Limiting Behavior of Mixed Coherent Systems With Lévy-Frailty Marshall–Olkin Failure Times

IF 1.3 4区 数学 Q3 MATHEMATICS, INTERDISCIPLINARY APPLICATIONS Applied Stochastic Models in Business and Industry Pub Date : 2024-10-08 DOI:10.1002/asmb.2893
Guido Lagos, Javiera Barrera, Pablo Romero, Juan Valencia
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Abstract

In this article, we show a limit result for the reliability function of a system—that is, the probability that the whole system is still operational after a certain given time—when the number of components of the system grows to infinity. More specifically, we consider a sequence of mixed coherent systems whose components are homogeneous and non-repairable, with failure-times governed by a Lévy-Frailty Marshall–Olkin (LFMO) distribution—a distribution that allows simultaneous component failures. We show that under integrability conditions the reliability function converges to the probability of a first-passage time of a Lévy subordinator process. To the best of our knowledge, this is the first result to tackle the asymptotic behavior of the reliability function as the number of components of the system grows. To illustrate our approach, we give an example of a parametric family of reliability functions where the system failure time converges in distribution to an exponential random variable, and give computational experiments testing convergence.

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具有 Lévy-Frailty Marshall-Olkin 故障时间的混合相干系统的极限行为
在本文中,我们展示了当系统组件数量增长到无穷大时,系统可靠性函数的极限结果,即整个系统在一定给定时间后仍能运行的概率。更具体地说,我们考虑了一连串混合相干系统,这些系统的组件是同质的、不可修复的,其失效时间受列维-弗雷迪-马歇尔-奥尔金(LFMO)分布控制--该分布允许组件同时失效。我们的研究表明,在可整性条件下,可靠性函数收敛于勒维从属过程的首次通过时间概率。据我们所知,这是第一个解决可靠性函数随系统组件数量增长而渐近的结果。为了说明我们的方法,我们举例说明了系统故障时间在分布上收敛于指数随机变量的可靠性函数参数族,并给出了测试收敛性的计算实验。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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来源期刊
CiteScore
2.70
自引率
0.00%
发文量
67
审稿时长
>12 weeks
期刊介绍: ASMBI - Applied Stochastic Models in Business and Industry (formerly Applied Stochastic Models and Data Analysis) was first published in 1985, publishing contributions in the interface between stochastic modelling, data analysis and their applications in business, finance, insurance, management and production. In 2007 ASMBI became the official journal of the International Society for Business and Industrial Statistics (www.isbis.org). The main objective is to publish papers, both technical and practical, presenting new results which solve real-life problems or have great potential in doing so. Mathematical rigour, innovative stochastic modelling and sound applications are the key ingredients of papers to be published, after a very selective review process. The journal is very open to new ideas, like Data Science and Big Data stemming from problems in business and industry or uncertainty quantification in engineering, as well as more traditional ones, like reliability, quality control, design of experiments, managerial processes, supply chains and inventories, insurance, econometrics, financial modelling (provided the papers are related to real problems). The journal is interested also in papers addressing the effects of business and industrial decisions on the environment, healthcare, social life. State-of-the art computational methods are very welcome as well, when combined with sound applications and innovative models.
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