Modeling Remaining Useful Life Dynamics in Reliability Engineering  book cover
1st Edition

Modeling Remaining Useful Life Dynamics in Reliability Engineering

  • Available for pre-order on May 16, 2023. Item will ship after June 6, 2023
ISBN 9781032168593
June 6, 2023 Forthcoming by CRC Press
200 Pages 60 B/W Illustrations

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Book Description

This book applies traditional reliability engineering methods to prognostics and health management (PHM), looking at remaining useful life (RUL) and its dynamics, to enable engineers to effectively and accurately predict machinery and systems useful lifespan. One of the key tools used in defining and implementing predictive maintenance policies is the RUL indicator. However, it is essential to account for the uncertainty inherent to the RUL, as otherwise predictive maintenance strategies can be incorrect. This can cause high costs or, alternatively, inappropriate decisions. Methods used to estimate RUL are numerous and diverse and, broadly speaking, fall into three categories: model-based, data-driven, or hybrid, which uses both. The author starts by building on established theory and looks at traditional reliability engineering methods through their relation to PHM requirements and presents the concept of RUL loss rate. Following on from this, the author presents an innovative general method for defining a nonlinear transformation enabling the mean residual life to become a linear function of time. He applies this method to frequently encountered time-to-failure distributions, such as Weibull and gamma, and degradation processes. Latest research results, including the author’s (some of which are previously unpublished), are drawn upon and combined with very classical work. Statistical estimation techniques are then presented to estimate RUL from field data, and risk-based methods for maintenance optimization are described, including the use of RUL dynamics for predictive maintenance.

The book ends with suggestions for future research, including links with machine learning and deep learning.

The theory is illustrated by industrial examples. Each chapter is followed by a series of exercises.


  • Provides both practical and theoretical background of RUL
  • Describes how the uncertainty of RUL can be related to RUL loss rate
  • Provides new insights into time-to-failure distributions
  • Offers tools for predictive maintenance

This book will be of interest to engineers, researchers and students in reliability engineering, prognostics and health management, and maintenance management.

Table of Contents





Chapter 1 Introduction


Chapter 2 Reminder of Reliability Engineering Fundamentals

2.1 Reliability, Failure Rate, RUL, MRL, and RUL Loss Rate

2.1.1 Reliability, Failure, and Failure Rate

2.1.2 RUL, MRL, MTTF, and RUL Loss Rate

2.2 Fundamental Relation between MRL, Reliability Function and Failure Rate

2.3 Confidence Interval for RUL Illustrations on Special Cases

2.4 Exercises


Chapter 3 The RUL Loss Rate for a Special Class of Time-to-Failure Distributions: MRL Linear Function of Time

