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arxiv: 2605.15823 · v1 · pith:54MQEB2Hnew · submitted 2026-05-15 · 📊 stat.AP · cs.IT· math.IT

Active Redundancy Allocation Strategy at Component and System Level

Bidhan Modok , Shovan Chowdhury , Amarjit Kundu This is my paper

Pith reviewed 2026-05-19 19:37 UTC · model grok-4.3

classification 📊 stat.AP cs.ITmath.IT
keywords active redundancycoherent systemsallocation strategystochastic orderingcopula dependencereliability optimizationlikelihood ratio orderreversed hazard rate
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The pith

Sufficient conditions establish optimal ways to allocate two heterogeneous active redundancies in coherent systems with dependent components.

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

The paper derives sufficient conditions that identify the best placement for two non-identical active spares, either attached to individual components or to the entire system, in coherent structures made of identical but possibly dependent parts. Dependence between components is captured by copulas through a distortion function, and the results hold for lifetimes from a broad parametric family rather than one fixed distribution. A sympathetic reader would care because the conditions let reliability engineers decide where to install the stronger spare to raise the chance the whole system survives longer, with the improvement proved via direct stochastic comparisons instead of case-by-case calculation.

Core claim

The authors show that sufficient conditions can be stated under which one of the two possible allocations of heterogeneous active redundancies is optimal. When the conditions hold, the resulting coherent systems satisfy likelihood ratio ordering for component-level allocations and reversed hazard ordering for system-level allocations. These ordering results are obtained for component lifetimes belonging to a general parametric family and when dependence is introduced through copulas with a distortion function. Several aging properties of the systems are also obtained as corollaries.

What carries the argument

Copula distortion function that encodes dependence among identical components, combined with likelihood ratio and reversed hazard rate stochastic orders to compare the reliability of the two allocation choices.

If this is right

  • Placing the stronger redundancy at the component level produces a system that is larger in the likelihood ratio order than the alternative placement.
  • The same ordering statements remain valid across an entire parametric family of lifetime distributions once the copula distortion is fixed.
  • System-level allocation yields reversed hazard rate ordering between the competing designs.
  • Additional aging properties, such as preservation of certain aging classes, follow directly from the allocation rules.

Where Pith is reading between the lines

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

  • The same sufficient conditions might be checked numerically for small coherent systems to produce explicit allocation tables for common structures such as series-parallel networks.
  • Engineers could embed the derived ordering criteria into optimization software that selects spare locations under measured dependence patterns.
  • Extending the distortion-function approach to non-identical components would remove one modeling restriction while keeping the ordering technique intact.

Load-bearing premise

The system is coherent, its components are identical, and their dependence can be represented by a copula through a distortion function.

What would settle it

A concrete coherent system with two heterogeneous spares where the stated sufficient conditions hold yet the likelihood ratio order between the component-level allocations fails to appear.

Figures

Figures reproduced from arXiv: 2605.15823 by Amarjit Kundu, Bidhan Modok, Shovan Chowdhury.

