Introduction

In today's world, the word "reliability" is evermore present in current conversations. Of course, in technical fields such as aeronautics, railways, space and so on, reliability is already widely used because it is a major parameter whether for performance, security, cost or brand image considerations.

For several years now, electronics and computers have entered the daily lives of human beings and televisions, mobile phones, tablets, computers and smartphones and so are present in many family homes. However, in recent years, they have also invaded the automotive sector and home appliances, and this will only increase in the coming years with the development of autonomous cars and trains, more electric aircraft and what do I know what the future holds? It is clear that security issues that arise are very important and therefore involve very thorough reliability analyses. However, the word "reliability" has entered deeply into our traditions. Do we not say about a person, an organization "that she or he is not reliable"?

We could then expect it to be a proper subject in the engineering sciences such as electronics, computer science and so on. Well, no, you will not find anything of that except for some universities where reliability of operation, reliability, safety, availability and maintainability, or the statistics applied to reliability are taught in a very theoretical way. And the situation is similar for most industries where reliability in terms of human or financial resources is poorly or at least superficially developed compared to the issues mentioned above.

We could also expect this word to be clearly defined, unequivocal, and not dependent on any particular activity or person. Of course, you will find the following definition in [AFN 11]:

"Ability of an entity to perform a required function under specified conditions for a given time interval".

We can be quite surprised at such a definition in the sense that, for most practical applications, the notions of failure rate and/or MTBF are used instead. This is particularly visible, for example, in the FIDES 2009 RevA guide [AFN 11] where we find this same definition while it is not used in the 465 pages of this methodological guide.

In the same way, these two reliability indicators, which are the failure rate and the MTBF, are often misunderstood, have several definitions which are different from each other, and use hypotheses to allow their estimation which are not always verified or even known by the user.

For a long time, human beings saw the universe as a fixed and purely deterministic world. Classical physics (Laplace) thought that it was possible to measure the properties of a system such as the temperature or speed of a body with all the desired precisions. It was enough to know the initial conditions and we could make any prediction.

The beginning of the 20th Century completely changed this vision. Indeed, in the world around us, the elementary particles that compose it and of which we are made behave randomly. Maxwell and Boltzmann were the first to exhibit this type of character with the famous velocity distribution of the molecules that bears their name. Considering the number of particles in a volume of a liter of air, for example (~6 × 109), no calculation is feasible individually. If this random nature in the current world can be seen as deterministic from calculations based on the average, this is generally not possible in reliability because of the number of failures that can be observed from a few units to a few tens at most. The randomness of the moments of failure must be preserved. Mathematically, it complicates things. For example, while it is easy to manipulate real variables, even a simple operation like addition becomes much more complex for a random variable since the sum of two random variables is defined as the convolution product of their probability density.

Another paradox is that the more reliable a product is, the more difficult it is to show this. Indeed, if the product is very reliable, the time to observe failures will be very long so it is very expensive or even completely unrealistic to demonstrate its reliability. This is further exacerbated by the speed of evolution of component technology, especially from an electronic point of view. For the integrated circuits for which it is the most significant, Moore's law [MOO 65], which states that the number of elementary transistors double every 18 months, stated in the 1960s, is still valid today. What is therefore paradoxical is that the new technologies of such components, with a different reliability, take less time to develop than to estimate the reliability of the current technology.

To conclude, therefore, we see that reliability is becoming increasingly important in the modern world while it remains a science that is not really considered as a subject to teach and is very often considered "obscure" or even overused in the industrial field.

I have heard many times:

"Reliability is not an exact science?"

"Come on, consult your crystal ball and tell me how much the MTBF of my product is."

The first assertion shows how little the level of scientific knowledge remains. Of course, reliability is not an exact science, but no other science is. Even mathematics, which often wrongly has this reputation, cannot claim this. It suffices to know Godel's incompleteness theorem to be convinced of this.

The second assertion borders on the ridiculous.

Purpose of this book

The main purpose of this book is to provide engineers with sound bases in the field of reliability. We will therefore mainly focus on the intrinsic reliability of components and in particular its estimation when the component is subjected to aging mechanisms without maintenance. This will be the subject of Chapter 1.

In this chapter, we will expose the main mathematical functions used for systems without maintenance, especially what they represent physically and why they were proposed. More mathematics, some of you will complain! It must be understood that mathematics is not a goal in itself; it is not there to complicate things but, on the contrary, to simplify them. In fact, nature around us is extremely complex and the only way to make predictions is to design a mathematical model representing physical reality with a margin of error that seems acceptable to us.

On the contrary, you will find in this book some mathematical demonstrations that can be too numerous and boring. I only propose demonstrations that can be tackled by an engineer and that have given me a broad understanding of the studied issue.

In Chapter 2, we will discuss a very important and often poorly understood part of the effect of maintenance on the reliability of a component with aging mechanisms that can be observed. This part is difficult from a mathematical point of view and I already want to apologize for the theoretical formalism which will certainly lack rigor. You will not find any notion of martingales, Borelian space, but this is not the main purpose in this book and I advise the more rigorous reader to refer to very enriching academic books such as [GAU 07] or [RIG 00].

This chapter will therefore provide essential mathematical concepts, although these often lead to non-explicit solutions and require the use of numerical resolutions. This last point, for the philosophy of this book, is boring because it is intended to be quite accessible to the common reliability engineer in the industrial field. Thus, we will offer the reader mathematical approximations depending on the industrial context to provide explicit approximate solutions, allowing them to better understand the impact of manipulated physical quantities.

In Chapter 3, we will see how these different types of maintenance for aging mechanisms are applied. In particular, the approximations proposed in Chapter 2 will be bases to the examples of proposed applications.

In Chapter 4, we will discuss the consequences of the type of maintenance for aging mechanisms from a "reliability" point of view in the context of aging mechanisms. It is clear that maintenance has a significant impact on it and doing no maintenance on our car, for example, can convince us of this. Very quickly, it will no longer be functional.

In Chapter 5, we will discuss the consequences of maintenance from a "spare parts inventory" perspective in the context of aging mechanisms. We will see that some types of maintenance, if advantageous in terms of reliability, can be very expensive. We will recall the modeling used for accidental type failures and will detail how this calculation can be performed for aging mechanisms.

Finally, in Chapter 6 by in Chapter 7, we will discuss the "security" aspects. Already very present in certain industrial fields such as aeronautics and nuclear, we saw in the introduction of this book that these aspects are already present today and even more in the near future for the automotive, railway industries and at a lower level for all home automation and appliances. Aging mechanisms and the impact of maintenance are not dealt with in the aeronautics field.

Finally, we propose in Chapter 7 an optimization of the maintenance strategy in case of wear-out failure mechanisms.

The hypothesis of an exponential distribution for each mechanism is regularly used. More serious, the impact of maintenance is not taken into account correctly even in reference documents such as...