# Mean time between failures

In engineering and telecommunication, the mean time between failures (MTBF) is the average time a system will operate without a failure. The MTBF is a commonly-quoted reliability statistic, and is usually expressed in hours (even intervals on the order of years are instead typically expressed in terms of thousands of hours).

MTBF is a measure of how reliable a product is. MTBF is usually given in units of hours; the higher the MTBF, the more reliable the product is.

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## Determining MTBF

It is generally calculated in one of several ways:

• An expected MTBF can be calculated on a statistical basis from the known failure rates of various components of the system. Specifically, it is the reciprocal of the sum of the failure rates of the components of the system (with the failure rates being expressed in failures per hour).
• Through empirical testing of a single part, the length of a performance measurement period can be divided by the number of failures that have occurred during that period.
• Through empirical testing of a group of items, the total functioning life of the population of items can be divided by the total number of failures within the population during the measurement period.

It should be noted by anyone comparing products on the basis of MTBF that:

• There is no standard measure of MTBF
• As stated above, it is often calculated and inferred rather than tested
• The MTBF applies only statistically (and cannot be taken as an expected lifetime)
• The MTBF applies only within the service life of a product (that is, after burn in and before the end of its service life). See Bathtub curve. After this time failure rates are not inferred or guaranteed in any way.
• The service life of a product is often shorter than its MTBF.
• MTBF may be quoted for individual components (for example, a chip) where the failure is defined as the component no longer working properly. This does not necessarily mean that the system of which the component is a part no longer works properly. Failure of the component may be covered in the system using redundancy or fault-tolerance. For example, the failure of a single DRAM IC within an error-correcting computer memory system may not cause a failure of the overall memory system. In general, adding more components to a system to provide redundancy will decrease the mean time between service being required, but increase the overall availability of the system (mean time before overall system failure).

## Statistical definition

The MTBF is often denoted by the Greek letter θ:

[itex]MTBF=\theta. \![itex]

The MTBF is the reciprocal of the failure rate,

[itex]MTBF=\theta =\frac{1}{\lambda}. \![itex]

Since failure rate and MTBF are simply reciprocals, both notations are found in the literature depending on which notation is most convenient for the application.

If one regards the time between failures (without the word mean) as a random variable, then the probability that the time between failures will be between two values a and b is given by an integral of the "failure density function" f, thus:

[itex]\int_a^b f(t)\,dt.[itex]

The MTBF is then the expected value of the time between failures, and can be computed using the same density function, thus:

[itex]MTBF=E(T)=\int_{0}^{\infty} tf(t)\, dt \![itex]

(Here, as is conventional in probability theory, we use capital T to represent the random variable that is the time between failures, and the lower-case t for the place-holder variable in the integral).

A common misconception about the MTBF is that it specifies the time (on average) when half of the items will fail. This is only true for certain symmetrical distributions. In many cases, such as the non-symmetrical exponential distribution, this is not true. For example, for an exponential failure distribution, the probability that an item will fail by the MTBF is approximately 0.63.

## Variations of MTBF

MTTF, mean-time-to-failure is sometimes used instead of MTBF in cases where a system is replaced after a failure, whereas MTBF denotes time between failures where the system is repaired.

Other variations of MTBF include mean-time-between-system-abort (MTBSA) or mean-time-between-critical-failure (MTBCF). Such nomenclature is used when it is desirable to differentiate among types of failures. For example, in an automobile, the failure of the FM radio does not prevent the primary operation of vehicle, so that it may be desirable to differentiate the failure rates of critical versus non-critical failures.

In some cases time is not accurate to predict a possible failure. Even if operating time is the same, two devices can be exposed to different levels of work. Typically this is the case of equipment or specific parts of equipment which rotate or electronic read-write devices where is more suitable to apply the cycle concept, and therefore mean cycles between failures (MCBF) instead of MTBF.

## Problems with MTBF

As of 1995, the use of MTBF in the aeronautical industry (and others) has been called into question due to the inaccuracy of its application to real systems and the nature of the culture that it engenders - many component MTBFs are given in databases, and often these values are horrendously inaccurate; its use has led to the negative exponential distribution being used much more than it should have been - it has been estimated that only 40% of components have failure rates described by this; it has also been corrupted into the notion of an "acceptable" level of failures, which removes the desire to get to the root cause of a problem and take measures to erase it. The British Royal Air Force is looking at other methods to describe reliability, such as Maintenance Free Operating Period (MFOP).

## References

• Federal Standard 1037C
• Blanchard, Benjamin S. (1992), Logistics Engineering and Management, Fourth Ed., Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
• Kapur, K.C., and Lamberson, L.R., (1977), Reliability in Engineering Design, John Wiley & Sons, New York.
• Knowles, D.I.,(1995), Should We Move Away From "Acceptable Failure Rate", Communications in Reliability Maintainability and Supportability, Vol. 2, No. 1, P. 23, International RMS Committee, USA
• Turner, T., Hockley, C., and Burdaky, R., (1997), The Customer Needs A Maintenance-Free Operating Period, 1997 Avionics Conference and Exhibition, No. 97-0819, P. 2.2, ERA Technology Ltd., Leatherhead, Surrey, UK

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