**Reliability metrics**

### Failure rate

Reliability can be quantified as MTBF (Mean Time Between Failures) for a repairable product and as MTTF (Mean Time To Failure) for a non-repairable product.

According to the theory behind the statistics of confidence intervals, the statistical average becomes the true average as the number of samples increase. So, a power supply with an MTBF of 50,000 hours does not mean that the power supply should last for an average of 50,000 hours because the MTBF of 50,000 hours, or 1 year for 1 module, becomes 50,000/2 for two modules and 50,000/4 for four modules. It is only when all the parts fail with the same failure mode that MTBF converges to MTTF.

If the MTBF is known, one can calculate the failure rate (l) as the inverse of the MTBF. The formula for l is:

Once a MTBF is calculated, what is the probability that any one particular module will be operational at time equal to the MTBF? For electronic components we have the following equation:

But when t = MTBF

This tells us that the

__probability that any one particular module will survive to its calculated MTBF is only 36.8%, i.e., there is 63.2% probability that a single device will break before the MTBF!__

### Bathtub curve

Over many years, and across a wide variety of mechanical and electronic components and systems, people have calculated empirical population failure rates as units age over time and repeatedly obtained a graph such as shown below. Because of the shape of this failure rate curve, it has become widely known as the "Bathtub" curve.

This curve (in blue) is widely used in reliability engineering as describing a particular form of the hazard function which comprises three parts:

- The first part is a decreasing failure rate, known as early failures.
- The second part is a constant failure rate, known as random failures.
- The third part is an increasing failure rate, known as wear-out failures.

## Reliability examples

Example: Suppose 10 devices are tested for 500 hours. During the test 2 failures occur.

The estimate of the MTBF is:

Whereas for MTTF is:

Another example: A router has an MTBF of 100,000 hours; what is the annual reliability? Annual reliability is the reliability for one year or 8,760 hours.

This means that the probability of no failure in one year is 91.6%; or, 91.6% of all units will survive one year.

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