Equivalence of the Hazard Rate and Usual Stochastic Orders for Parallel Systems

TL;DR

This paper explores the equivalence of hazard rate and stochastic orders in parallel systems, providing characterization results for systems with heterogeneous components and multiple-outlier models, with implications for bounds on system reliability.

Contribution

It establishes conditions under which hazard rate and stochastic orders are equivalent for different parallel system configurations, extending reliability comparison methods.

Findings

Equivalence of hazard rate and stochastic orders for certain parallel systems.

Conditions under which these orders are equivalent for heterogeneous components.

Bounds for hazard rate and survival functions of parallel systems.

Abstract

In this paper, we investigate stochastic comparisons of parallel systems, and obtain two characterization results in this regard. First, we compare a parallel system with independent heterogeneous components to a parallel system with homogeneous components, and establish some certain assumptions under which the hazard rate and usual stochastic orders between the lifetimes of two parallel systems are equivalent. Next, we turn our attention to two parallel systems with their component lifetimes following multiple-outlier model and prove that under some specified assumptions, the $p$-larger order between the vectors of scale parameters is equivalent to the hazard rate order as well as the usual stochastic order between the lifetimes of these systems. The results established here are applicable to compute an upper bound for the hazard rate function and a lower bound for the survival function of a parallel systems consisting of heterogeneous components.