Mean time between failures

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: : \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 : \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 indicating the time until failure. Thus, it can be written as : \text{MTBF} = \mathbb{E}\{T\} = \int_0^\infty tf_T(t)\, dt where f_T(t) is the probability density function of T. Equivalently, the MTBF can be expressed in terms of the reliability function R_T(t) as : \text{MTBF} = \int_0^\infty R(t)\, dt . The MTBF and 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. == Application ==

The MTBF value can be used as a system reliability parameter or to compare different systems or designs. This value should only be understood conditionally as the “mean lifetime” (an average value), and not as a quantitative identity between working and failed units. Since MTBF can be expressed as “average life (expectancy)”, many engineers assume that 50% of items will have failed by time t = MTBF. This inaccuracy can lead to bad design decisions. Furthermore, probabilistic failure prediction based on MTBF implies the total absence of systematic failures (i.e., a constant failure rate with only intrinsic, random failures), which is not easy to verify. Assuming no systematic errors, the probability the system survives during a duration, T, is calculated as exp^(-T/MTBF). Hence the probability a system fails during a duration T, is given by 1 - exp^(-T/MTBF). MTBF value prediction is an important element in the development of products. Reliability engineers and design engineers often use reliability software to calculate a product's MTBF according to various methods and standards (MIL-HDBK-217F, Telcordia SR332, Siemens SN 29500, FIDES, UTE 80-810 (RDF2000), etc.). The Mil-HDBK-217 reliability calculator manual in combination with RelCalc software (or other comparable tool) enables MTBF reliability rates to be predicted based on design. A concept which is closely related to MTBF, and is important in the computations involving MTBF, is the mean down time (MDT). MDT can be defined as mean time which the system is down after the failure. Usually, MDT is considered different from MTTR (Mean Time To Repair); in particular, MDT usually includes organizational and logistical factors (such as business days or waiting for components to arrive) while MTTR is usually understood as more narrow and more technical. == Application of MTBF in manufacturing ==

MTBF serves as a crucial metric for managing machinery and equipment reliability. Its application is particularly significant in the context of total productive maintenance (TPM), a comprehensive maintenance strategy aimed at maximizing equipment effectiveness. MTBF provides a quantitative measure of the time elapsed between failures of a system during normal operation, offering insights into the reliability and performance of manufacturing equipment. By integrating MTBF with TPM principles, manufacturers can achieve a more proactive maintenance approach. This synergy allows for the identification of patterns and potential failures before they occur, enabling preventive maintenance and reducing unplanned downtime. As a result, MTBF becomes a key performance indicator (KPI) within TPM, guiding decisions on maintenance schedules, spare parts inventory, and ultimately, optimizing the lifespan and efficiency of machinery. This strategic use of MTBF within TPM frameworks enhances overall production efficiency, reduces costs associated with breakdowns, and contributes to the continuous improvement of manufacturing processes. ==MTBF and MDT for networks of components==

Two components c_1,c_2 (for instance hard drives, servers, etc.) may be arranged in a network, in series or in parallel. The terminology is here used by close analogy to electrical circuits, but has a slightly different meaning. We say that the two components are in series if the failure of either causes the failure of the network, and that they are in parallel if only the failure of both causes the network to fail. The MTBF of the resulting two-component network with repairable components can be computed according to the following formulae, assuming that the MTBF of both individual components is known: :\text{mtbf}(c_1 ; c_2) = \frac{1}{\frac{1}{\text{mtbf}(c_1)} + \frac{1}{\text{mtbf}(c_2)}} = \frac{\text{mtbf}(c_1)\times \text{mtbf}(c_2)} {\text{mtbf}(c_1) + \text{mtbf}(c_2)}\;, where c_1 ; c_2 is the network in which the components are arranged in series. For the network containing parallel repairable components, to find out the MTBF of the whole system, in addition to component MTBFs, it is also necessary to know their respective MDTs. Then, assuming that MDTs are negligible compared to MTBFs (which usually stands in practice), the MTBF for the parallel system consisting from two parallel repairable components can be written as follows: and likewise :\text{mdt}(c_1\parallel\dots\parallel c_n) = \left(\sum_{k=1}^n \frac 1{\text{mdt}(c_k)}\right)^{-1}\;, since the formula for the mdt of two components in parallel is identical to that of the mtbf for two components in series. == Variations of MTBF ==

There are many variations of MTBF, such as mean time between system aborts (MTBSA), mean time between critical failures (MTBCF) or mean time between unscheduled removal (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. It is recommended to use Mean time to failure (MTTF) instead of MTBF in cases where a system is replaced after a failure ("non-repairable system"), since MTBF denotes time between failures in a system which can be repaired. MTBF considering censoring In fact the MTBF counting only failures with at least some systems still operating that have not yet failed underestimates the MTBF by failing to include in the computations the partial lifetimes of the systems that have not yet failed. With such lifetimes, all we know is that the time to failure exceeds the time they've been running. This is called censoring. In fact with a parametric model of the lifetime, the likelihood for the experience on any given day is as follows: :L = \prod_i \lambda(u_i)^{\delta_i} S(u_i), where :u_i is the failure time for failures and the censoring time for units that have not yet failed, :\delta_i = 1 for failures and 0 for censoring times, :S(u_i) = the probability that the lifetime exceeds u_i, called the survival function, and :\lambda(u_i) = f(u)/S(u) is called the hazard function, the instantaneous force of mortality (where f(u) = the probability density function of the distribution). For a constant exponential distribution, the hazard, \lambda, is constant. In this case, the MBTF is :MTBF = 1 / \hat\lambda = \sum u_i / k, where \hat\lambda is the maximum likelihood estimate of \lambda, maximizing the likelihood given above and k = \sum \sigma_i is the number of uncensored observations. We see that the difference between the MTBF considering only failures and the MTBF including censored observations is that the censoring times add to the numerator but not the denominator in computing the MTBF. ==See also==