scholarly journals The Residual Lifetime of Surviving Components of Coherent System under Periodical Inspections

Mathematics ◽  
2020 ◽  
Vol 8 (12) ◽  
pp. 2181
Author(s):  
Zhouxia Guo ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Reliability Function ◽  
Stochastic Orders ◽  
Residual Lifetime ◽  
Coherent System ◽  
Numerical Examples ◽  
Aging Properties ◽  
Theoretical Results

In this manuscript, we gain a mixture representation for reliability function of the residual lifetime of unfailed components in a coherent system under periodical inspections, given that the number of failed components before time t1 is r(≥0), but the system is still operating at time t1, and the system eventually failed at time t2(>t1). Some aging properties and stochastic orders of the residual lifetime on survival components are also established. Finally, some numerical examples and graphs are given in order to confirm the theoretical results.

On inactivity times of failed components of coherent system under double monitoring

2021 ◽  
pp. 1-18
Author(s):  
Zhouxia Guo ◽  
Jiandong Zhang ◽  
Rongfang Yan
Keyword(s):  
Sufficient Conditions ◽  
Reliability Function ◽  
Coherent System ◽  
Stochastic Comparison ◽  
Coherent Systems ◽  
Stochastic Behavior ◽  
Numerical Examples ◽  
Comparison Results ◽  
Aging Properties ◽  
Theoretical Results

Abstract This article discusses the stochastic behavior and reliability properties for the inactivity times of failed components in coherent systems under double monitoring. A mixture representation of reliability function is obtained for the inactivity times of failed components, and some stochastic comparison results are also established. Furthermore, some sufficient conditions are developed in terms of the aging properties of the inactivity times of failed components. Finally, some numerical examples are presented to illustrate the theoretical results.

ON RELATIVE AGING COMPARISONS OF COHERENT SYSTEMS WITH IDENTICALLY DISTRIBUTED COMPONENTS

2020 ◽  
pp. 1-15 ◽  
Author(s):  
Nil Kamal Hazra ◽  
Neeraj Misra
Keyword(s):  
Hazard Rate ◽  
Sufficient Conditions ◽  
Stochastic Orders ◽  
Coherent System ◽  
Coherent Systems ◽  
Cumulative Hazard ◽  
Numerical Examples ◽  
Distributed Components ◽  
Rate Functions ◽  
Important Notion

The relative aging is an important notion which is useful to measure how a system ages relative to another one. Among the existing stochastic orders, there are two important orders describing the relative aging of two systems, namely, aging faster orders in the cumulative hazard and the cumulative reversed hazard rate functions. In this paper, we give some sufficient conditions under which one coherent system ages faster than another one with respect to the aforementioned stochastic orders. Further, we show that the proposed sufficient conditions are satisfied for k-out-of-n systems. Moreover, some numerical examples are given to illustrate the applications of proposed results.

A Note on the Mixture Representation of the Conditional Residual Lifetime of a Coherent System

2013 ◽  
Vol 50 (02) ◽  
pp. 475-485 ◽  
Author(s):  
Xiuying Feng ◽  
Shuhong Zhang ◽  
Xiaohu Li
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Reliability Function ◽  
Residual Lifetime ◽  
Coherent System ◽  
Distributed Components ◽  
Residual Lifetimes ◽  
Independent And Identically Distributed

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

scholarly journals Mean Residual Lifetime Frailty Models: A Weighted Perspective

2021 ◽  
Vol 2021 ◽  
pp. 1-21
Author(s):  
Mashael A. Alshehri ◽  
Mohamed Kayid
Keyword(s):  
Residual Life ◽  
Sufficient Conditions ◽  
Frailty Model ◽  
Random Parameter ◽  
Stochastic Orders ◽  
Residual Lifetime ◽  
Mean Residual Life ◽  
Aging Properties ◽  
Life Model ◽  
Multiplicative Mean

The mean residual life frailty model and a subsequent weighted multiplicative mean residual life model that requires weighted multiplicative mean residual lives are considered. The expression and the shape of a mean residual life for some semiparametric models and also for a multiplicative degradation model are given in separate examples. The frailty model represents the lifetime of the population in which the random parameter combines the effects of the subpopulations. We show that for some regular dependencies of the population lifetime on the random parameter, some aging properties of the subpopulations’ lifetimes are preserved for the population lifetime. We indicate that the weighted multiplicative mean residual life model generates positive dependencies of this type. The copula function associated with the model is also derived. Necessary and sufficient conditions for certain aging properties of population lifetimes in the model are determined. Preservation of stochastic orders of two random parameters for the resulting population lifetimes in the model is acquired.

scholarly journals A Note on the Mixture Representation of the Conditional Residual Lifetime of a Coherent System

