Attainability of the Upper Bounds for the Mean Past Lifetime of Parallel System Components

Among reliability systems, one of the basic systems is a parallel system. In this article, we consider a parallel system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that the system has failed by time t, with t being 100pth percentile of F(t = F ^?1(p), 0 < p < 1), we characterize the probability distributions based on the mean past lifetime of the components of the system. These distributions are described in the form of a specific shape on the left of t and arbitrary continuous function on the right tail.     

Some aspects of the mean past lifetime of a parallel system under double regularly checking

In this paper, a parallel system consisting of a finite number of identical components with independent lifetimes having a common distribution function is considered, when the failure time of the system is restricted to a finite interval (double regularly checking). Under these conditions, the mean past lifetime (MPL) of the system is presented and some of its properties are derived. It is shown that the underlying distribution function can be recovered from the proposed MPL. Then, a consistent estimator for MPL is presented and some of its properties are studied. This estimator also could be used for the single monitoring case or ordinary MPL. Finally, some properties of the MPL of a parallel system with nonidentical components are discussed.     

MRL and MPL Functions of Parallel Systems with INID Components under Double Monitoring

This article investigates parallel systems with n independent but not identically distributed (INID) components. Under the condition that, at time t ₁ (t ₁ > 0) the total number of failures of the components is r (r < n), and at time t ₂ (t ₂ ≥ t ₁) the sys-tem is still working or it has failed, the mean residual life (MRL) function of the parallel system and the mean past lifetime (MPL) function of the components are conducted. Some representations and corresponding properties of the MRL function and the MPL function under several specific conditions are obtained.     

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.     

Some representation theorems in terms of mean residual lifetime and mean inactivity time of coherent systems

Abstract

In this paper, we consider a k-out-of-n system consisting of n identical components with independent lifetimes. We show that when the underlying distribution function F(t) is absolutely continuous, then it can be univocally determined by some particular mean residual lives or mean inactivity times of the system. It is then shown that these results may be extended to coherent (or mixed) systems.     

Reliability modeling of systems with two dependent degrading components based on gamma processes

Abstract

Many engineering systems have multiple components with more than one degradation measure which is dependent on each other due to their complex failure mechanisms, which results in some insurmountable difficulties for reliability work in engineering. To overcome these difficulties, the system reliability prediction approaches based on performance degradation theory develop rapidly in recent years, and show their superiority over the traditional approaches in many applications. This paper proposes reliability models of systems with two dependent degrading components. It is assumed that the degradation paths of the components are governed by gamma processes. For a parallel system, its failure probability function can be approximated by the bivariate Birnbaum–Saunders distribution. According to the relationship of parallel and series systems, it is easy to find that the failure probability function of a series system can be expressed by the bivariate Birnbaum–Saunders distribution and its marginal distributions. The model in such a situation is very complicated and analytically intractable, and becomes cumbersome from a computational viewpoint. For this reason, the Bayesian Markov chain Monte Carlo method is developed for this problem that allows the maximum likelihood estimates of the parameters to be determined in an efficient manner. After that, the confidence intervals of the failure probability of systems are given. For an illustration of the proposed model, a numerical example about railway track is presented.     

Edgeworth expansions for nonparametric distribution estimation with applications

In this paper, we will investigate the nonparametric estimation of the distribution function F of an absolutely continuous random variable. Two methods are analyzed: the first one based on the empirical distribution function, expressed in terms of i.i.d. lattice random variables and, secondly, the kernel method, which involves nonlattice random vectors dependent on the sample size n; this latter procedure produces a smooth distribution estimator that will be explicitly corrected to reduce the effect of bias or variance. For both methods, the non-Studentized and Studentized statistics are considered as well as their bootstrap counterparts and asymptotic expansions are constructed to approximate their distribution functions via the Edgeworth expansion techniques. On this basis, we will obtain confidence intervals for F(x) and state the coverage error order achieved in each case.     

Some properties of hazard rate functions of systems with two components

In this paper we compare the hazard rate functions of two parallel systems, each of which consists of two independent components with exponential distribution functions. The paper gives various conditions under which there exists a hazard rate ordering between the two parallel systems. It is also shown that some of these conditions are both sufficient and necessary. In particular, it is proven that if the vector consisting of the two hazard rates of the two exponential components in one parallel system weakly supmajorizes the counterpart of the other parallel system, then the first parallel system is greater than the second parallel system in the hazard rate ordering. This paper further compares the hazard rate functions of two parallel systems when both systems have components following a certain bivariate exponential distribution.     

