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Failure Rate Properties of Parallel Systems

Idir Arab, Milto Hadjikyriakou, Paulo Eduardo Oliveira

TL;DR

The paper develops and analyzes iterated tail distributions $\overline{T}_{X,s}$ to study $s\!- ext{IFR}$ and $s\!- ext{IFRA}$ ageing notions, including their monotonicity and stochastic-order properties. It proves that hereditary preservation of these orders is not guaranteed in general, but provides a sign-variation criterion that facilitates $s\!- ext{IFR}$/$s\!- ext{IFRA}$ comparisons and yields practical results for Gamma, Weibull, and exponential-parallel systems. The authors establish new results on the behavior of parallel systems, showing that iterated IFR properties are preserved under certain constructions and demonstrating extended aging results for exponential lifetimes. They also supply counterexamples to clarify limitations of inherited ordering and present applications to parallel systems, including an exploration of a Kochar–Xu conjecture with counterexamples for $s\ge2$ and open cases for $s=1$. Overall, the work advances the understanding of ageing, stochastic orders, and reliability of parallel-component systems using iterated tail methods.

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.

Failure Rate Properties of Parallel Systems

TL;DR

The paper develops and analyzes iterated tail distributions to study and ageing notions, including their monotonicity and stochastic-order properties. It proves that hereditary preservation of these orders is not guaranteed in general, but provides a sign-variation criterion that facilitates / comparisons and yields practical results for Gamma, Weibull, and exponential-parallel systems. The authors establish new results on the behavior of parallel systems, showing that iterated IFR properties are preserved under certain constructions and demonstrating extended aging results for exponential lifetimes. They also supply counterexamples to clarify limitations of inherited ordering and present applications to parallel systems, including an exploration of a Kochar–Xu conjecture with counterexamples for and open cases for . Overall, the work advances the understanding of ageing, stochastic orders, and reliability of parallel-component systems using iterated tail methods.

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.

Paper Structure

This paper contains 7 sections, 28 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hereditary monotonicity properties
  4. Simple failure rate monotonicity properties
  5. Nonhereditary of the $s\!-\!{\rm IFR}$ ordering
  6. A criterium for $s\!-\!{\rm IFR}$ ordering and a first application
  7. Failure rate ordering of exponentially distributed parallel systems

Key Result

Lemma 2.1

The tails $\overline{T}_{X,s}$ may be represented as The $s-$iterated distribution moments are given by

Theorems & Definitions (41)

  • Definition 2.1
  • Lemma 2.1: Lemma 2 and Remark 3 in Arab and Oliveira AO18
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Theorem 2.1
  • Corollary 2.1
  • Theorem 2.2
  • Remark 2.1
  • Theorem 2.3
  • ...and 31 more