# Failure rate

The failure rate is a parameter for the reliability of an object. It indicates how many objects fail on average in a period of time. It is given in 1 / time, i.e. failure per time period. If the failure rate is constant over time, in this case it is usually denoted by the constant λ , the reciprocal value is the mean service life MTTF , for repairable objects the mean time between two failures MTBF . A special unit for the failure rate is FIT Failure In Time with the unit “failures per 10 9 hours”.

In the event time analysis, the failure rate is also referred to in the statistics as the hazard rate ( English  hazard : danger, chance, risk). From this it is possible to determine the probability that a certain event (e.g. death of a person, sale of goods, decay of a radioactive element) will occur at a fixed point in time. One also speaks of a momentary tendency to change state .

Two for comparison failure rates, a quotient can be formed, the Hazard ratio ( English hazard ratio , in short HR ), which maps the risk of hazard rate compared to the other. This hazard ratio is used in particular in randomized controlled trials to compare two or more drugs.

## Failure rate and reliability

Apart from the sign , the failure rate is the quotient of the time derivative of the reliability and the reliability itself:

${\ displaystyle h (t) = - {\ frac {\ frac {dR (t)} {dt}} {R (t)}}}$.

Conversely, the reliability can be determined by the survival function , also known as the reliability function: ${\ displaystyle R (t)}$

${\ displaystyle R (t) = e ^ {- {\ int \ limits _ {0} ^ {t} h (x) dx}}}$

Example: If an object lasts with a constant failure rate, this corresponds to an exponential distribution of the reliability function and in this case , on average 100 hours, the failure rate is λ = 1 / 360000s. ${\ displaystyle h (t) = \ lambda}$

The inverse of the failure rate is the Mills quotient .

## Changes in failure rate

The failure rate initially depends on whether the object is in use or not. For example, the failure rate per operating hour is given for motors. The failure rate strongly depends on the environment, in particular on the temperature. According to the RGT rule , the failure rate doubles for a temperature increase of around 10 ° C. Temperature cycles (hot-cold) massively increase the failure rate. Vibrations, radiation (sunlight, cosmic radiation), moisture or chemical substances (e.g. salty air) also increase the failure rate. This is consciously exploited in aging tests such as the Highly Accelerated Life Test .

The failure rate also depends on the age of the object. The failure rate typically follows a bathtub curve : At the beginning of life, the failure rate is high due to "teething troubles": production errors and switch-on stress. Objects that have survived this phase then initially show a lower failure rate. For this reason, objects - especially in electronics - are subjected to temperature stress after manufacture before testing in order to read out objects that have already passed their teething troubles (“burn-in”).

After that, the failure rate remains constant for quite a long time, this is the bottom of the bathtub. This constant failure rate is the basis of most reliability calculations because it is mathematically easy to handle.

With increasing age, the failure rate increases again as a result of "old age diseases": mechanical wear, chemical decomposition of the materials, insulation breakdown in electrical systems, the effect of UV radiation or neutron bombardment on the strength of the material.

After all, the failure rate depends on the quality of maintenance.

## Determination of the failure rate

Failure rate measurements on incandescent lamps
above: curve of the functioning specimens over time
below : failure rate, this results in a bathtub profile of failure probability that is also typical for many other products

The failure rate cannot be measured on a single object. It is estimated from observations on a large number of identical objects. In such a statistical experiment the empirical distribution function of the service life is determined. The empirical distribution function is a step function with one step for each determined failure time.

The failure rate at a certain time is then given by the number of objects that fail in a certain time interval (e.g. one day) divided by the number of good objects at the beginning of the time interval.

For example 10,000 light bulbs are measured (picture). On the 19th day, 9,600 bulbs remained, and that day five bulbs failed. So the failure rate on day 19 was 5/9600/24 ​​= 21.7 per million hours = 21,700 FIT.

From a statistical point of view, it is of equal importance whether the failure rate is specified in failure per hour of a certain object or in number of failed objects per hour of a large number.

This measurement is often carried out under increased temperature stress and, in particular, under temperature cycles or under irradiation in order to shorten the service life and get results more quickly.

This can be used to create catalogs of the failure rate of the components, such as B. the US Forces MIL-HDBK-217. The failure rates contained therein are given for different areas of application (buildings, vehicles, ships, helicopters, ...) and temperatures.

The engineers can also correct these failure rates or estimate them from the experience of the repair shop.

Mathematical models can also predict the failure rate, e.g. B. by calculating crack growth on turbine blades.

## Systems of objects

In a system of objects, the failure rate of the system is calculated as the sum of the failure rate of the individual elements. It is assumed that the loss of any element will cause the system to fail, which is not the case if the system contains redundancy (see MTBF).

For example, a flashing lamp consists of

• 20 resistors: 20 x 0.1 FIT
• 3 transistors: 3 x 1 FIT
• 2 capacitors: 2 x 0.5 FIT
• 1 battery: 200 FIT.

The total failure rate is the sum of all failure rates and thus 206 FIT. The mean lifespan is therefore 554 years. However, this value for the mean service life only applies provided that the battery is replaced regularly: The battery has a small failure rate at the beginning, but this increases sharply with increasing age.

## Connections

If the probability density for failure is at the time , then the function determines ${\ displaystyle f (t)}$${\ displaystyle t}$

${\ displaystyle h (t) = {\ frac {f (t)} {1-F (t)}}}$

with the lifetime as a real variable, the failure rate at a point in time t. The failure probability is: ${\ displaystyle t}$${\ displaystyle \ lambda}$${\ displaystyle F (t)}$

${\ displaystyle F (t) = \ int \ limits _ {0} ^ {t} f (\ tau) \, \ mathrm {d} \ tau}$.

Alternatively, the failure rate in the context of survival function can be expressed as: ${\ displaystyle R (t) \;}$

${\ displaystyle h (t) = {\ frac {f (t)} {1-F (t)}} = {\ frac {f (t)} {R (t)}} = {\ frac {{\ frac {\ mathrm {d}} {\ mathrm {d} t}} F (t)} {R (t)}} = {\ frac {{\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ left (1-R (t) \ right)} {R (t)}} = - {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ ln \ left (R (t) \ right)}$.

This results in:

${\ displaystyle h (t) = - {\ frac {\ mathrm {d}} {\ mathrm {d} t}} \ ln (1-F (t))}$

and thus

${\ displaystyle F (t) = 1-e ^ {- \ int \ limits _ {0} ^ {t} h (\ tau) \, \ mathrm {d} \ tau}}$.
Different failure rates in the exponential distribution

For the exponential distribution (which is important in event-time analysis)

${\ displaystyle f (t) = \ lambda e ^ {- \ lambda t}}$,

and the following relationship applies:

${\ displaystyle F (t) = \ int _ {0} ^ {t} \ lambda e ^ {- \ lambda \ tau} \, \ mathrm {d} \ tau = 1-e ^ {- \ lambda t}}$.

The exponential distribution thus results in a failure rate that is constant over time:

${\ displaystyle h (t) = {\ frac {f (t)} {R (t)}} = {\ frac {\ lambda e ^ {- \ lambda t}} {e ^ {- \ lambda t}} } = \ lambda}$.

## Lending

The default rate is determined by the classified loans, multiplied by their probability of default .

## literature

• Arno Meyna, Bernhard Pauli: Reliability Technology. Quantitative assessment procedures . 2nd Edition. Hanser, 2010, ISBN 978-3-446-41966-7 .