# Failure rate and mtbf relationship

### The Bathtub Curve and Product Failure Behavior (Part 2 of 2)

failure rate and MTBF (Mean Time Between Failures) data for electronic and The following relationship holds true since a component or system must either. In a situation like this we can say that widgets have a constant failure rate (in this case, ), which results in an exponential failure distribution. The "density. Eq. () is the formal relation between λ(t) and f(t). In general, one .. As it turns out, for constant failure rate, the MTBF for the “maintained” device is. MTBF = 1/.

However, in each application it's important to evaluate whether the devices in question really do have constant failure rates. If they don't, then use of the exponential distribution may be misleading.

Thus, we have Of course, if t0 equals 0 the first term is simply 1, and we have the cumulative failure distribution which is the probability that a functioning widget will fail at any time during the next t units of time. By the way, for any failure distribution not just the exponential distributionthe "rate" at any time t is defined as In other words, the "failure rate" is defined as the rate of change of the cumulative failure probability divided by the probability that the unit will not already be failed at time t.

## Failure rate

Now, we might ask what is the mean time to fail for a device with an arbitrary failure density f t? We just need to take the weighted average of all time values from zero to infinity, weighted according to the density.

Thus the mean time to fail is Of course, the denominator will ordinarily be 1, because the device has a cumulative probability of 1 of failing some time from 0 to infinity. Thus it is a characteristic of probability density functions that the integrals from 0 to infinity are 1. Thus the mean time to fail for an exponential system is the inverse of the rate.

Now let's try something a little more interesting. Suppose we manufacture a batch of dual-redundant widgets, hoping to improve their reliability in service. A dual-widget is said to be failed only when both sub-widgets have failed.

### Annualized failure rate - Wikipedia

What is the failure density for a dual-widget? This can be derived in several different ways, but one simple way is to realize that the probability of both sub-widgets being failed by time t is so this is the cumulative failure distribution F t for dual-widgets.

This is often prohibitively expensive or impractical, so that the previous data sources are often used instead. Units[ edit ] Failure rates can be expressed using any measure of time, but hours is the most common unit in practice.

Other units, such as miles, revolutions, etc. The Failures In Time FIT rate of a device is the number of failures that can be expected in one billion device-hours of operation.

## Annualized failure rate

This term is used particularly by the semiconductor industry. Additivity[ edit ] Under certain engineering assumptions e. This permits testing of individual components or subsystemswhose failure rates are then added to obtain the total system failure rate.

A test can be performed to estimate its failure rate. Ten identical components are each tested until they either fail or reach hours, at which time the test is terminated for that component.

The level of statistical confidence is not considered in this example. The results are as follows: Estimated failure rate is.