Abstract

To measure the magnitude among random variables, we can apply a partial order connection defined on a distribution class, which contains the symmetry. In this paper, based on majorization order and symmetry or asymmetry functions, we carry out stochastic comparisons of lifetimes of two series (parallel) systems with dependent or independent heterogeneous Marshall–Olkin Topp Leone G (MOTL-G) components under random shocks. Further, the effect of heterogeneity of the shape parameters of MOTL-G components and surviving probabilities from random shocks on the reliability of series and parallel systems in the sense of the usual stochastic and hazard rate orderings is investigated. First, we establish the usual stochastic and hazard rate orderings for the lifetimes of series and parallel systems when components are statistically dependent. Second, we also adopt the usual stochastic ordering to compare the lifetimes of the parallel systems under the assumption that components are statistically independent. The theoretical findings show that the weaker heterogeneity of shape parameters in terms of the weak majorization order results in the larger reliability of series and parallel systems and indicate that the more heterogeneity among the transformations of surviving probabilities from random shocks according to the weak majorization order leads to larger lifetimes of the parallel system. Finally, several numerical examples are provided to illustrate the main results, and the reliability of series system is analyzed by the real-data and proposed methods.

Highlights

  • Order statistics plays a critical role in many research areas such as reliability theory, survival analysis, actuarial science, and auction theory

  • Li and Li [28] carried out stochastic comparisons between two series systems equipped with starting devices regarding the hazard rate and the dispersive and usual stochastic orderings when the components are Gumbel–Hougaard–Copula-dependent

  • In accordance with Theorem 6, we conclude that more heterogeneity among the transformations of surviving probabilities from random shocks, according to the weak majorization order, implies larger lifetime of the parallel systems consisting of two dependent heterogeneous components in the sense of the usual stochastic order

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Summary

Introduction

Order statistics plays a critical role in many research areas such as reliability theory, survival analysis, actuarial science, and auction theory. In practical reliability engineering, the lifetime of components may be statistically interdependent as all components of a system usually function in a common environment In this regard, Li and Li [28] carried out stochastic comparisons between two series (parallel) systems equipped with starting devices regarding the hazard rate and the dispersive and usual stochastic orderings when the components are Gumbel–Hougaard–Copula-dependent. To the best of our knowledge, no research work has been done on stochastic comparisons of the lifetimes of series and parallel systems with dependent heterogeneous MOTL-G components subject to random shocks. We will establish sufficient conditions to stochastically compare two series (parallel) systems with dependent heterogeneous MOTL-G components bearing independent random shocks. (1) The effect of the heterogeneity of the shape parameters of MOTL-G components and surviving probabilities from random shocks on the reliability of series and parallel systems with dependent heterogeneous components is investigated. Before proceeding to the main results, we first recall some concepts of stochastic orders, Majorization orders, and copula, which will be used in the sequel

Stochastic Order
Majorization Order
Archimedean Copula
Some Useful Lemmas
Main Results
Comparison of Two Series Systems
Comparison of Two Parallel Systems
Comparison of Two Parallel Systems in Heterogeneous Independent Case
Applications of the Real Data in the Series System Reliability
Analysis of the Obtained Results
Innovation of Research
Scientific and Practical Conclusions
Future Topic
Methods
Full Text
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