Ismihan Bayramoglu

The mean residual life function of a k-out-of-n structure at the system level

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 k-out-of-n system consisting of n identical components with independent lifetimes having a common distribution function F. Under the condition that, at time t, all the components of the system are working, we propose a new definition for the mean residual life (MRL) function of the system, and obtain several properties of that system.

A note on the mean residual life function of a parallel system

Abstract One of the most important types of system structures is the parallel structure. In the present article, we propose a definition for the mean residual life function of a parallel system and obtain some of its properties. The proposed definition measures the mean residual life function of a parallel system consisting of n identical and independent components under the condition that n - i, i = 0, 2, …, n - 1, components of the system are working and other components of the system have already failed. It is shown that, for the case where the components of the system have increasing hazard rate, the mean residual life function of the system is a nonincreasing function of time. Finally, we will obtain an upper bound for the proposed mean residual life function.

Reliability and Mean Residual Life of Complex Systems With Two Dependent Components Per Element

The reliability and mean residual life of complex systems are discussed. These systems consist of n elements each having two s -dependent subcomponents. The reliability of such systems involves the distributions of bivariate order statistics, and are connected with a bivariate binomial distribution. The mean residual life function of complex systems with intact components at time t is also discussed. Some examples and graphical representations are given.

The Reliability of Coherent Systems Subjected to Marshall–Olkin Type Shocks

In the classical Marshall-Olkin model, the system is subjected to two types of shocks coming at random times, and destroying components of the system. In statistics and reliability engineering literature, there are numerous papers dealing with various extensions of this model. However, none of these works takes into account the system structure, i.e., in existing shock models usually the system structure is not considered. In this work, we consider a new shock model involving the system structure. More precisely, we consider a coherent system which is subjected to Marshall-Olkin type shocks. We investigate the reliability, and mean time to failure (MTTF) of such systems subjected to shocks coming at random times. Numerical examples and graphs are provided, and an extension to a general model is discussed.

On Marshall–Olkin type distribution with effect of shock magnitude

Abstract In classical Marshall–Olkin type shock models and their modifications a system of two or more components is subjected to shocks that arrive from different sources at random times and destroy the components of the system. With a distinctive approach to the Marshall–Olkin type shock model, we assume that if the magnitude of the shock exceeds some predefined threshold, then the component, which is subjected to this shock, is destroyed; otherwise it survives. More precisely, we assume that the shock time and the magnitude of the shock are dependent random variables with given bivariate distribution. This approach allows to meet requirements of many real life applications of shock models, where the magnitude of shocks is an important factor that should be taken into account. A new class of bivariate distributions, obtained in this work, involve the joint distributions of shock times and their magnitudes. Dependence properties of new bivariate distributions have been studied. For different examples of underlying bivariate distributions of lifetimes and shock magnitudes, the joint distributions of lifetimes of the components are investigated. The multivariate extension of the proposed model is also discussed.

Baker- Lin-Huang Type Bivariate Distributions Based on Order Statistics

Baker (2008) introduced a new class of bivariate distributions based on distributions of order statistics from two independent samples of size n. Lin and Huang (2010) discovered an important property of Baker’s distribution and showed that the Pearson’s correlation coefficient for this distribution converges to maximum attainable value, i.e., the correlation coefficient of the Fréchet upper bound, as n increases to infinity. Bairamov and Bayramoglu (2013) investigated a new class of bivariate distributions constructed by using Baker’s model and distributions of order statistics from dependent random variables, allowing higher correlation than that of Baker’s distribution. In this article, a new class of Baker’s type bivariate distributions with high correlation are constructed based on distributions of order statistics by using an arbitrary continuous copula instead of the product copula.

Mean residual life and inactivity time of a coherent system subjected to Marshall-Olkin type shocks

We consider coherent systems subjected to Marshall-Olkin type shocks coming at random times and destroying components of the system. The paper combines two important models, coherent systems and Marshall-Olkin type shocks and studies the mean residual life (MRL) and the mean inactivity time (MIT) functions of coherent systems that is subjected to random shocks. The considered models and theoretical results are supported with examples and graphical representations.

Order statistics of dependent sequences consisting of two different sets of exchangeable variables

We consider two different sets of exchangeable samples which are assumed to be dependent. A single set of observations is obtained from these two dependent samples. The distribution of single order statistic, and the joint distribution of the minimum and an arbitrary order statistic are derived. The results are illustrated in the context of reliability problem.

On Extreme Residual Lives after the Failure of the System

The concept of residual lifetime has attracted considerable research interest in reliability theory. It is useful for evaluating the dynamic behavior of a system. In this paper, we study the extreme residual lives, that is, the minimum and maximum residual lives of the remaining components after the failure of the system. The system is assumed to have an arbitrary structure. We obtain signature-based distributional and ordering results for the extreme residual lives.

Joint distribution of new sample rank of bivariate order statistics

Let , be independent copies of bivariate random vector with joint cumulative distribution function and probability density function . For , the vector of order statistics of and , respectively, is denoted by . Let , , be a new sample from , which is independent from . Let be the rank of order statistics in a new sample and be the rank of order statistics in a new sample . We derive the joint distribution of discrete random vector and a general scheme wherein the distributions of new and old samples are different is considered. Numerical examples for given well-known distribution are also provided.