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.     

Reliability Properties of Systems with Two Exchangeable Log-Logistic Components

In this article, we study the reliability properties of systems under bivariate log-logistic model which comes out from a particular stress-strength analysis. For this model, we obtain basic reliability characteristics of series and parallel systems and investigate their properties. We also derive distribution and moments of cold standby system under the abovementioned exchangeable model.     

Reliability Properties of Reversed Residual Lifetime

Abstract

If the random variable X denotes the lifetime (X ≥ 0, with probability one) of a unit, then the random variable X _t  = (t ? X|X ≤ t), for a fixed t > 0, is known as `time since failure', which is analogous to the residual lifetime random variable used in reliability and survival analysis. The reversed hazard rate function, which is related to the random variable X _t , has received the attention of many researchers in the recent past [(cf. Shaked, M., Shanthikumar, J. G., 1994 Shaked, M. and Shanthikumar, J. G. 1994. Stochastic Orders and Their Applications New York: Academic Press.  [Google Scholar]). Stochastic Orders and Their Applications. New York: Academic Press]. In this paper, we define some new classes of distributions based on the random variable X _t and study their interrelations. We also define a new ordering based on the mean of the random variable X_t and establish its relationship with the reversed hazard rate ordering.     

Comparison of Several Confidence Intervals for Median Residual Lifetime with Left-truncated and Right-censored Data

For left-truncated and right-censored data, the technique proposed by Brookmeyer and Crowley (1982) is extended to construct a point-wise confidence interval for median residual lifetime. This procedure is computationally simpler than the score type confidence interval in Jeong et al. (2008) and empirical likelihood ratio confidence interval in Zhou and Jeong (2011). Further, transformations of the estimator are applied to improve the approximation to the asymptotic distribution for small sample sizes. A simulation study is conducted to investigate the accuracy of these confidence intervals and the implementation of these confidence intervals to two real datasets is illustrated.     

On Properties of Percentile Bootstrap Confidence Intervals for Period Using Periodogram

Periodic functions have many applications in astronomy. They can be used to model the magnitude of light intensity of the period variable stars that their brightness vary with time. Because the data related to the astronomical applications are commonly observed at the time points that are not regularly spaced, the use of the periodogram as a good tool for estimating period is highlighted. Our bootstrap inference about period is based on maximizing the periodogram and consists of percentile two-sided bootstrap confidence intervals construction for the true period. We also obtain their coverage levels theoretically, and discuss the benefit of double-bootstrap confidence intervals for the parameter by which the coverage levels are substantially improved. Precisely, we show that the coverage error of single-bootstrap confidence intervals is of order n ^?1, decreasing to order n ^?2 when applying double-bootstrap methods. The simulation study given here is a numerical assessment of the theoretical work.     

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.     

On Signatures of Series and Parallel Systems Consisting of Modules with Arbitrary Structures

The signature of a system is a useful concept not only in the analysis of binary coherent systems but also in network reliability. Computation of system signature is a well-defined combinatorial problem. This article is concerned with the computation of signature vectors of series and parallel systems consisting of modules. We derive simple formulas for the signature and minimal signature of series and parallel systems based on signatures and minimal signatures of modules with given structures. We present computational results to illustrate the findings.     

On the quantile past lifetime of the components of the parallel systems

ABSTRACT

Let T_{1: n} ? T_{2: n} ? ??? ? T_{n: n} be ordered lifetimes of components of a parallel system. In this article, the α-quantile past lifetime from the failure of the component with lifetime T_{r: n} provided that the system has failed at or before time t has been introduced. Then, some properties of this measure have been studied.     

The percentile residual life up to time t0: Ordering and aging properties

Motivated by practical issues, a new stochastic order for random variables is introduced by comparing all their percentile residual life functions until a certain instant. Some interpretations of these stochastic orders are given, and various properties of them are derived. The relationships to other stochastic orders are studied and also an application in reliability theory is described. Finally, we present some characterization results of the decreasing percentile residual life up to time t₀ aging notion.     

Characterizations Based on Cumulative Residual Entropy of First-Order Statistics

Two different distributions may have equal cumulative residual entropy (CRE), thus a distribution cannot be determined by its CRE. In this article, we explore properties of the CRE and study conditions under which the CRE of the first-order statistics can uniquely determines the parent distribution. Weibull family is characterized through ratio of the CRE of the first-order statistics to its expectation. We have also some bounds for the CRE of residual lifetime of a series system.     

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.     

On Upshifted Reversed Mean Residual Life Order

Recently, the concept of reversed mean residual life order based on the mean of the random variable X _t  = (t ? X | X ≤ t), t > 0, called the reversed residual life, defined for the nonnegative random variable X, has been introduced in the literature. In this paper, a stochastic order based on the shifted version of the reversed mean residual life is proposed, based on the reversed mean residual life function for a random variable X with support (l _X , ∞), where l _X may be negative infinity, and its properties are studied. Closure under the Poisson shock model and properties for spare allocation are also discussed.     

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 Nonparametric Mean Residual Life Estimator in Length-biased Sampling

In this article, we discuss nonparametric estimation of a mean residual life function from length-biased data. Precisely, we prove strong uniform consistency and weak converge of the nonparametric mean residual life estimator in length-biased setting.     

Weighted extensions of generalized cumulative residual entropy and their applications

Abstract

In this paper, we consider weighted extensions of generalized cumulative residual entropy and its dynamic(residual) version. Our results include linear transformations, stochastic ordering, bounds, aging class properties and some relationships with other reliability concepts. We also define the conditional weighted generalized cumulative residual entropy and discuss some properties of its. For these concepts, we obtain some characterization results under some assumptions. Finally, we provide an estimator of the new information measure using empirical approach. In addition, we study large sample properties of this estimator.     

Reliability Research of Dependent Failure Systems Using Copula

The computation of reliability characteristics of a system that consists of dependent components sometimes becomes difficult, especially when a specific type of dependence is not identified. In this paper, some systems with arbitrary dependent components are studied using copula. In the system, the components are dependent on each other and the dependent relations may be either linear or nonlinear correlation. The efficient formulas are presented to compute the reliability characteristics, such as reliability function, failure rate and meantime to failure of series, parallel and k-out-of-n systems. The reliability functions of dependant systems are compared with independent system. At last, the numerical examples are presented to illustrate the results obtained in this paper.     

On the generalized cumulative residual entropy weighted distributions

Recently, Feizjavadian and Hashemi (2015 Feizjavadian, S.H., Hashemi, R. (2015). Mean residual weighted versus the length-biased Rayleigh distribution. J. Stat. Comput. Simul. 85:2823–2838.[Taylor &; Francis Online], [Web of Science ®] , [Google Scholar]) introduced and studied the mean residual weighted (MRW) distribution as an alternative to the length-biased distribution, by using the concepts of the mean residual lifetime and the cumulative residual entropy (CRE). In this article, a new sequence of weighted distributions is introduced based on the generalized CRE. This sequence includes the MRW distribution. Properties of this sequence are obtained generalizing and extending previous results on the MRW distribution. Moreover, expressions for some known distributions are given, and finite mixtures between the new sequence of weighted distributions and the length-biased distribution are studied. Numerical examples are given to illustrate the new results.     

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.     

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.