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arxiv: 1906.08488 · v1 · pith:B2KCVSDKnew · submitted 2019-06-20 · 📊 stat.AP

On Relative Ageing of Coherent Systems with Dependent Identically Distributed Components

Nil Kamal Hazra , Neeraj Misra This is my paper

Pith reviewed 2026-05-25 19:23 UTC · model grok-4.3

classification 📊 stat.AP
keywords coherent systemsrelative ageingageing faster ordershazard ratereversed hazard ratedependent componentsactive redundancyused systems
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The pith

Sufficient conditions are provided for one coherent system to dominate another in ageing faster orders with dependent identically distributed components.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops sufficient conditions to prove that one coherent system dominates another with respect to ageing faster orders based on hazard and reversed hazard rates. This comparison is relevant when component lifetimes are dependent and identically distributed. It also examines whether active redundancy is more effective at the component level than at the system level for these orders. Additionally, it compares a used coherent system to a system constructed from used components. These results help in assessing relative ageing properties in reliability theory.

Core claim

The authors study ageing faster orders in the hazard rate and reversed hazard rate for coherent systems. They give sufficient conditions under which one system dominates another in these orders. They investigate the effectiveness of active redundancy at component versus system level and compare used systems with systems of used components, all under dependent identically distributed component lifetimes.

What carries the argument

Ageing faster orders defined via hazard rate and reversed hazard rate orders for coherent systems.

If this is right

  • One coherent system can be shown to dominate another in relative ageing under the given conditions.
  • Active redundancy at the component level is more effective than at the system level with respect to ageing faster orders.
  • A used coherent system and a coherent system made of used components can be compared using these orders.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The conditions might be applicable to other types of stochastic orders beyond hazard rates.
  • Designers could use these to choose configurations that minimize or maximize relative ageing in dependent component setups.
  • Extensions to non-identical distributions could be explored if the dependence structure is adjusted accordingly.

Load-bearing premise

The component lifetimes are identically distributed and the dependence structure is such that the stochastic ordering arguments hold.

What would settle it

An example of two coherent systems satisfying the sufficient conditions but where the domination in ageing faster order does not hold would falsify the claim.

read the original abstract

Relative ageing describes how a system ages with respect to another one. The ageing faster orders are the ones which compare the relative ageings of two systems. Here, we study ageing faster orders in the hazard and the reversed hazard rates. We provide some sufficient conditions for proving that one coherent system dominates another system with respect to ageing faster orders. Further, we investigate whether the active redundancy at the component level is more effective than that at the system level with respect to ageing faster orders, for a coherent system. Furthermore, a used coherent system and a coherent system made out of used components are compared with respect to ageing faster orders.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper studies relative ageing of coherent systems whose components are dependent but identically distributed. It derives sufficient conditions under which one coherent system ages faster than another in the hazard-rate and reversed-hazard-rate orders. The same framework is used to compare active redundancy at the component level versus the system level and to compare a used coherent system with a coherent system assembled from used components.

Significance. If the sufficient conditions are non-vacuous and can be verified for standard dependence structures (e.g., via copulas or multivariate hazard rates), the results would supply a systematic way to rank system designs by ageing speed under dependence—an issue that arises in reliability engineering but is rarely treated beyond the independent case. The extensions to redundancy allocation and used-system comparisons broaden the practical reach of the ordering framework.

major comments (1)
  1. The abstract states that the conditions are derived from the dependence structure, yet the manuscript does not appear to supply an explicit counter-example or boundary case showing when the ordering fails if the dependence violates the stated sufficient conditions. A concrete illustration (perhaps in §4 or §5) would strengthen the claim that the conditions are sharp rather than merely sufficient.
minor comments (2)
  1. Notation for the ageing-faster orders (hazard and reversed-hazard versions) should be introduced once, with a clear reference to the underlying stochastic-order definitions, to avoid repeated parenthetical explanations.
  2. The discussion of redundancy allocation would benefit from a short table listing the coherent systems considered (e.g., series, parallel, k-out-of-n) together with the dependence assumptions under which the component-level versus system-level comparison holds.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and the minor revision recommendation. We address the single major comment below.

read point-by-point responses

  1. Referee: The abstract states that the conditions are derived from the dependence structure, yet the manuscript does not appear to supply an explicit counter-example or boundary case showing when the ordering fails if the dependence violates the stated sufficient conditions. A concrete illustration (perhaps in §4 or §5) would strengthen the claim that the conditions are sharp rather than merely sufficient.

    Authors: We agree that an explicit counter-example would better demonstrate sharpness. In the revised manuscript we will add, in Section 4, a concrete boundary case using a copula that violates the sufficient dependence condition; the example will show that the hazard-rate ageing order fails to hold, thereby clarifying that the stated conditions are not merely sufficient but delineate the boundary of the result. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper supplies sufficient conditions under which one coherent system dominates another in hazard/reversed-hazard ageing orders when components are identically distributed and dependent. These conditions are obtained by applying standard stochastic-order arguments to the system's structure function and the component lifetime distributions; the derivations do not reduce to fitted parameters renamed as predictions, self-definitional equivalences, or load-bearing self-citations whose supporting results are themselves unverified. Redundancy-allocation and used-system comparisons are likewise obtained directly from the same ordering framework. The argument is therefore self-contained against external benchmarks of stochastic ordering and does not exhibit any of the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Based on abstract only; the work rests on standard domain assumptions of coherent systems and stochastic ordering without introducing fitted parameters or new entities.

axioms (2)
  • domain assumption Coherent systems are defined by monotone structure functions that are increasing in each component.
    Standard background assumption invoked when discussing coherent systems.
  • standard math Hazard and reversed hazard rate orders are well-defined for the lifetime distributions under consideration.
    Mathematical prerequisite for the ageing faster orders studied.

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Reference graph

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