3.1 Characterizing the Special Family of Distributions

3.2 Limiting Cases

3.2.1 Exponential Distribution

3.2.2 Dirac Distribution

3.2.3 Uniform Distribution

3.3 Synopsis

3.4 RUL Distribution, Coefficient of Variation of TTF and RUL

3.4.1 Coefficient of Variation of the Time to Failure

3.4.2 Coefficient of Variation of the RUL

3.5 Confidence Interval for RUL and Relation to RUL Loss Rate

3.6 Higher-Order Moments and Moment-Generating Function

3.6.1 Moment-Generating Function

3.6.2 Moment-Generating Function for Special TTF Family

3.7 Cumulative Hazard Function

3.8 Exercises


Chapter 4 Generalization to an MRL Piecewise-Linear Function of Time

4.1 Reliability Function, MRL, and Failure Rate

4.2 Exercises


Chapter 5 Generalization to a Wide Class of Lifetime Distributions

5.1 Introduction: Generalization Method

5.2 Nonlinear Time Transformation

5.3 Confidence Interval for RUL

5.4 Application to Weibull Distribution

5.4.1 Derivation of the Nonlinear Transformation

5.4.2 Derivation of Confidence Intervals for the RUL

5.5 Application to Gamma Distribution

5.6 Application to the Lognormal Distribution

5.7 Application to the Pareto Distribution

5.8 Application to Continuous Degradation Processes

5.8.1 Problem Statement

5.8.2 RUL for Wiener Process with Drift

5.8.3 RUL for Gamma Process

5.9 Exercises


Chapter 6 Properties of the dg Metric

6.1 Introduction

6.2 Derivative of g(t)

6.3 Differential Equation for MRL

6.3.1 Example: The Rayleigh Distribution

6.3.2 Other Example: The Gamma Distribution

6.4 Second Derivative of g(t)

6.4.1. Weibull Distribution

6.4.2. Gamma Distribution

6.5 Upper Bound for the Average RUL Loss Rate

6.6 Exercises


Chapter 7 Multiple Failure or Degradation Modes

7.1 Introduction

7.2 General Formulation

7.3 Illustration in Special Case

7.4 Exercises


Chapter 8 Statistical Estimation Aspects

8.1 Introduction

8.2 Nonparametric Estimation

8.3 Parametric Estimation—Illustration on Two Case Studies

8.3.1 Parametric Estimation: The Maximum Likelihood Estimation Method Maximum Likelihood Estimator Desirable Properties of the Maximum Likelihood Estimator Derivation of a Confidence Interval

8.3.2 First Case Study: Light-Emitting Diodes (LEDs)

8.3.3 Second Case Study: Redundant System

8.4 Surrogate Model for the g(t) Transformation

8.5 Bayesian Estimation

8.6 Exercises


Appendix 8 A: ARAMIS Data Challenge

Chapter 9 Implications for Maintenance Optimization

9.1 Introduction

9.2 Maintenance Decisions: Balancing Costs and Risks

9.3 Quantifying Costs and Risks

9.3.1 Rejuvenation and Maintenance Efficiency

9.3.2 Minimal Maintenance (ABAO)

9.3.3 Perfect Maintenance (AGAN)

9.4 Predictive Maintenance

9.5 Exercises


Chapter 10 Advanced Topics and Further Research

10.1 The Gini Index

10.2 Entropy and the k Parameter

10.2.1 Entropy

10.2.2 Differential Entropy Differential Entropy as a Function of k and μ Parameters

10.3 Perspectives for Future Research

10.3.1. Complex Systems

10.3.2. Dynamic Maintenance Policy

10.3.3. Signal Processing

10.3.4. Physics and Machine Learning



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Pierre Dersin graduated from the Massachusetts Institute of Technology (MIT) with a Ph.D. in Electrical Engineering after receiving a Master’s degree in Operations Research also from MIT. He also holds math & E.E.degrees from Université Libre de Bruxelles ( Belgium).

Since 2019, he has been Adjunct Professor at Luleå University of Technology (Sweden) in the Operations & Maintenance Engineering Division.

In January 2022, he founded a small consulting company, Eumetry sas, in Louveciennes, France, in the fields of RAMS, PHM and AI, just after retiring from ALSTOM where he had spent more than 30 years.

With Alstom, he was RAM (Reliability-Availability-Maintainability) Director from 2007 to 2021 and founded the "RAM Center of Excellence".In 2015, he launched the predictive maintenance activity and became PHM (Prognostics & Health Management) Director of ALSTOM Digital Mobility, and then ALSTOM Digital & Integrated Systems, St-Ouen, France.

Prior to joining Alstom, he worked in the USA on the reliability of large electric power networks, as part of the Large Scale System Effectiveness Analysis Program sponsored by the US Department of Energy, from MIT and Systems Control, Inc, and later, with FABRICOM (Suez Group), on fault detection and diagnostics in industrial systems.

He has contributed a number of communications and publications in scientific conferences and journals in the fields of RAMS, PHM, AI, automatic control and electric power systems (including Engineering Applications of AI, IEEE Transactions on Automatic Control, IEEE Transactions on Power Apparatus & Systems, ESREL, RAMS Symposia, French Lambda-Mu Symposia, the 2012 IEEE-PHM Conference, the 2014 European Conference of the PHM Society ( keynote speaker) and WSC 2013).

He serves on the IEEE Reliability Society AdCom and the IEE Digital Reality Initiative,and chairs the IEEE Reliability Society Technical Committee on Systems of Systems. He is a contributor of four chapters in the "Handbook of RAMS in Railways: Theory & Practice" (CRC Press, Taylor & Francis),2018, including a chapter on "PHM in Railways ‘(Ch.6). In January 2020, he was awarded the Alan 0. Plait Award for the best tutorial at the RAMS conference, " Designing for Availability in Systems, and Systems of Systems".

His main research interests focus on the confluence between RAMS and PHM, as well as complex systems resilience and asset management.