2013 ◽  
Vol 50 (2) ◽  
pp. 475-485 ◽  
Author(s):  
Xiuying Feng ◽  
Shuhong Zhang ◽  
Xiaohu Li
Keyword(s):  
Order Statistics ◽  
Stochastic Ordering ◽  
Reliability Function ◽  
Residual Lifetime ◽  
Coherent System ◽  
Distributed Components ◽  
Residual Lifetimes ◽  
Independent And Identically Distributed

This paper builds a mixture representation of the reliability function of the conditional residual lifetime of a coherent system in terms of the reliability functions of conditional residual lifetimes of order statistics. Some stochastic ordering properties for the conditional residual lifetime of a coherent system with independent and identically distributed components are obtained, based on the stochastically ordered coefficient vectors.

scholarly journals Edge detection with trigonometric polynomial shearlets

2021 ◽  
Vol 47 (1) ◽  
Author(s):  
Kevin Schober ◽  
Jürgen Prestin ◽  
Serhii A. Stasyuk
Keyword(s):  
Edge Detection ◽  
Trigonometric Polynomial ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Numerical Examples ◽  
Special Cases ◽  
Periodic Characteristic ◽  
Boundary Curves ◽  
Theoretical Results ◽  
De La Vallée Poussin

AbstractIn this paper, we show that certain trigonometric polynomial shearlets which are special cases of directional de la Vallée Poussin-type wavelets are able to detect step discontinuities along boundary curves of periodic characteristic functions. Motivated by recent results for discrete shearlets in two dimensions, we provide lower and upper estimates for the magnitude of the corresponding inner products. In the proof, we use localization properties of trigonometric polynomial shearlets in the time and frequency domain and, among other things, bounds for certain Fresnel integrals. Moreover, we give numerical examples which underline the theoretical results.

scholarly journals Stability Analysis of Impulsive Control Systems with Finite and Infinite Delays

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Xuling Wang ◽  
Xiaodi Li ◽  
Gani Tr. Stamov
Keyword(s):  
Control Systems ◽  
Impulsive Control ◽  
Sufficient Conditions ◽  
Delay Systems ◽  
Stability Criteria ◽  
Numerical Examples ◽  
Infinite Delays ◽  
Impulsive Control Systems ◽  
Stable Matrix ◽  
Theoretical Results

This paper studies impulsive control systems with finite and infinite delays. Several stability criteria are established by employing the largest and smallest eigenvalue of matrix. Our sufficient conditions are less restrictive than the ones in the earlier literature. Moreover, it is shown that by using impulsive control, the delay systems can be stabilized even if it contains no stable matrix. Finally, some numerical examples are discussed to illustrate the theoretical results.

scholarly journals Stochastic orders for a multivariate Pareto distribution

2021 ◽  
Vol 29 (1) ◽  
pp. 53-69
Author(s):  
Luigi-Ionut Catana
Keyword(s):  
Pareto Distribution ◽  
Random Variables ◽  
Stochastic Orders ◽  
Distribution Family ◽  
Theoretical Results

Abstract In this article we give some theoretical results for equivalence between different stochastic orders of some kind multivariate Pareto distribution family. Weak multivariate orders are equivalent or imply different stochastic orders between extremal statistics order of two random variables sequences. The random variables in this article are not neccesary independent.

scholarly journals Nonparametric bayes estimation of the reliability function of a coherent system

2021 ◽  
Vol 54 (2) ◽  
pp. 183-206
Author(s):  
AKM Fazlur Rahman ◽  
Edsel A. Pena
Keyword(s):  
Structure Function ◽  
Dirichlet Process ◽  
Synthetic Data ◽  
Reliability Function ◽  
Bayes Estimation ◽  
Coherent System ◽  
System Failure ◽  
Nonparametric Bayes ◽  
Lifetime Distributions ◽  
Safety Considerations

Complex coherent systems are the engines driving forward our technological world. A coherent system is composed of components, which could be modules or sub-systems, that interact with each other according to some structure function. For purposes of maintenance and safety considerations, it is of critical importance to gain knowledge of the distribution of the system lifetime, with this distribution being a function of the distributions of the components lifetimes. Since the monitoring of a system ceases upon system failure, at system failure some components will be failed, while others, depending on the structure function, will still be functioning with their lifetimes right-censored by the system lifetime. This paper deals with the estimation of the system lifetime distribution. The inferential framework is nonparametric Bayesian, with partition-based Dirichlet processes (PBDP) assigned as priors on the components lifetime distributions. PBDP are more general than the usual Dirichlet process (DP) priors and are particularly suited as priors in settings with censored data. The resulting estimator of the system life distribution, which is a function of the nonparametric Bayes estimators of the components lifetime distributions, is compared in terms of bias and variance with a product-limit type estimator proposed by Doss, et. al. (Ann. Statist., 1989), which can be obtained as a limit of the proposed estimator. These comparisons, which are facilitated through computer simulations, demonstrate that the proposed estimator possesses some robustness. The proposed estimator is illustrated using a synthetic data for a parallel system with five components.

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