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Stochastic comparisons of imperfect maintenance models for a gamma deteriorating system

Sophie Mercier, Inma T. Castro

TL;DR

Two imperfect repair models for a degrading system, with deterioration level modeled by a non homogeneous gamma process, are compared, where the two repair models provide identical expected deterioration levels at maintenance times.

Abstract

This paper compares two imperfect repair models for a degrading system, with deterioration level modeled by a non homogeneous gamma process. Both models consider instantaneous and periodic repairs. The first model assumes that a repair reduces the degradation of the system accumulated from the last maintenance action. The second model considers a virtual age model and assumes that a repair reduces the age accumulated by the system since the last maintenance action. Stochastic comparison results between the two resulting processes are obtained. Furthermore, a specific case is analyzed, where the two repair models provide identical expected deterioration levels at maintenance times. Finally, two optimal maintenance strategies are explored, considering the two models of repair.

Stochastic comparisons of imperfect maintenance models for a gamma deteriorating system

TL;DR

Two imperfect repair models for a degrading system, with deterioration level modeled by a non homogeneous gamma process, are compared, where the two repair models provide identical expected deterioration levels at maintenance times.

Abstract

This paper compares two imperfect repair models for a degrading system, with deterioration level modeled by a non homogeneous gamma process. Both models consider instantaneous and periodic repairs. The first model assumes that a repair reduces the degradation of the system accumulated from the last maintenance action. The second model considers a virtual age model and assumes that a repair reduces the age accumulated by the system since the last maintenance action. Stochastic comparison results between the two resulting processes are obtained. Furthermore, a specific case is analyzed, where the two repair models provide identical expected deterioration levels at maintenance times. Finally, two optimal maintenance strategies are explored, considering the two models of repair.
Paper Structure (13 sections, 12 theorems, 48 equations, 10 figures)

This paper contains 13 sections, 12 theorems, 48 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Technical reminders
  3. The two models of imperfect repairs
  4. First model: Arithmetic Reduction of Deterioration of order 1 (ARD1)
  5. Second model: Arithmetic Reduction of (virtual) Age of order 1 (ARA1)
  6. Comparison of the moments
  7. Stochastic comparison between $Y_{t}^{(1)}$ and $Z_{t}^{(2)}$
  8. Mostly equivalent imperfect repair models
  9. Maintenance strategies
  10. The reward function
  11. $(n,T)$ policy
  12. (M,T) policy
  13. Conclusions and perspective

Key Result

Lemma 1

Let $X$ and $Y$ be gamma distributed random variables with parameters $\left( a_{1},b_{1}\right)$ and $\left( a_{2},b_{2}\right)$, respectively, where $a_{i},b_{i}>0$ for $i=1,2$. Then:

Figures (10)

  • Figure 1: Expectations of $Y_{t}^{(1)}$ and $Z_{t}^{(2)}$ with $b=1$, $T=1$
  • Figure 2: Variances of $Y_{t}^{(1)}$ and $Z_{t}^{(2)}$ with $b=1$, $T=1$
  • Figure 3: Plots of $log(f_{Y_t^{(1)}}/f_{Z_t^{(2)}})$ for $A(t)=t^{\alpha}$
  • Figure 4: Plots of $\int_{x}^{\infty} \bar{F}_{Z_t^{(2)}}(u)du-\int_{x}^{\infty}\bar{F}_{Y_t^{(1)}}(u) du$ for $A(t)=t^{\alpha}$
  • Figure 5: Plots of $\int_{0}^{x} F_{Z_t^{(2)}}(u)du-\int_{0}^{x}F_{Y_t^{(1)}}(u) du$ for $A(t)=t^{\alpha}$
  • ...and 5 more figures

Theorems & Definitions (40)

  • Definition 1
  • Definition 2
  • Lemma 1
  • Remark 1
  • Proposition 1
  • proof
  • Remark 2
  • Remark 3
  • Proposition 2
  • proof
  • ...and 30 more