Failure rate is the frequency with which an engineered system or component fails, expressed for example in failures per hour. It is often denoted by the Greek letter λ (lambda) and is important in reliability theory. In practice, the reciprocal rate MTBF is more commonly expressed and used for high quality components or systems. FreQuency is a music video game developed by Harmonix and published by SCEI. It was released in November 2001. ... Look up failure in Wiktionary, the free dictionary. ... The hour (symbol: h) is a unit of time. ... The Greek alphabet is an alphabet that has been used to write the Greek language since about the 9th century BCE. It was the first alphabet in the narrow sense, that is, a writing system using a separate symbol for each vowel and consonant alike. ... Lambda (upper case Λ, lower case λ) is the 11th letter of the Greek alphabet. ... Reliability theory developed apart from the mainstream of probability and statistics, and was used originally as a tool to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers. ... In engineering and telecommunication, the mean time between failures (MTBF) is the average time a system will operate without a failure. ...

Failure rate is usually time dependent, and an intuitive corollary is that both rates change over time versus the expected life cycle of a system. For example, as an automobile grows older, the failure rate in its fifth year of service may be many times greater than its failure rate during its first year of service—one simply does not expect to replace an exhaust pipe, overhaul the brakes, or have major power plant-transmission problems in a new vehicle. So in the special case when the likelihood of failure remains constant with respect to time (for example, in some product like a brick or protected steel beam), failure rate is simply the inverse of the mean time between failure (MTBF), expressed for example in hours per failure. MTBF is an important specification perameter in all aspects of high importance engineering design— such as naval architecture, aerospace engineering, automotive design, etc. —in short, any task where failure in a key part or of the whole of a system needs be minimized and severely curtailed, particularly where lives might be lost if such factors are not taken into account. These factors account for many safety and maintenance practices in engineering and industry practices and government regulations, such as how often certain inspections and overhauls are required on an Aircraft. A similar ratio used in the transport industries, especially in railways and trucking is 'Mean Distance Between Failure', a variation which attempts to correlate actual loaded distances to similar reliability needs and practices. Failure rates and their projective manifestations are important factors in insurance, business, and regulation practices as well as fundamental to design of safe systems throughout a national or international economy. A theorem is a statement which can be proven true within some logical framework. ... For the philosophical/theological concept of a prime mover (that is, a self-existent being that is the ultimate cause or mover of all things), see cosmological argument. ... Look up Transmission in Wiktionary, the free dictionary. ... Mean-time-between-failure (MTBF) is the average time between failures, the reciprocal of the failure rate in the special case when failure rate is constant. ... It has been suggested that this article or section be merged into Naval Architecture. ... Aerospace engineering is the branch of engineering concerning aircraft, spacecraft and related topics. ... Mechanical engineering is the application of physical principles to the creation of useful devices, objects and machines. ... Engineering is the design, analysis, and/or construction of works for practical purposes. ... Look up aircraft in Wiktionary, the free dictionary. ... Transportation redirects here. ... The driver of this DAF tractor with an auto-transport semi-trailer prepares to offload Skoda Octavia cars in Cardiff, Wales For further uses of the word truck, see Truck (disambiguation). ... Positive linear correlations between 1000 pairs of numbers. ...

Failure rate in the discrete sense

In words appearing in an experiment, the failure rate can be defined as In mathematics, a probability distribution is called discrete, if it is fully characterized by a probability mass function. ...

The total number of failures within an item population, divided by the total time expended by that population, during a particular measurement interval under stated conditions. (MacDiarmid, et al.)

Here failure rate λ(t) can be thought of as the probability that a failure occurs in a specified interval, given no failure before time t. It can be defined with the aid of the reliability function or survival function R(t), the probability of no failure before time t, as: Probability is the extent to which something is likely to happen or be the case[1]. Probability theory is used extensively in areas such as statistics, mathematics, science, philosophy to draw conclusions about the likelihood of potential events and the underlying mechanics of complex systems. ... Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. ...

where t₁ (or t) and t₂ are respectively the beginning and ending of a specified interval of time spanning Δt. Note that this is a conditional probability, hence the R(t) in the denominator. This article defines some terms which characterize probability distributions of two or more variables. ...

