Some new results on stochastic comparisons of coherent systems using signatures

AbstractWe consider coherent systems with independent and identically distributed components. While it is clear that the system’s life will be stochastically larger when the components are replaced with stochastically better components, we show that, in general, similar results may not hold for hazard rate, reverse hazard rate, and likelihood ratio orderings. We find sufficient conditions on the signature vector for these results to hold. These results are combined with other well-known results in the literature to get more general results for comparing two systems of the same size with different signature vectors and possibly with different independent and identically distributed component lifetimes. Some numerical examples are also provided to illustrate the theoretical results.

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Stochastic comparisons of coherent systems under different random environments

Journal of Applied Probability ◽

10.1017/jpr.2018.30 ◽

2018 ◽

Vol 55 (2) ◽

pp. 459-472 ◽

Cited By ~ 2

Author(s):

Ebrahim Amini-Seresht ◽

Yiying Zhang ◽

Narayanaswamy Balakrishnan

Keyword(s):

Hazard Rate ◽

Sufficient Conditions ◽

Random Environments ◽

Reliability Engineering ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Numerical Examples ◽

Reversed Hazard Rate ◽

Collaborative Environment ◽

Theoretical Results

Abstract For many practical situations in reliability engineering, components in the system are usually dependent since they generally work in a collaborative environment. In this paper we build sufficient conditions for comparing two coherent systems under different random environments in the sense of the usual stochastic, hazard rate, reversed hazard rate, and likelihood ratio orders. Applications and numerical examples are provided to illustrate all the theoretical results established here.

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ON RELATIVE AGING COMPARISONS OF COHERENT SYSTEMS WITH IDENTICALLY DISTRIBUTED COMPONENTS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964820000066 ◽

2020 ◽

pp. 1-15 ◽

Cited By ~ 1

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.

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ON THE RESIDUAL AND PAST LIFETIMES OF COHERENT SYSTEMS UNDER RANDOM MONITORING

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964820000078 ◽

2020 ◽

pp. 1-16

Author(s):

Ebrahim Amini-Seresht ◽

Maryam Kelkinnama ◽

Yiying Zhang

Keyword(s):

Hazard Rate ◽

Sufficient Conditions ◽

System Structure ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Distortion Functions ◽

Aging Properties ◽

Observation Times ◽

Theoretical Results ◽

Residual Lifetimes

This paper discusses stochastic comparisons for the residual and past lifetimes of coherent systems with dependent and identically distributed (d.i.d.) components under random monitoring in terms of the hazard rate, the reversed hazard rate, and the likelihood ratio orders. Some stochastic comparisons results are also established on the residual lifetimes of coherent systems under random observation times when all of the components are alive at that time. Sufficient conditions are established in terms of the aging properties of the components and the distortion functions induced from the system structure and dependence among components lifetimes. Numerical examples are provided to illustrate the theoretical results as well.

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COMPARISONS OF SAMPLE RANGES ARISING FROM MULTIPLE-OUTLIER MODELS: IN MEMORY OF MOSHE SHAKED

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964817000468 ◽

2017 ◽

Vol 33 (1) ◽

pp. 28-49

Author(s):

Narayanaswamy Balakrishnan ◽

Jianbin Chen ◽

Yiying Zhang ◽

Peng Zhao

Keyword(s):

Hazard Rate ◽

Stochastic Order ◽

Sufficient Conditions ◽

Hazard Rates ◽

Proportional Hazard ◽

Stochastic Comparisons ◽

Hazard Rate Order ◽

Numerical Examples ◽

Reversed Hazard Rate ◽

Usual Stochastic Order

In this paper, we discuss the ordering properties of sample ranges arising from multiple-outlier exponential and proportional hazard rate (PHR) models. The purpose of this paper is twofold. First, sufficient conditions on the parameter vectors are provided for the reversed hazard rate order and the usual stochastic order between the sample ranges arising from multiple-outlier exponential models with common sample size. Next, stochastic comparisons are separately carried out for sample ranges arising from multiple-outlier exponential and PHR models with different sample sizes as well as different hazard rates. Some numerical examples are also presented to illustrate the results established here.

