Failure Rate, Reliability & Probability

With adequate data, it can be shown that, on the average, a component fails
after a certain period of time. This is called the average failure rate and
is represented by **u** with units of faults/time. The easiest method for representing
failure probability of a component is its reliability, expressed as an exponential
(Poisson) distribution:

where R(t) is the reliability, i.e. the probability that the component will not fail within the time interval (0, t).

There are other distributions available to represent equipment failures, but these require more detailed information on the device and a more detailed analysis. For most situations the exponential distribution is adequate.

It is worthwhile to note that the above equation assumes a constant failure rate. As t increases, R goes to 0. The speed at which this occurs is dependent on the value of the failure rate u, i.e. the higher the failure rate, the faster the reliability decreases.

Once the reliability is defined, the failure probability (i.e. unreliability), P(t), follows:

The failure density function f(t) is defined as the derivative of the failure probability,

The area under the complete failure density function is unity.

The failure density function is used to determine the probability P, of at least one failure in the time period t0 to t1:

The integral represents the fraction of the total area under the failure density function between time t0 and t1.

Typical plots of the functions are shown in the Figure. Whereas the reliability of the device is initially unity, it falls off exponentially with time and asymptotically approaches zero. The failure probability, on the other hand, does the reverse. Thus new devices start life with high reliability and end with a high failure probability.

The time interval between 2 failures if the component is called the mean time between failures (MTBF) and is given by the first moment if the failure density function:

A considerable assumption in the exponential distribution is the assumption
of a constant failure rate. Real devices demonstrate a failure rate curve
that exhibits a typical **“bathtub” failure rate** as shown in the
Figure.

For a new device, the failure rate is initially high owing
to manufacturing defects, material defects, etc. This period is called **infant
mortality**. Following this is a period of relatively constant failure
rate. This is the period during which the exponential distribution is most
applicable.
Finally, as the device ages, the failure rate eventually increases. Click
here for more discussion on **Revealed vs. Unrevealed Failures**.

The Table lists typical failure rate data for a variety of types of process equipment. Large variations between these numbers and specific equipment can be expected.

However, this table demonstrates a very fundamental principle: the more complicated the device, the higher the failure rate. Thus switches and thermocouples have low failure rates; gas-liquid chromatographs have high failure rates.