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Assessment of the maintenance cost and analysis of availability measures in a finite life cycle for a system subject to competing failures

Nuria Caballé, Inma T. Castro

TL;DR

This work addresses maintenance optimization for a finite-horizon system subject to two dependent failure modes: gradual degradation modeled by a gamma process and sudden shocks modeled by a DSPP whose intensity depends on the degradation level. A condition-based maintenance policy with periodic inspections is analyzed, and recursive Markov renewal equations are developed to compute the expected transient cost, its variance, and key performance measures such as availability, reliability, and interval reliability. The study compares recursive methods to Monte Carlo simulations, identifies optimal inspection intervals $T$ and preventive thresholds $M$, and demonstrates that the optimal policy yields substantial availability and reliable operation within the life cycle. The results offer a practical framework for transient reliability and cost analysis in finite horizons, with potential extensions to more complex degradation mechanisms and dependence structures.

Abstract

This paper deals with the assessment of the performance of a system under a finite planning horizon. The system is subject to two dependent causes of failure: internal degradation and sudden shocks. We assume that internal degradation follows a gamma process. When the deterioration level of the degradation process exceeds a predetermined value, a degradation failure occurs. Sudden shocks arrive at the system following a doubly stochastic Poisson process (DSPP). A sudden shock provokes the total breakdown of the system. A condition-based maintenance (CBM) with periodic inspection times is developed. To evaluate the maintenance cost, recursive methods combining numerical integration and Monte Carlo simulation are developed to evalute the expected cost rate and its standard deviation. Also, recursive methods to calculate some transient measures of the system are given. Numerical examples are provided to illustrate the analytical results.

Assessment of the maintenance cost and analysis of availability measures in a finite life cycle for a system subject to competing failures

TL;DR

This work addresses maintenance optimization for a finite-horizon system subject to two dependent failure modes: gradual degradation modeled by a gamma process and sudden shocks modeled by a DSPP whose intensity depends on the degradation level. A condition-based maintenance policy with periodic inspections is analyzed, and recursive Markov renewal equations are developed to compute the expected transient cost, its variance, and key performance measures such as availability, reliability, and interval reliability. The study compares recursive methods to Monte Carlo simulations, identifies optimal inspection intervals and preventive thresholds , and demonstrates that the optimal policy yields substantial availability and reliable operation within the life cycle. The results offer a practical framework for transient reliability and cost analysis in finite horizons, with potential extensions to more complex degradation mechanisms and dependence structures.

Abstract

This paper deals with the assessment of the performance of a system under a finite planning horizon. The system is subject to two dependent causes of failure: internal degradation and sudden shocks. We assume that internal degradation follows a gamma process. When the deterioration level of the degradation process exceeds a predetermined value, a degradation failure occurs. Sudden shocks arrive at the system following a doubly stochastic Poisson process (DSPP). A sudden shock provokes the total breakdown of the system. A condition-based maintenance (CBM) with periodic inspection times is developed. To evaluate the maintenance cost, recursive methods combining numerical integration and Monte Carlo simulation are developed to evalute the expected cost rate and its standard deviation. Also, recursive methods to calculate some transient measures of the system are given. Numerical examples are provided to illustrate the analytical results.
Paper Structure (11 sections, 7 theorems, 88 equations, 12 figures, 5 tables)

This paper contains 11 sections, 7 theorems, 88 equations, 12 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Framework of the problem
  3. Expected transient cost analysis
  4. Performance measures of the system
  5. Numerical examples
  6. Asymptotic expected cost rate analysis
  7. Expected transient cost rate analysis for a fixed $T$
  8. Analysis of the expected cost rate in the life cycle for a fixed $M$
  9. Transient two-dimensional expected cost rate analysis
  10. Performance measures for optimal values of $T$ and $M$
  11. Conclusions and further works

Key Result

Theorem 1

For $t<T$, the expected transient cost, $E\left[C_T^M(t)\right]$ is given by where $\bar{F}_{\sigma_{M_s}}(u)$ and $f_{\sigma_{M_s}}(u)$ denote the survival function and density function of $\sigma_{M_s}$ given by (distribucion_sigmaz), $\bar{F}_{\sigma_{L}-\sigma_{M_s}}$ and $I(x,y)$ the survival functions given by (supervivencia_sigma_z2_sigma_z1) and (funcionI), and $f_1( where with initial c

Figures (12)

  • Figure 3: A realization of a renewal cycle sequence.
  • Figure 5: Mesh and contour plots for the asymptotic expected cost rate.
  • Figure 6: Expected cost rate in the life cycle for different values of $M$.
  • Figure 7: Expected transient cost rate and asymptotic expected cost rate at time $t_f=50$$t.u.$ for different values of $M$.
  • Figure 8: Expected cost rate in the life cycle and standard deviation versus $M$.
  • ...and 7 more figures

Theorems & Definitions (12)

  • Theorem 1
  • proof
  • Corollary 1
  • Theorem 2
  • proof
  • Corollary 2
  • Theorem 3
  • proof
  • Theorem 4
  • proof
  • ...and 2 more