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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.