Figure 1
Figure 1. Figure 1: Multiple redundant system Reliability engineers adopt active redundancy to improve system reliability, particularly in situations where the time between component failure and the recovery time is not permitted. On the other hand, design engineers ensure reliability optimization in terms of adopting optimal allocation policies of a given number of active redundancies to the components of coherent systems. I… view at source ↗
Figure 2
Figure 2. Figure 2: Curves for Example 3.1 The next theorem shows that the same ordering holds between Xc and X ∗ c , but in reverse order, under different conditions of qθ(u) and F(t, b). The conceptual interpretation of these sufficient conditions, as well as the structure of the proof, is analogous to that of Theorem 3.1 and is therefore omitted here. Theorem 3.2 Let us consider two coherent systems with redundancy at comp… view at source ↗
Figure 3
Figure 3. Figure 3: Curves for Example 3.2 Theorems 3.3 and 3.4 demonstrate sufficient conditions for hr ordering between Xc and X ∗ c and determine the best allocation choice of redundancies. The hr ordering helps compare two systems conditioned on the event that the systems have survived upon a specified time point; in other words, the failure (hazard) rates of two used systems can be compared using hr ordering. In this sen… view at source ↗
Figure 4
Figure 4. Figure 4: Curves for Example 3.3 The proof of Theorem 3.4 is similar to Theorem 3.4, and hence is omitted. Theorem 3.4 Let (F, b, qθ1 ) and (F, b ∗ , qθ2 ) , described above, be two coherent systems with lifetimes Xc and X ∗ c respectively. Now if θ1 ≥ θ2 and for all u ≥ 0, i) F(t; b) is log-convex in b, ii) h˜(t; b) is convex (concave) in b and iii) (1−u)q ′ θ (u) qθ(u) is decreasing (increasing) in u and decreasin… view at source ↗
Figure 5
Figure 5. Figure 5: Curve of [PITH_FULL_IMAGE:figures/full_fig_p010_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Curves for Example 3.5 The proof of 3.6 is similar to Theorem 3.5 and hence is omitted. Theorem 3.6 Let two coherent system are having model (F, b, qθ1 ) with lifetime Xc and model (F, b ∗ , qθ2 ) with lifetime X ∗ c . For θ1 ≥ θ2 if i) F(t; b) is log-convex in b, 11 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Curve of F ∗ c (y) Fc(y) Now the question arises, whether Theorem 3.3−3.6 can be improved to the strongest lr ordering or not. Theorem 3.7 provides sufficient conditions for which a coherent system with CLR can be shown to perform better than another similar system with respect to lr ordering. The result is a significant improvement upon the few existing results in the present context. The following defini… view at source ↗
Figure 8
Figure 8. Figure 8: Curves for Example 3.7 In reliability analysis, both the ordering and aging behavior of a system over its lifetime are of significant interest. Aging refers to the mathematical characterization of a system’s degra￾dation, or, in some cases, improvement over time. Various aging classes have been explored extensively in the literature. The following theorem shows that, under a specific reversed hazard rate c… view at source ↗
Figure 9
Figure 9. Figure 9: Curves for Example 3.8 4 Allocation of Active Redundancy at System Level Due to the superiority of active redundancy at component level of a system having in￾dependent components, comparisons at system level redundancy is relatively scarced in the literature. However, this superiority does not universally extend to stronger stochastic order 14 [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Curve of F¯ c(y) − F¯ s(y) In this section, stochastic comparison results are presented for two coherent systems of did original components having cdfs F (t, b0) and F (t, b∗ 0 ) with redundancy at the system level (see [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Curve of F¯ ∗ s (y) − F¯ s(y) Theorem 4.2 Let us consider two coherent systems with redundancy at system level having model (F, b, qθ) with lifetime Xs and model (F, b ∗ , qθ) with lifetime X ∗ s . if i) F(t; b) is decreasing and log-convex in b, ii) (1−u)q ′ θ (u) 1−qθ(u) is decreasing in u and θ, Then, for b, b ∗ and b ∗ ∈ D+ if b ⪯m b ∗ , it follows that Xs ≥st X ∗ s . Theorem 4.3 demonstrates that whe… view at source ↗
Figure 12
Figure 12. Figure 12: Curve of F ∗ s (x) Fs(x) The next theorem implies that if the lifetime distribution of a redundant system belongs to DRHR class, the lifetime of a coherent system comprising dependent components, connected through a copula with a specific characteristic of distortion function also belongs to the DRHR aging class. Example 3.8 justifies the conditions. Theorem 4.4 Let us consider a coherent system with redu… view at source ↗
Figure 13
Figure 13. Figure 13: indicates that, uq ′ θ (u) 1−qθ(u) is increasing in u and θ. Again, consider the Pareto (Type I) distribution with cdf F(x; b) = 1 − ( b x ) α , x ≥ b, α > 0, which satisfies condition ii) [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

Researchers and practitioners in the field of reliability engineering and optimization frequently use active redundancy techniques to intensify the performance of systems. In this article, we study allocation strategies of non-matching active redundancies (spares) in coherent systems consisting of possibly dependent and identical components for achieving better system reliability. The dependence of the components is modeled through copulas using the distortion function. Sufficient conditions are derived to establish optimal allocation strategies for two heterogeneous active redundancies at the component or system levels. Moreover, the results are true for the component lifetimes following a general family of parametric distributions. The results guarantee the likelihood ratio (reversed hazard) ordering between the coherent systems at the component level (system level) active redundancies. Some aging properties are also established in this endeavor. Several examples are provided to demonstrate the theoretical results.

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

2 major / 2 minor

Summary. The paper derives sufficient conditions for optimal allocation of two heterogeneous active redundancies in coherent systems whose identical components may be dependent, with dependence modeled via copulas and a distortion function. It claims these conditions apply to a general family of parametric lifetime distributions and guarantee likelihood-ratio ordering (component-level allocation) or reversed-hazard ordering (system-level allocation) between the resulting coherent systems, together with some aging properties; several examples are supplied to illustrate the results.

