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We introduce a five-parameter continuous model, called the McDonald log-logistic distribution, to extend the two-parameter log-logistic distribution. Some structural properties of this new distribution such as reliability measures and entropies are obtained. The model parameters are estimated by the method of maximum likelihood using L-BFGS-B algorithm. A useful characterization of the distribution is proposed which does not require explicit closed form of the cumulative distribution function and also connects the probability density function with a solution of a first order differential equation. An application of the new model to real data set shows that it can give consistently better fit than other important lifetime models.

The reliability study of k-out-of-n systems is of interest both from theoretical and practical points of view. Applications of such models can be seen in many real-world phenomena, including telecommunication, transmission, transportation, manufacturing, and services. A probabilistic study of a real-world k-out-of-n system often helps to develop an optimal strategy for maintaining high system-level reliability. There are many investigations devoted to the reliability-centric analysis of such systems. We consider a mathematical model of a repairable k-out-of-n system that works until k of its n components have failed. During the system's life cycle, its components are repaired with the help of a single repair facility. It is supposed that the components' lifetimes have an exponential distribution and their repair times have a general distribution. The proposed model is intended to be applied to the description of operation of unmanned rotorcraft high-altitude platforms and to be validated with the help of an experimental prototype. For the considered system, we propose an algorithm for calculation of the reliability function, and for special cases, k = 2 and k = 3, its closed-form representation is given. A numerical investigation is performed for special cases. The obtained results are a first step toward the sensitivity analysis of reliability characteristics of k-out-of-n systems to the shape of the repair time distributions of their components.

In this article, mathematical modeling for the evaluation of reliability is studied using two methods. One of the methods, is developed based on possibility theory. The performance of the reliability of the system is of prime concern. In view of this, the outcomes for the failure are required to evaluate with utmost care. In possibility theory, the reliability information data determined from decision-making experts are subjective. The same method is also related to the survival possibilities as against the survival probabilities. The other method is the one that is developed using the concept of approximation of closed interval including the piecewise quadratic fuzzy numbers. In this method, a decision-making expert is not sure of his/her estimates of the reliability parameters. Numerical experiments are performed to illustrate the efficiency of the suggested methods in this research. In the end, the paper is concluded with some future research directions to be explored for the proposed approach.

With the rise of the semiconductor industry, passive components have become a foundation of the electronics industry and propelled the development of peripheral equipment and industries. This paper focused on improving the lifetime performance of products to increase product value and achieve the green production goals of energy conservation and waste reduction. We first established a relative lifetime performance index for passive component capacitors. Next, we derived the cumulative distribution function and reliability function based on the probability density function of relative lifetime to investigate the properties of the relative lifetime performance index. The results indicated that the reliability increased with the index value and that the probability of the product lifetime exceeding the minimum requirement also increased with the index value. Due to the fact that the index contains unknown parameters, using point estimates to estimate product lifetime performance may lead to misjudgment caused by sampling errors. For this reason, we propose a uniformly minimum-variance unbiased estimator for the index and derived its probability density function. In addition, we establish a fuzzy testing model based on the confidence interval to determine whether the lifetime performance of a product reaches the required performance level. Finally, we present a numerical example of passive components to demonstrate application of the proposed model. This example further demonstrates how the proposed model improves product lifetime performance, which in turn increases product value and also achieves the green objectives of energy conservation and waste reduction.

In the present work, we study threshold reliability systems consisting of three different kinds of independent components. Within these structures, the components belonging to the same type, share a common reliability and the same weight. The reliability attributes of the aforementioned systems are investigated in some detail. Explicit expressions for calculating their reliability function, the mean time to failure and the corresponding Birnbaum’s importance measures are established. For illustration purposes, several numerical results are presented, while some concluding remarks about the impact of the design parameters of the underlying structures are delivered. A short discussion for potential future work is also developed.