Failure rate in the continuous sense

Exponential failure density functions

By calculating the failure rate for smaller and smaller intervals of time Δt, the interval becomes infinitesimally small. This results in the hazard function, which is the instantaneous failure rate at any point in time: In mathematics, a continuous function is one in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... Download high resolution version (1300x975, 118 KB) See the image on the commons for gnuplot source. ... Download high resolution version (1300x975, 118 KB) See the image on the commons for gnuplot source. ...

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Continuous failure rate depends on a failure distribution, F(t), which is a cumulative distribution function that describes the probability of failure prior to time t, In probability theory, the cumulative distribution function (abbreviated cdf) completely describes the probability distribution of a real-valued random variable, X. For every real number x, the cdf is given by where the right-hand side represents the probability that the random variable X takes on a value less than...

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The failure distribution function is the integral of the failure density function, f(x), In mathematics, a probability density function (pdf) serves to represent a probability distribution in terms of integrals. ...

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The hazard function can be defined now as

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There are many failure distributions (see List of important probability distributions). A common failure distribution is the exponential failure distribution, In mathematics and statistics, a probability distribution, more properly called a probability density, assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

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which is based on the exponential density function. This leads to a constant hazard rate. For other distributions, such as the Weibull distribution, log-normal distribution, or bathtub curve, the hazard function is not constant, which means that the failure rate varies with time. In probability theory and statistics, the exponential distributions are a class of continuous probability distribution. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... In probability and statistics, the log-normal distribution is the probability distribution of any random variable whose logarithm is normally distributed. ... In reliability theory, the bathtub curve is the phenomenon that the fraction of products failing in a given timespan is usually high early in the lifecycle, low in the middle, and rising strongly towards the end. ...

Failure rate data

Failure rate data can be obtained in several ways. The most common means are: In general, data consist of propositions that reflect reality. ...

• Historical data about the device or system under consideration.

Many organizations maintain internal databases of failure information on the devices or systems that they produce, which can be used to calculate failure rates for those devices or systems. For new devices or systems, the historical data for similar devices or systems can serve as a useful estimate.

• Government and commercial failure rate data.

Handbooks of failure rate data for various components are available from government and commercial sources. MIL-HDBK-217, Reliability Prediction of Electronic Equipment, is a military standard that provides failure rate data for many military electronic components. Several failure rate data sources are available commercially that focus on commercial components, including some non-electronic components.

• Testing.

The most accurate source of data is to test samples of the actual devices or systems in order to generate failure data. This is often prohibitively expensive or impractical, so that the previous data sources are often used instead.

U.S. Military Standards are general requirements for military personnel to comply with, whether it be Grooming Standards (a certain military image must be maintained), or the design of new weapons. ...

Units

Failure rates can be expressed using any measure of time, but hours is the most common unit in practice. Other units, such as miles, revolutions, etc., can also be used in place of "time" units.

Failure rates are often expressed in engineering notation as failures per million, or 10⁶, especially for individual components, since their failure rates are often very low. Engineering notation is scientific notation in which the powers of ten are limited to those where the exponent is a multiple of three, i. ...

The Failures In Time (FIT) rate of a device is the number of failures that can be expected in one billion (10⁹) hours of operation. This term is used particularly by the semiconductor industry. One thousand million (1,000,000,000) is the natural number following 999,999,999 and preceding 1,000,000,001. ... A semiconductor is a solid whose electrical conductivity can be controlled over a wide range, either permanently or dynamically. ...

