Introduction

The terminology degradation process refers to many types of models in reliability, which correspond to various kinds of stochastic processes used for deterioration modeling. This book is restricted to the case of a continuous set of possible deterioration levels, to be opposed for instance to multi-state models, where only a finite number of deterioration levels is possible (see, for instance, [COM 99, HOU 99, AND 02] for reviews on these multi-state models with application in epidemiology). Also, only univariate models are considered, which means that one single measurable quantity is assumed to be observed over time. In this book, the terms degradation model or degradation process refer to this context.

Even within this apparently restrictive context, degradation models have been much studied for several decades. Nowadays, there exists a large number of publications on this topic in scientific journals. However, only a few books deal with it. Some monographs contain one single chapter on degradation data and cannot cover a wide range of degradation models. For instance, Chapter 13 in [MEE 98] deals with the degradation path model (see section I.3 below for details) while Chapter 11 in [NEL 02] and Chapter 3 in [BAG 04] focus on some accelerated degradation models. One can however quote [FIN 13], which is mostly devoted to shocks and burn-in models and does not cover the range of this book. Beside these monographs, some books gather some material from conferences in mathematical reliability (see [NIK 10] or [RYK 10], for instance). They clearly contain interesting papers on degradation models but they do not share the same unity as a monograph and they do not allow us either to go as deeply into these models. Other books exist that mostly focus on the study of maintenance policies and their optimization, see [NAG 07].

Consequently, a monograph uniquely devoted to this topic is lacking. This book aims at filling this gap. Well-known results together with new results are provided here in a unique rigorous mathematical framework. We hope that it will be useful for researchers and PhD students in applied mathematics, but also for research engineers dealing with degradation modeling and maintenance. It can also be used as a basis for teaching applied stochastic processes.

After this brief overview, we now go on with the core of this introduction. We first explain why it is interesting to study degradation models. A few real data sets are presented in the following for illustrative purposes. The most classical degradation models are then introduced, which are nearly all based on specific Lévy processes or on some of their extensions. Basic definitions and properties are next provided for such mathematical objects. Finally, an outline of the book is sketched in the last section.

I.1. From lifetime data to degradation path observations

Historically, the first reliability studies [BAR 65, BAR 75] were focused on lifetime data analysis, where equipment (or component) lifetimes were directly modeled through random variables. These studies belong to the so-called survival analysis that had (and still has) a major role in reliability engineering [MEE 98] but also in many other fields such as biology and health [KLE 03], demography, actuarial science, economics and social sciences (see, for instance, the bibliographical notes of Chapter 1 in [LAW 03] for further references).

However, purely lifetime-based reliability analysis is not always an effective means to analyze the available data. First, samples of lifetime data can be very small or highly censored. Indeed, observing an item up to failure can be very long and costly, especially for highly reliable products. For some applications, lifetime data are even not observed at all due to the severity of the failure (such as in nuclear engineering, for instance). Second, the development of on-line monitoring together with the increasing use of sensors for safety assessment now allow us to have more and more available information. These data collected on-line can correspond to some covariate process, which has some influence on the lifetime data. Most of the time, they are however the reflection of some deterioration (or degradation) mechanism, which cannot be handled with lifetime models. This includes, for instance, processes of wear, corrosion, crack growth, vibration, etc. Often, this degradation is measured through a scalar indicator, which tends to increase over time. This scalar indicator is hereafter called degradation measurement. In most cases, there exists a prescribed critical level above which a piece of equipment is considered not to be able to consistently perform its intended mission any more, either for safety or technical reasons. The time required to reach this critical level corresponds to the equipment lifetime, which of course remains of the first interest, even considering degradation data. However, the degradation data bring some additional information on the equipment status when compared to dealing only with failure data. The advantage of considering degradation data toward lifetime data has, for instance, been highlighted from an example by Meeker and Escobar [MEE 98, page 335].

From a probabilistic point of view, although lifetime data stand for a sample of non-negative random variables, successive degradation data represent observations of a stochastic process, whose parameters (eventually functional) can be fit from the data. A main interest is that once fitted, it next allows us to make some prediction over the future of the equipment. As an example, given its present deterioration level, the distribution of its (future) residual life can be derived (as well as many other quantities of interest in an industrial context). This is of major importance to define complex and powerful preventive maintenance policies that can now be based on the effective status of the equipment and on its known future (random) evolution. See [VAN 09] for a large survey on maintenance policies devoted to the gamma process (a specific degradation process studied in Chapter 2), with also some references on other processes such as the Wiener process (studied in Chapter 1). See also [AHM 12] and [JAR 06] for an overview on condition-based maintenance policies in a more general setting. Beyond the definition of efficient maintenance policies, a last interest of degradation models is that they can be used for test planning and burn-in modeling, see sections 5 and 6 in [YE 15] for a survey on these two topics as well as [FIN 13] for the second topic.

For a better understanding of which kind of data sets we are thinking about, we now introduce a few real data sets from the literature, which will be analyzed in the final chapter of the book, with the tools developed therein.

I.2. A few real data sets

The first data set is called Takeda device data in the literature and it has been first studied by Takeda and Suzuki [TAK 83] (see also [STI 89] or [LU 97], for instance). These authors have measured the performances of a certain kind of semiconductors, called metallized and oxidized semiconductors field-effect transistors. Such semiconductors are subjected to one particular type of degradation, namely, the hot carrier degradation. For more details, the reader could refer to the monograph by Leblebici and Kang [LEB 93], especially Chapters 1 and 2. The critical level is fixed at 15%. Five specimens were observed at 35 instants, giving a total of 170 increments. All the observation times (and thus, in particular, the first ones) are the same for all units. The data set contains positive (126), null (32) and negative (12) increments. The data has been reproduced in Figure I.1 in a log-log scale, together with the critical level (dot line). The critical level is assumed to be equal to 15% of transconductance degradation [LU 97]. Clearly, after a log-log transformation, degradation appears to be linearly increasing with respect to the time in average. We can notice that three units among the five specimens are unfailed at the end of the experience. Hence, an interesting problem is the estimation of the remaining lifetime distribution.

Figure I.1. Takeda device data

The second data set has been introduced by Meeker and Escobar [MEE 98], known in the literature as laser data. A set of 15 gallium arsenide laser devices is tested at 80° and periodically observed. More precisely, the performance characteristics of the devices are observed each 250 h until 4,000 h (observation times are identical for all units). A device is considered to be failed when its performance characteristic reaches the predefined critical level equal to 10. The data are plotted in Figure I.2. For this data set, we will be able to fit two different degradation parametric models. Hence, naturally, the problem of selection model arises: which model seems to be more suitable? After estimating the parameters for these two models, we will offer a discussion about this problem.

Figure I.2. Laser data

The third and last data set deal with crack growth. Hudak et al. [HUD 78] have observed 21 metallic specimens, each subjected to loading cycles with crack length recorded every 104 cycles. Initial crack length was 0.9 inches for each specimen. The average degradation is clearly nonlinear a contrario to the previous data set. The data are plotted in Figure I.3. Such a behavior is typical for crack growth (see, for instance, another famous data which is the Virkler data, [VIR 79]). Hence, the choice of a parametric degradation model is not obvious....