Failure Rate Properties of Parallel Systems

TL;DR

This paper investigates failure rate properties and stochastic ordering in parallel systems, focusing on how tail weight iteration affects distribution characteristics and aging properties of systems with exponential components.

Contribution

It introduces a new criterion for stochastic ordering based on sign variation analysis and explores hereditary failure rate properties under iteration procedures.

Findings

Failure rate monotonicity can be non-hereditary under iteration.

A new sign variation criterion effectively characterizes stochastic orderings.

Parallel systems with exponential components exhibit specific aging properties.

Abstract

We study failure rate monotonicity and generalized convex transform stochastic ordering properties of random variables, with a concern on applications. We are especially interested in the effect of a tail weight iteration procedure to define distributions, which is equivalent to the characterization of moments of the residual lifetime at a given instant. For the monotonicity properties, we are mainly concerned with hereditary properties with respect to the iteration procedure providing counter-examples showing either that the hereditary property does not hold or that inverse implications are not true. For the stochastic ordering, we introduce a new criterium, based on the analysis of the sign variation of a suitable function. This criterium is then applied to prove ageing properties of parallel systems formed with components that have exponentially distributed lifetimes.