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On the Conditional Residual Life and Inactivity Time of Coherent Systems

Journal of Applied Probability ◽

10.1239/jap/1421763323 ◽

2014 ◽

Vol 51 (4) ◽

pp. 990-998 ◽

Cited By ~ 2

Author(s):

A. Parvardeh ◽

N. Balakrishnan

Keyword(s):

Residual Life ◽

Coherent System ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Distributed Components ◽

Inactivity Time ◽

Independent And Identically Distributed

In this paper we derive mixture representations for the reliability functions of the conditional residual life and inactivity time of a coherent system with n independent and identically distributed components. Based on these mixture representations we carry out stochastic comparisons on the conditional residual life, and the inactivity time of two coherent systems with independent and identical components.

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Stochastic Comparisons of Residual Lifetimes and Inactivity Times of Coherent Systems

Journal of Applied Probability ◽

10.1239/jap/1378401240 ◽

2013 ◽

Vol 50 (3) ◽

pp. 848-860 ◽

Cited By ~ 5

Author(s):

Nitin Gupta

Keyword(s):

Likelihood Ratio ◽

Sufficient Conditions ◽

Structure Functions ◽

Coherent System ◽

Stochastic Comparison ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Likelihood Ratio Ordering ◽

Necessary And Sufficient ◽

Residual Lifetimes

Under the assumption of independent and identically distributed (i.i.d.) components, the problem of the stochastic comparison of a coherent system having used components and a used coherent system has been considered. Necessary and sufficient conditions on structure functions have been provided for the stochastic comparison of a coherent system having used/inactive i.i.d. components and a used/inactive coherent system. As a consequence, for r-out-of-n systems, it has been shown that systems having used i.i.d. components stochastically dominate used systems in the likelihood ratio ordering.

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ORDERING CONSECUTIVE k-OUT-OF-n:F SYSTEMS

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964811000301 ◽

2012 ◽

Vol 26 (2) ◽

pp. 147-158

Author(s):

Gaofeng Da ◽

Ben Zheng ◽

Taizhong Hu

Keyword(s):

Likelihood Ratio ◽

Sufficient Condition ◽

Stochastic Comparisons ◽

Independent Components ◽

Distributed Components ◽

F System ◽

Independent And Identically Distributed

Stochastic comparisons of linear (circular) consecutive k-out-of-n:F systems with independent components are studied. A sufficient condition is given under which the lifetime of a circular consecutive k-out-of-n:F system with independent and nonidentically distributed components is stochastically decreasing in n for fixed k. Furthermore, the likelihood ratio orderings of the lifetimes of linear (circular) consecutive k-out-of-n:F systems with independent and identically distributed components are also established, and some counterexamples are given to show that these orderings are not true in general.

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Stochastic Comparisons of Residual Lifetimes and Inactivity Times of Coherent Systems

Journal of Applied Probability ◽

10.1017/s0021900200009888 ◽

2013 ◽

Vol 50 (03) ◽

pp. 848-860 ◽

Cited By ~ 3

Author(s):

Nitin Gupta

Keyword(s):

Likelihood Ratio ◽

Sufficient Conditions ◽

Structure Functions ◽

Coherent System ◽

Stochastic Comparison ◽

Coherent Systems ◽

Stochastic Comparisons ◽

Likelihood Ratio Ordering ◽

Necessary And Sufficient ◽

Residual Lifetimes

Under the assumption of independent and identically distributed (i.i.d.) components, the problem of the stochastic comparison of a coherent system having used components and a used coherent system has been considered. Necessary and sufficient conditions on structure functions have been provided for the stochastic comparison of a coherent system having used/inactive i.i.d. components and a used/inactive coherent system. As a consequence, forr-out-of-nsystems, it has been shown that systems having used i.i.d. components stochastically dominate used systems in the likelihood ratio ordering.

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

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964821000152 ◽

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.

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