Franck Bayle - Product Maturity 1

with
	Ea is the activation energy.
	Kb is the Boltzmann constant.

In most industrial applications, the focus is on the reliability observed on the ground, which must take into account the maintenance actions carried out. Maintenance can take several forms, depending on the level at which it is being performed (components, products, etc.). At the component level, maintenance is generally referred to as “perfect”, also known as “corrective maintenance”, since defective components are replaced with new ones.

At the product level, maintenance may be referred to as “preventive”. This is the case with cars, for example, where engine oil, various filters, etc., are changed on a regular basis without any failures having been observed. More generally, there is “minimal” maintenance at the product level, as replacing the defective component effectively restores the reliability of the product to the level it had before the failure.

Therefore, maintenance has an important effect on product reliability, as illustrated by the following figure.

Figure 1.4. Example of a car that has not been maintained. For a color version of this figure, see www.iste.co.uk/bayle/maturity1.zip

For further details on the effect of maintenance on reliability and its (rather difficult) modeling, the reader is invited to refer to Rigdon and Basu (2000), Gaudoin and Ledoux (2007) and Bayle (2019).

For most industrial applications, the objective of reliability is MTBF. There is much confusion surrounding this acronym; indeed, MTBF may signify:

– Mean time before failures:

In this case, failure instants were observed on “n” components (or products) assumed to be identical. This is equivalent to MTTF (Mean Time To Failure), as there are no maintenance actions. This can be illustrated by Figure 1.5.

– Mean time between failures:

This refers to the mean time between two consecutive failures. If there are two failures, this means there was a maintenance action, as illustrated in the following figure.

Figure 1.5. MTBF (mean time between failures)

Figure 1.6. MTBF (mean time between failures). For a color version of this figure, see www.iste.co.uk/bayle/maturity1.zip

When there are maintenance actions, the concept of failure rate has no meaning after the first failure. Hence, time between failures (TBF) and time to repair (TTR) are used. MTBF is therefore defined here by:

[1.8]

NOTE.– In practice, the TTR is often very short compared to the TBF; thus, the numerical expression of equation [1.8]can be written as:

[1.9]

Moreover, if the product is mature (no youth or aging failure), then .

According to these hypotheses, equation [1.8]can be written as:

[1.10]

This equation is often found in the literature but is only numerically true under certain hypotheses (exponential distribution), which must be verified.

Product specifications always include a reliability objective. There are two main industrial applications:

– The first is less common, requiring a probability of success. This probability, which is a function of the product use time, is therefore generally provided after the product becomes operational. The unilateral lower bound of this probability is generally used as the reliability objective. This is due to the fact that it applies to one or several products for which operational failure is to be excluded (e.g. Ariane rockets or certain military weapons).

– The second covers all other applications (avionics, motor vehicles, rail, etc.) where the mean number of failures is examined. This is the well-known MTBF.

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