The research presents the reliability. It is defined as the probability of accomplishing any part of the system within a specified time and under the same circumstances. On the theoretical side, the reliability, the reliability function, and the cumulative function of failure are studied within the one-parameter Raleigh distribution. This research aims to discover many factors that are missed the reliability evaluation which causes constant interruptions of the machines in addition to the problems of data. The problem of the research is that there are many methods for estimating the reliability function but no one has suitable qualifications for most of these methods in the data such as the presence of anomalous values or extreme values or the appropriate distribution of these data is unknown. Therefore, the data need methods through which can be dealt with this problem. Two of the estimation methods have been used: the robust (estimator M) method and the nonparametric Kernel method. These estimation methods are derived to arrive at the formulas of their capabilities. A comparison of these estimations is made using the simulation method as it is implemented. Simulation experiments using different sample sizes and each experiment is repeated (1000) times to achieve the objective. The results are compared by using one of the most important statistical measures which is the mean of error squares (MSE). The best estimation method has been reached is the robust (M estimator) method. It has been shown that the estimation of the reliability function gradually decreases with time, and this is identical to the properties of this function.

A nanocomponent is a collection of atoms arranged to a specific design in order to achieve a desired function with an acceptable performance and reliability. The type of atoms, the manner in which they are arranged within the nanocomponent, and their interrelationship have a direct effect on the nanocomponent's reliability (survival) function. In this paper we propose models based on the notion of a copula that are used to describe the relationship between the atoms of a nanocomponent. Having defined these models, we go on to construct a ‘nanocomponent’ model in order to obtain the reliability function of a nanocomponent.

Assessing the reliability of two production lines, whether individually or simultaneously, is of significant importance for improving manufacturing processes and guaranteeing superior product quality. This paper examines the reliability assessment of two production lines utilizing joint progressively Type-II censored samples derived from the XLindley distribution. In addition to estimating the unknown parameters, the reliability functions for each production line, as well as for both production lines simultaneously, are analyzed. Both classical likelihood-based and Bayesian methodologies are employed for estimation purposes. The maximum likelihood method is applied to obtain point estimates for the unknown parameters and reliability functions, while Bayesian analysis is performed under the squared error loss function, employing the Markov Chain Monte Carlo technique to generate samples from the posterior distribution. The approximate confidence intervals, percentile bootstrap confidence intervals, and the highest posterior density credible intervals for the unknown parameters and various reliability functions are discussed. The problem of selecting the optimal censoring plan is also considered. A comprehensive simulation study is conducted to assess the performance of the proposed methods, comparing the accuracy and efficiency of the estimates across different censoring schemes. Furthermore, the applicability of the proposed methodology is demonstrated through the analysis of two real-world data sets, underscoring its practical utility within the field of reliability. Finally, three criteria, namely A-optimality, D-optimality, and F-optimality, are considered to determine the optimal censoring plan.

In this paper we are concerned with modelling the reliability of a system subject to external shocks. In a run shock model, the system fails when a sequence of shocks above a threshold arrive in succession. Nevertheless, using a single threshold to measure the severity of a shock is too critical in real practice. To this end, we develop a generalized run shock model with two thresholds. We employ a phase-type distribution to model the damage size and the inter-arrival time of shocks, which is highly versatile and may be used to model many quantitative features of random phenomenon. Furthermore, we use the Markovian property to construct a multi-state system which degrades with the arrival of shocks. We also provide a numerical example to illustrate our results.

The reliability of software has a tremendous influence on the reliability of systems. Software dependability models are frequently utilized to statistically analyze the reliability of software. Numerous reliability models are based on the nonhomogeneous Poisson method (NHPP). In this respect, in the current study, a novel NHPP model established on the basis of the new power function distribution is suggested. The mathematical formulas for its reliability measurements were found and are visually illustrated. The parameters of the suggested model are assessed utilizing the weighted nonlinear least-squares, maximum-likelihood, and nonlinear least-squares estimation techniques. The model is subsequently verified using a variety of reliability datasets. Four separate criteria were used to assess and compare the estimating techniques. Additionally, the effectiveness of the novel model is assessed and evaluated with two foundation models both objectively and subjectively. The implementation results reveal that our novel model performed well in the failure data that we examined.