Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems

Ghobad Barmalzan; Ali Akbar Hosseinzadeh; Narayanaswamy Balakrishnan

doi:10.3934/jimo.2021101

Article Contents

2022, Volume 18, Issue 5: 3029-3053. Doi: 10.3934/jimo.2021101

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Stochastic comparisons of series-parallel and parallel-series systems with dependence between components and also of subsystems

1.
Department of Statistics, University of Zabol, Sistan and Baluchestan, Iran
2.
Department of Mathematics, University of Zabol, Sistan and Baluchestan, Iran
3.
Department of Mathematics and Statistics, McMaster University, Hamilton, Canada

^* Corresponding author: Ghobad Barmalzan, Email Address:ghbarmalzan@uoz.ac.ir
^* Corresponding author: Ghobad Barmalzan, Email Address:ghbarmalzan@uoz.ac.ir

Received: September 30, 2020

Revised: December 31, 2020

Early access: May 2021

Published: November 2022

The first author is supported by University of Zabol, grant number: GR-UOZ:3389

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Abstract

In this paper, we consider series-parallel and parallel-series systems comprising dependent components that are drawn from a heterogeneous population consisting of $ m $ different subpopulations. The components within each subpopulation are assumed to be dependent, and the subsystems themselves are also dependent, with their joint distribution being modeled by an Archimedean copula. We consider a very general setting in which we assume that the subpopulations have different Archimedean copulas for their dependence. Under such a general setup, we discuss the usual stochastic, hazard rate and reversed hazard rate orders between these systems and present a number of numerical examples to illustrate all the results established here. Finally, some concluding remarks are made. The results established here extend the recent results of Fang et al. (2020) in which they have assumed all the subsystems to be independent.

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Mathematics Subject Classification: Primary: 60E15; Secondary: 90B25.

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Figure 1. Plots of the reliability functions under Clayton copula for the components and copulas considered in Parts (ⅰ)-(ⅲ) for the subsystems, from left to right

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Figure 2. Plots of the reliability functions under Clayton copula for the components and copulas considered in Parts (ⅰ)-(ⅲ) for the subsystems, from left to right

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Figure 3. Plots of the distribution functions of parallel-series systems under Ali-Mikhail-Haq copula for the components and copulas considered in Parts (ⅰ)- (ⅲ) for the subsystems, from left to right

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Figure 4. Plots of the distribution functions of parallel-series systems under Ali-Mikhail-Haq copula for the components and copulas considered in Parts (ⅰ)-(ⅲ) for the subsystems, from left to right

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