability) are also considered.     

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.     

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.     

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.     

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.     

Optimal number of components in a load-sharing system for maximizing reliability

A $k$-out-of-$n$:G load sharing system is a cluster of $n$ components designed to withstand a certain amount of load in field operation, working only if no fewer than $k$ components work. Previous research on a load sharing system has focused on predicting the time-independent reliability from the stress–strength model or estimating the unknown parameters of the time-dependent reliability for a given load sharing rule. Differently, in this paper, we consider the problem of determining the optimal $n$ to maximize the reliability of both $n$-out-of-$n$:G and ($n$–$1$)-out-of-$n$:G load sharing systems. Since the load of each component decreases in $n$, the proportional hazard model is employed to relate the component failure rate with the load, assuming that the components, which have exponential distributions for given loads, are independent of each other. We then derive a sufficient condition under which a smaller number of components each withstanding a high load is preferred to a larger number of components each withstanding a small load. A numerical example is given for the rocket propulsion system to illustrate the result.