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Comparisons of policies based on relevation and replacement by a new one unit in reliability

Belzunce, F., Martínez-Riquelme, C., Mercader, J. A., Ruiz, J. M

TL;DR

This work studies when replacing a failed unit by a new one versus continuing under a relevation policy (replacing with a same-age unit) yields more favorable reliability outcomes. Modeling both policies as counting processes within an elementary pure birth framework, the authors develop multivariate stochastic and hazard-rate order results under ageing assumptions such as $IFR/DFR$ and $NBU/NWU$. They prove that, under $NBU/NWU$, the new-unit policy produces failure times that are stochastically larger than those under relevation, implying fewer failures by a given time, while IFR/DFR conditions yield corresponding dynamic hazard-rate orderings. The paper then specializes these results to minimal repair and generalized Yule birth processes, showing how ordering depends on the ageing characteristics of the life distributions and illustrating practical implications for maintenance strategies in reliability engineering. Overall, the results provide a rigorous multivariate framework to compare repair and replacement policies and guide policy choice based on aging properties.

Abstract

The purpose of this paper is to study the role of the relevation transform, where a failed unit is replaced by a used unit with the same age as the failed one, as an alternative to the policy based on the replacement by a new one. In particular, we compare the stochastic processes arising from a policy based on the replacement of a failed unit by a new one and from the one in which the unit is being continuously subjected to a relevation policy. The comparisons depend on the aging properties of the units under repair.

Comparisons of policies based on relevation and replacement by a new one unit in reliability

TL;DR

This work studies when replacing a failed unit by a new one versus continuing under a relevation policy (replacing with a same-age unit) yields more favorable reliability outcomes. Modeling both policies as counting processes within an elementary pure birth framework, the authors develop multivariate stochastic and hazard-rate order results under ageing assumptions such as and . They prove that, under , the new-unit policy produces failure times that are stochastically larger than those under relevation, implying fewer failures by a given time, while IFR/DFR conditions yield corresponding dynamic hazard-rate orderings. The paper then specializes these results to minimal repair and generalized Yule birth processes, showing how ordering depends on the ageing characteristics of the life distributions and illustrating practical implications for maintenance strategies in reliability engineering. Overall, the results provide a rigorous multivariate framework to compare repair and replacement policies and guide policy choice based on aging properties.

Abstract

The purpose of this paper is to study the role of the relevation transform, where a failed unit is replaced by a used unit with the same age as the failed one, as an alternative to the policy based on the replacement by a new one. In particular, we compare the stochastic processes arising from a policy based on the replacement of a failed unit by a new one and from the one in which the unit is being continuously subjected to a relevation policy. The comparisons depend on the aging properties of the units under repair.
Paper Structure (7 sections, 5 theorems, 54 equations, 2 figures)

This paper contains 7 sections, 5 theorems, 54 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries on relevation, stochastic comparisons and ageing properties
  3. Comparison of policies based on relevation and replacement by a new one unit
  4. Applications
  5. Minimal repair policies
  6. Generalized Yule birth processes
  7. Conclusions

Key Result

Theorem 3.2

Let us consider the processes, $N$ and $N'$, based on a sequence of distributions functions $\{F_n;n\ge 1\}$, corresponding to a sequence of independent continuous random variables, $\{X_n; n\ge 1\}$, with arrival times $\{T_n;n\ge 1\}$ and $\{T'_n; n\ge 1\}$, respectively. If $X_i$ is NBU [NWU], fo and

Figures (2)

  • Figure 1: Plot of the survival functions of $T_1 \# T_2$ (dashed line) and $T_1 + T_2$ (continuous line).
  • Figure 2: Survival functions for the arrival times 1 to 4, of a minimal repair process (blue) and an age process for $K=0.5$, 1 and 2 (black), according to \ref{['surv-nbu']}.

Theorems & Definitions (10)

  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • Theorem 3.5
  • proof
  • Theorem 4.1
  • Example 4.2