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Failure rateFailure 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. 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 planttransmission 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 parameter 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. Additional recommended knowledge
Failure rate in the discrete senseIn words appearing in an experiment, the failure rate can be defined as
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: where t_{1} (or t) and t_{2} 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. Failure rate in the continuous senseBy calculating the failure rate for smaller and smaller intervals of time , the interval becomes infinitely small. This results in the hazard function, which is the instantaneous failure rate at any point in time: Continuous failure rate depends on a failure distribution, , which is a cumulative distribution function that describes the probability of failure prior to time t, The failure distribution function is the integral of the failure density function, f(x), The hazard function can be defined now as There are many failure distributions (see List of important probability distributions). A common failure distribution is the exponential failure distribution, which is based on the exponential density function. For an exponential failure distribution the hazard rate is a constant with respect to time (known as "memoryless"). For other distributions, such as a Weibull distribution or a lognormal distribution, the hazard function is not constant with respect to time. Failure rate dataFailure rate data can be obtained in several ways. The most common means are:
UnitsFailure 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 10^{6}, especially for individual components, since their failure rates are often very low. The Failures In Time (FIT) rate of a device is the number of failures that can be expected in one billion (10^{9}) hours of operation. This term is used particularly by the semiconductor industry. AdditivityUnder 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. ExampleSuppose 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:
Estimated failure rate is or 799.8 failures for every million hours of operation. See also
References
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This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Failure_rate". A list of authors is available in Wikipedia. 