Significance. If the stated ordering results hold under the given sufficient conditions, the work would supply concrete, distribution-family-wide guidelines for redundancy allocation that incorporate component dependence—an extension beyond the independent-component literature that is directly relevant to reliability engineering practice.

major comments (2)
  1. [Section 3 (main results and proofs)] The central claim that the distortion function preserves the required monotonicity for likelihood-ratio ordering (component level) or reversed-hazard ordering (system level) is load-bearing. The proofs appear to rely on convexity or monotonicity properties induced by the distortion function on the survival copula or cumulative hazard; it is not clear whether these properties continue to hold for common non-Archimedean or tail-dependent copulas outside the paper’s examples. A concrete counter-example or an explicit restriction on the admissible copula class would be needed to confirm that the stochastic comparison cannot reverse when the marginal conditions are met.
  2. [Theorem 3.2 and surrounding discussion] The assertion that the results hold for a “general family of parametric distributions” is stated without an explicit characterization of that family or verification that the sufficient conditions on the marginals remain compatible with the copula-induced distortion for all members of the family. If the family is intended to be all distributions with monotone hazard rates, this should be stated and checked against the distortion step.
minor comments (2)
  1. [Preliminaries] Notation for the distortion function and the associated copula is introduced without a self-contained definition or reference to a standard source; a brief appendix recalling the relevant copula properties would improve readability.
  2. [Section 4 (examples)] In the numerical examples, the reported reliability values or ordering conclusions should be accompanied by the specific parameter values of the copula and the marginal distributions so that readers can reproduce the comparisons.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments, which help clarify the scope and applicability of our results. We respond to each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: [Section 3 (main results and proofs)] The central claim that the distortion function preserves the required monotonicity for likelihood-ratio ordering (component level) or reversed-hazard ordering (system level) is load-bearing. The proofs appear to rely on convexity or monotonicity properties induced by the distortion function on the survival copula or cumulative hazard; it is not clear whether these properties continue to hold for common non-Archimedean or tail-dependent copulas outside the paper’s examples. A concrete counter-example or an explicit restriction on the admissible copula class would be needed to confirm that the stochastic comparison cannot reverse when the marginal conditions are met.

    Authors: We agree that the monotonicity preservation under the distortion function is central to the stochastic ordering results. Our proofs in Section 3 establish the orders under the assumption that the copula-induced distortion function satisfies the requisite monotonicity and convexity conditions. These hold for the Archimedean families (e.g., Clayton, Gumbel-Hougaard) used in the examples. To address the concern for non-Archimedean or tail-dependent copulas, we will revise the manuscript to include an explicit restriction: the results apply to copulas whose associated distortion functions preserve the necessary monotonicity properties. We will add discussion of applicability to common copula families and note limitations for certain tail-dependent structures. revision: partial

  2. Referee: [Theorem 3.2 and surrounding discussion] The assertion that the results hold for a “general family of parametric distributions” is stated without an explicit characterization of that family or verification that the sufficient conditions on the marginals remain compatible with the copula-induced distortion for all members of the family. If the family is intended to be all distributions with monotone hazard rates, this should be stated and checked against the distortion step.

    Authors: The phrase 'general family of parametric distributions' is intended to encompass common parametric families (exponential, Weibull, gamma) whose hazard or reversed-hazard functions satisfy the monotonicity conditions needed for the ordering results to hold under the given marginal assumptions. We acknowledge that an explicit characterization and compatibility check are required. In the revision we will define the family precisely as those parametric distributions for which the survival functions meet the stated monotonicity requirements that remain compatible with the copula distortion, and we will verify this for the main examples in the updated discussion of Theorem 3.2. revision: yes

Circularity Check

0 steps flagged

No circularity: sufficient conditions derived from standard stochastic ordering and copula properties

full rationale

The paper derives sufficient conditions for optimal allocation of two heterogeneous active redundancies in coherent systems, establishing likelihood ratio ordering at the component level and reversed hazard ordering at the system level. Dependence is modeled via copulas and a distortion function on identical components, with results claimed to hold for a general family of parametric lifetime distributions. No derivation step reduces by construction to a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation chain. The proofs rely on external mathematical properties of monotonicity, convexity, and preservation under distortion functions, making the central claims self-contained against standard benchmarks in reliability theory rather than tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 3 axioms · 0 invented entities

The paper rests on standard domain assumptions from reliability theory and copula modeling without introducing new free parameters or invented entities visible in the abstract.

axioms (3)
  • domain assumption The system is coherent and the components are identical.
    Required setting for the allocation results stated in the abstract.
  • domain assumption Component dependence is modeled through copulas using the distortion function.
    Central modeling choice used to obtain the ordering results.
  • domain assumption Component lifetimes follow a general family of parametric distributions.
    Scope under which the sufficient conditions are claimed to hold.

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Works this paper leans on

26 extracted references · 26 canonical work pages

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