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.     

E-Bayesian estimation for system reliability and availability analysis based on exponential distribution

The main purpose of this article is to introduce the E-Bayesian approach to gain flexibility in the reliability-availability system estimation. This approach will be used in series systems, parallel systems, and k-out-of-m systems, based on exponential distribution under squared error loss function, when time is continuous. We use three prior distributions to investigate its impact on the E-Bayesian approach, those prior distributions cover a big spectrum of possibilities. We show in real examples and also by simulations, how the procedure behaves. In the simulation study also we explore the impact on this estimation approach, when the number of components of the system increases.     

Approximate tests in unbalanced two-way random models without interaction

In the presence of non-normality, we consider testing for the significance of the variance components in the unbalanced two-way random model without interaction. The approximate test is based on the F-statistic for this model. The asymptotic distribution of the F-statistic is derived as the number of treatments tends to infinity while the number of observations for a treatment in any block takes value from a finite set of positive integers. Robustness of the approximate test is given.     

A note on the mean residual life function of a coherent system with exchangeable or nonidentical components

This article investigates some properties of the mean residual life function of (n−k+1)-out-of-n systems, when the lifetimes of the system components are independent random variables but not necessarily identically distributed and when the joint distribution of the component lifetimes is exchangeable, extending the results of Asadi and Goliforushani (2008) [On the mean residual life function of coherent systems. IEEE Transactions on Reliability 57 (4) 574-580] for the case of independent and identically distributed components. The extension to a coherent system with exchangeable components is also given.     

On confidence intervals for the mean past lifetime function under random censorship

The mean past lifetime (MPL) function (also known as the expected inactivity time function) is of interest in many fields such as reliability theory and survival analysis, actuarial studies and forensic science. For estimation of the MPL function some procedures have been proposed in the literature. In this paper, we give a central limit theorem result for the estimator of MPL function based on a right-censored random sample from an unknown distribution. The limiting distribution is used to construct normal approximation-based confidence interval for MPL. Furthermore, we use the empirical likelihood ratio procedure to obtain confidence interval for the MPL function. These two intervals are compared with each other through simulation study in terms of coverage probability. Finally, a couple of numerical example illustrating the theory is also given.     

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 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.     

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.     

Improving the bridge structure by using linear failure rate distribution

In this paper, a system of five components is studied; one of these components is a bridge network component. Each of these components has a non-constant failure rate. The system components have linear failure rate lifetime distribution. The given system is improved by using three methods: reduction, warm standby with perfect switch and warm standby with imperfect switch. The reliability equivalence factors of the bridge structure system are obtained. The γ-fractiles are obtained to compare the original system with these improved systems. Finally, we present numerical results to show the difference between these methods.     

A note on the mean past and the mean residual life of a (n − k + 1)-out-of-n system under multi monitoring

In the study of the reliability of technical systems, k-out-of-n systems play an important role. In the present paper, we consider a (n − k + 1)-out-of-n system consisting of n identical components such that the lifetimes of components are independent and have a common distribution function F. It is assumed that the number of monitoring is l and the total number of failures of the components at time t _i is m _i , i = 1, . . . , l − 1. Also at time t _l (t ₁ < . . . < t _l ) the system have failed or the system is still working. Under these conditions, the mean past lifetime, the mean residual lifetime of system and their properties are investigated.     

On characterizations of the generalized Pareto distributions based on progressively censored order statistics

It is well-known, in the literature, that most of the characterization results on exponential distribution are based on the solution of Cauchy functional equation and integrated Cauchy functional equation. In the present paper, we consider the functional equation $$F(x) = F(xy) + F(xQ(y)), \quad x, xQ(y) \in [0, \theta),\; y \in [0,1],$$ where F and Q satisfy certain conditions, to give some new characterization results on the generalized Pareto distributions based on progressively Type-II right censored order statistics. We prove the main results without restricting to distributions that are absolutely continuous with respect to Lebesgue measure.     

A model for k-out-of-m load-sharing systems

ABSTRACT

In a load-sharing system, the failure of a component affects the residual lifetime of the surviving components. We propose a model for the load-sharing phenomenon in k-out-of-m systems. The model is based on exponentiated conditional distributions of the order statistics formed by the failure times of the components. For an illustration, we consider two component parallel systems with the initial lifetimes of the components having Weibull and linear failure rate distributions. We analyze one data set to show that the proposed model may be a better fit than the model based on sequential order statistics.