Additivity

Under certain engineering assumptions, the failure rate for a complex system is simply the sum of the individual failure rates of its components, as long as the units are consistent, e.g. failures per million hours. This permits testing of individual components or subsystems, whose failure rates are then added to obtain the total system failure rate. Engineering is the design, analysis, and/or construction of works for practical purposes. ... System (from Latin systēma, in turn from Greek sustēma) is a set of entities, real or abstract, comprising a whole where each component interacts with or is related to at least one other component. ... For the Macintosh operating system, which was called System up to version 7. ...

Example

Suppose it is desired to estimate the failure rate of a certain component. A test can be performed to estimate its failure rate. Ten identical components are each tested until they either fail or reach 1000 hours, at which time the test is terminated for that component. (The level of statistical confidence is not considered in this example.) The results are as follows: In statistics, a confidence interval (CI) for a population parameter is an interval between two numbers with an associated probability p which is generated from a random sample of an underlying population, such that if the sampling was repeated numerous times and the confidence interval recalculated from each sample according...

Estimated failure rate is,

,

or 799.8 failures for every million hours of operation.

See also

• Failure
• Failure mode
• Reliability
• Reliability theory
• Reliability theory of aging and longevity
• Reliability engineering
• Survival analysis
• Weibull distribution
• MTBF
• Annualized failure rate
• Burn in

Look up failure in Wiktionary, the free dictionary. ... A failure mode is a characterization of the way a product or process fails. ... Reliability concerns quality or consistency. ... Reliability theory developed apart from the mainstream of probability and statistics, and was used originally as a tool to help nineteenth century maritime insurance and life insurance companies compute profitable rates to charge their customers. ... Reliability Theory of Aging and Longevity is a scientific approach aimed to gain theoretical insights into mechanisms of biological aging and species survival patterns by applying a general theory of systems failure, known as reliability theory. ... Reliability engineering is the discipline of ensuring that a system will be reliable when operated in a specified manner. ... Survival analysis is a branch of statistics which deals with death in biological organisms and failure in mechanical systems. ... In probability theory and statistics, the Weibull distribution (named after Waloddi Weibull) is a continuous probability distribution with the probability density function where and is the shape parameter and is the scale parameter of the distribution. ... In engineering and telecommunication, the mean time between failures (MTBF) is the average time a system will operate without a failure. ... Annualized failure rate, or short AFR is the reciprocal of the MTBF expressed in years and percent. ... Burn in is that process by which components of a system are exercised prior to being placed in service (and often, prior to the system being completely assembled from those components). ...

References

Print

• Blanchard, Benjamin S. (1992), Logistics Engineering and Management, Fourth Ed., pp 26-32, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
• Ebeling, Charles E., (1997), An Introduction to Reliability and Maintainability Engineering, pp 23-32, McGraw-Hill Companies, Inc., Boston.
• Federal Standard 1037C
• Kapur, K.C., and Lamberson, L.R., (1977), Reliability in Engineering Design, pp 8-30, 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
• MacDiarmid, Preston; Morris, Seymour; et. al., (no date), Reliability Toolkit: Commercial Practices Edition, pp 35-39, Reliability Analysis Center and Rome Laboratory, Rome, New York.
• 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

Federal Standard 1037C entitled Telecommunications: Glossary of Telecommunication Terms is a U.S. Federal Standard, issued by the General Services Administration pursuant to the Federal Property and Administrative Services Act of 1949, as amended. ...

Online

• Mondro, Mitchell J, (June 2002), "Approximation of Mean Time Between Failure When a System has Periodic Maintenance", IEEE Transactions on Reliability, v 51, no 2. (available from MITRE Corp.)
• Reliability Prediction of Electronic Equipment, MIL-HDBK-217F(2), (DOD download site.)
• Bathtub curve issues by ASQC.
• MTBF Software Tool

External links

• MTBF and reliability prediction Knowledge Base
• Google Answers (TM) question on MTBF.
• Usenet FAQ about MTBF.
• Reliability and Availability Basics
• MTBF FAQ and MTBF prediction software
• Product failure behaviour and wear out
• Burn in and reliability
• MTBF Calculator