# Mean time between failures

Mean time between failures

Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a system during operation.[1] MTBF can be calculated as the arithmetic mean (average) time between failures of a system. The MTBF is typically part of a model that assumes the failed system is immediately repaired (MTTR), as a part of a renewal process. This is in contrast to the mean time to failure (MTTF), which measures average time to failures with the modeling assumption that the failed system is not repaired (infinite repair rate).

The definition of MTBF depends on the definition of what is considered a system failure. For complex, repairable systems, failures are considered to be those out of design conditions which place the system out of service and into a state for repair. Failures which occur that can be left or maintained in an unrepaired condition, and do not place the system out of service, are not considered failures under this definition.[2] In addition, units that are taken down for routine scheduled maintenance or inventory control, are not considered within the definition of failure.

## Overview

For each observation, downtime is the instantaneous time it went down, which is after (i.e. greater than) the moment it went up, uptime. The difference (downtime minus uptime) is the amount of time it was operating between these two events.

MTBF value prediction is an important element in the development of products. Reliability engineers / design engineers, often utilize Reliability Software to calculate products' MTBF according to various methods/standards (MIL-HDBK-217F, Telcordia SR332, Siemens Norm, FIDES,UTE 80-810 (RDF2000), etc.). However, these "prediction" methods are not intended to reflect fielded MTBF as is commonly believed. The intent of these tools is to focus design efforts on the weak links in the design.

## Formal definition of MTBF

By referring to the figure above, the MTBF is the sum of the operational periods divided by the number of observed failures. If the "Down time" (with space) refers to the start of "downtime" (without space) and "up time" (with space) refers to the start of "uptime" (without space), the formula will be:

$\text{Mean time between failures} = \text{MTBF} =\frac{\sum{(\text{start of downtime} - \text{start of uptime})}}\text{number of failures}. \!$

The MTBF is often denoted by the Greek letter θ, or

$\text{MTBF} = \theta. \!$

The MTBF can be defined in terms of the expected value of the density function ƒ(t)

$\text{MTBF} = \int_0^\infty tf(t)\, dt \!$

where ƒ is the density function of time until failure – satisfying the standard requirement of density functions –

$\int_0^\infty f(t)\, dt=1. \!$

## Variations of MTBF

There are many variations of MTBF, such as mean time between system aborts (MTBSA) or mean time between critical failures (MTBCF) or mean time between unit replacement (MTBUR). Such nomenclature is used when it is desirable to differentiate among types of failures, such as critical and non-critical failures. For example, in an automobile, the failure of the FM radio does not prevent the primary operation of the vehicle. Mean time to failure (MTTF) is sometimes used instead of MTBF in cases where a system is replaced after a failure, since MTBF denotes time between failures in a system which is repaired.

## Notes

1. ^ Jones, James V., Integrated Logistics Support Handbook, page 4.2
2. ^ Colombo, A.G., and Sáiz de Bustamante, Amalio: Systems reliability assessment – Proceedings of the Ispra Course held at the Escuela Tecnica Superior de Ingenieros Navales, Madrid, Spain, September 19–23, 1988 in collaboration with Universidad Politecnica de Madrid, 1988

## References

• Jones, James V., Integrated Logistics Support Handbook, McGraw–Hill Professional, 3rd edition (June 8, 2006), ISBN 0071471685

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