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Condition-based maintenance for a system subject to multiple degradation processes with stochastic arrival intensity

L. Bautista, Inma T. Castro, L. Landesa

TL;DR

This work studies condition-based maintenance for a system subject to multiple degradation processes, modeling the arrival of degradation events with a shot-noise Cox process and the growth of deterioration with a homogeneous gamma process. Heterogeneity across degradation paths is captured by letting the inverse scale parameter be Uniform$(a,b)$, yielding a random-effects gamma process $X_h(t)$. The combined arrival-growth dynamics are analyzed via displaced Cox process theory, providing expressions for the expected intensity $\mathbb{E}[\lambda_L(t)]$ and the expected count $\mathbb{E}[N_L(t)]$ of degradations crossing the failure threshold, with an IFR-compatible time-to-failure distribution for $W_{[1]}$. A periodic inspection maintenance policy is optimized by minimizing the asymptotic cost rate $C(T,M)$, incorporating preventive and corrective replacements, inspections, and downtime; numerical results show how heterogeneity and model parameters influence the optimal policy, offering insights for CBM design in complex degradation environments. The framework is extensible to alternative Cox processes and dependence structures, though real-data validation remains a limitation.

Abstract

In this work, a system subject to different deterioration processes is analysed. The arrival of the degradation processes to the system is modelled using a shot-noise Cox process. The degradation processes grow according to an homogeneous gamma process. The system fails when a degradation process exceeds a failure threshold. The combined process of initiation and growth of the degradation processes is modelled and the system reliability is obtained. Heterogeneities are also integrated in the model assuming that the inverse of the scale parameter follows a uniform distribution. A maintenance strategy is implemented in this system and the state of the system is checked in inspection times. If the system is working at inspection time, a preventive replacement is performed if the deterioration level of a degradation process exceeds a certain threshold. A corrective replacement is performed if the system is down at inspection time. Under this maintenance strategy, the expected cost rate is obtained. Sensitivity analysis on the main parameters of the gamma process is performed.

Condition-based maintenance for a system subject to multiple degradation processes with stochastic arrival intensity

TL;DR

This work studies condition-based maintenance for a system subject to multiple degradation processes, modeling the arrival of degradation events with a shot-noise Cox process and the growth of deterioration with a homogeneous gamma process. Heterogeneity across degradation paths is captured by letting the inverse scale parameter be Uniform, yielding a random-effects gamma process . The combined arrival-growth dynamics are analyzed via displaced Cox process theory, providing expressions for the expected intensity and the expected count of degradations crossing the failure threshold, with an IFR-compatible time-to-failure distribution for . A periodic inspection maintenance policy is optimized by minimizing the asymptotic cost rate , incorporating preventive and corrective replacements, inspections, and downtime; numerical results show how heterogeneity and model parameters influence the optimal policy, offering insights for CBM design in complex degradation environments. The framework is extensible to alternative Cox processes and dependence structures, though real-data validation remains a limitation.

Abstract

In this work, a system subject to different deterioration processes is analysed. The arrival of the degradation processes to the system is modelled using a shot-noise Cox process. The degradation processes grow according to an homogeneous gamma process. The system fails when a degradation process exceeds a failure threshold. The combined process of initiation and growth of the degradation processes is modelled and the system reliability is obtained. Heterogeneities are also integrated in the model assuming that the inverse of the scale parameter follows a uniform distribution. A maintenance strategy is implemented in this system and the state of the system is checked in inspection times. If the system is working at inspection time, a preventive replacement is performed if the deterioration level of a degradation process exceeds a certain threshold. A corrective replacement is performed if the system is down at inspection time. Under this maintenance strategy, the expected cost rate is obtained. Sensitivity analysis on the main parameters of the gamma process is performed.
Paper Structure (12 sections, 6 theorems, 80 equations, 8 figures, 8 tables)

This paper contains 12 sections, 6 theorems, 80 equations, 8 figures, 8 tables.

Table of Contents

  1. Introduction
  2. Arrival processes
  3. The stochastic process of growth
  4. Random effects
  5. Combined process of arrival and growth
  6. Time to the system failure
  7. Maintenance policy
  8. Expected downtime in a replacement cycle
  9. Numerical examples
  10. $\beta$ deterministic
  11. Random effects model
  12. Conclusions, future extensions and limitations of this work

Key Result

Lemma 1

Let $\lambda^*({\color{black} s})$, with ${\color{black} s} \geq 0$, be the random intensity function of the counting process $\left\{N^*({\color{black} s}), {\color{black} s} \geq 0\right\}$ having $N^*(0)=0$. Then

Figures (8)

  • Figure 1: Density function $f_{X_h(t)}$ for $\alpha=1$, $t=5$ and $\beta^{-1}$ uniform in $(1,b)$.
  • Figure 2: Expectation and variance of $X_h(t)$ versus $t$ for $\alpha=1$ and $\beta^{-1}$ uniform in $(1,b)$
  • Figure 3: $Var(X_h(t))/E(X_h(t))$ versus $t$ with $\beta^{-1}$ uniform in $(1,b)$
  • Figure 4: Trajectories and log likelihood
  • Figure 5: Trajectories and log likelihood
  • ...and 3 more figures

Theorems & Definitions (11)

  • Lemma 1
  • Example 1
  • Definition 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 1 more