Mean time between failures (MTBF) is the predicted elapsed time between inherent failures of a mechanical or electronic system during normal system operation. MTBF can be calculated as the arithmetic mean (average) time between failures of a system. The term is used for repairable systems while mean time to failure (MTTF) denotes the expected time to failure for a non-repairable system. The definition of MTBF depends on the definition of what is considered a 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. In addition, units that are taken down for routine scheduled maintenance or inventory control are not considered within the definition of failure. The higher the MTBF, the longer a system is likely to work before failing.
Overview Mean time between failures (MTBF) describes the expected time between two failures for a repairable system. For example, three identical systems starting to function properly at time 0 are working until all of them fail. The first system fails after 100 hours, the second after 120 hours and the third after 130 hours. The MTBF of the systems is the average of the three failure times, which is 116.667 hours. If the systems were non-repairable, then their MTTF would be 116.667 hours. In general, MTBF is the "up-time" between two failure states of a repairable system during operation as outlined here:
For each observation, the "down time" is the instantaneous time it went down, which is after (i.e. greater than) the moment it went up, the "up time". The difference ("down time" minus "up time") is the amount of time it was operating between these two events. By referring to the figure above, the MTBF of a component is the sum of the lengths of the operational periods divided by the number of observed failures:
MTBF = ∑ ( start of downtime − start of uptime ) number of failures . {\displaystyle {\text{MTBF}}={\frac {\sum {({\text{start of downtime}}-{\text{start of uptime}})}}{\text{number of failures}}}.}
In a similar manner, mean down time (MDT) can be defined as
MDT = ∑ ( start of uptime − start of downtime ) number of failures . {\displaystyle {\text{MDT}}={\frac {\sum {({\text{start of uptime}}-{\text{start of downtime}})}}{\text{number of failures}}}.}
Mathematical description The MTBF is the expected value of the random variable T {\displaystyle T} indicating the time until failure. Thus, it can be written as
MTBF = E { T } = ∫ 0 ∞ t f T ( t ) d t {\displaystyle {\text{MTBF}}=\mathbb {E} \{T\}=\int _{0}^{\infty }tf_{T}(t)\,dt}
where f T ( t ) {\displaystyle f_{T}(t)} is the probability density function of T {\displaystyle T} . Equivalently, the MTBF can be expressed in terms of the reliability function R T ( t ) {\displaystyle R_{T}(t)} as
MTBF = ∫ 0 ∞ R ( t ) d t {\displaystyle {\text{MTBF}}=\int _{0}^{\infty }R(t)\,dt} . The MTBF and T {\displaystyle T} have units of time (e.g., hours). Any practically-relevant calculation of the MTBF assumes that the system is working within its "useful life period", which is characterized by a relatively constant failure rate (the middle part of the "bathtub curve") when only random failures are occurring. In other words, it is assumed that the system has survived initial setup stresses and has not yet approached its expected end of life, both of which often increase the failure rate. Assuming a constant failure rate λ {\displaystyle \lambda } implies that T {\displaystyle T} has an exponential distribution with parameter λ {\displaystyle \lambda } . Since the MTBF is the expected value of T {\displaystyle T} , it is given by the reciprocal of the failure rate of the system,
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