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Failure rate
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==Examples== ===Decreasing failure rates=== A decreasing failure rate describes cases where early-life failures are common<ref>{{Cite book | doi = 10.1007/978-1-84800-986-8_1 | chapter = Introduction | first = Maxim | last = Finkelstein| title = Failure Rate Modelling for Reliability and Risk | series = Springer Series in Reliability Engineering | pages = 1–84 | year = 2008 | isbn = 978-1-84800-985-1 }}</ref> and corresponds to the situation where <math>h(t)</math> is a [[decreasing function]]. This can describe, for example, the period of [[infant mortality]] in humans, or the early failure of a [[transistor]]s due to manufacturing defects. Decreasing failure rates have been found in the lifetimes of spacecraft - Baker and Baker commenting that "those spacecraft that last, last on and on."<ref>{{Cite journal | last1 = Baker | first1 = J. C. | last2 = Baker | first2 = G. A. S. . | doi = 10.2514/3.28040 | title = Impact of the space environment on spacecraft lifetimes | journal = Journal of Spacecraft and Rockets | volume = 17 | issue = 5 | pages = 479 | year = 1980 | bibcode = 1980JSpRo..17..479B }}</ref><ref>{{Cite book | doi = 10.1002/9781119994077.ch1 | chapter = On Time, Reliability, and Spacecraft | first1 = Joseph Homer | last1 = Saleh | first2 =Jean-François | last2 =Castet| title = Spacecraft Reliability and Multi-State Failures | pages = 1 | year = 2011 | isbn = 9781119994077 }}</ref> The hazard rate of aircraft air conditioning systems was found to have an exponentially decreasing distribution.<ref name="proschan">{{Cite journal | last1 = Proschan | first1 = F. | title = Theoretical Explanation of Observed Decreasing Failure Rate | doi = 10.1080/00401706.1963.10490105 | journal = Technometrics | volume = 5 | issue = 3 | pages = 375–383 | jstor = 1266340| year = 1963 }}</ref> ===Renewal processes=== In special processes called [[renewal process]]es, where the time to recover from failure can be neglected, the likelihood of failure remains constant with respect to time. For a [[renewal process]] with DFR renewal function, inter-renewal times are concave.{{clarify|date=December 2024}}<ref name="brown1980" /><ref name="shanthikumar">{{Cite journal | last1 = Shanthikumar | first1 = J. G. | doi = 10.1214/aop/1176991910 | title = DFR Property of First-Passage Times and its Preservation Under Geometric Compounding | journal = The Annals of Probability | volume = 16 | issue = 1 | pages = 397–406 | year = 1988 | jstor = 2243910| doi-access = free }}</ref> Brown conjectured the converse, that DFR is also necessary for the inter-renewal times to be concave,<ref>{{Cite journal | last1 = Brown | first1 = M. | title = Further Monotonicity Properties for Specialized Renewal Processes | doi = 10.1214/aop/1176994317 | journal = The Annals of Probability | volume = 9 | issue = 5 | pages = 891–895 | year = 1981 | jstor = 2243747| doi-access = free }}</ref> however it has been shown that this conjecture holds neither in the discrete case<ref name="shanthikumar" /> nor in the continuous case.<ref>{{Cite journal | last1 = Yu | first1 = Y. | title = Concave renewal functions do not imply DFR interrenewal times | doi = 10.1239/jap/1308662647 | journal = Journal of Applied Probability | volume = 48 | issue = 2 | pages = 583–588 | year = 2011 | arxiv = 1009.2463 | s2cid = 26570923 }}</ref> ===Coefficient of variation=== When the failure rate is decreasing the [[coefficient of variation]] is ⩾ 1, and when the failure rate is increasing the coefficient of variation is ⩽ 1.{{clarify|date=December 2024}}<ref>{{Cite journal | last1 = Wierman | first1 = A. | author-link1 = Adam Wierman| last2 = Bansal | first2 = N. | last3 = Harchol-Balter | first3 = M.|author3-link=Mor Harchol-Balter | title = A note on comparing response times in the M/GI/1/FB and M/GI/1/PS queues | doi = 10.1016/S0167-6377(03)00061-0 | journal = Operations Research Letters | volume = 32 | pages = 73–76 | url = http://users.cms.caltech.edu/~adamw/papers/fbnote.pdf| year = 2004 }}</ref> Note that this result only holds when the failure rate is defined for all t ⩾ 0<ref>{{cite book | title = Analysis of Queues: Methods and Applications | first = Natarajan | last =Gautam | publisher = CRC Press | year = 2012 | page = 703 | isbn = 978-1439806586}}</ref> and that the converse result (coefficient of variation determining nature of failure rate) does not hold. ===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 10<sup>−6</sup>, 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<sup>9</sup>) device-hours of operation<ref> Xin Li; Michael C. Huang; Kai Shen; Lingkun Chu. [http://www.cs.rochester.edu/~kshen/papers/usenix2010-li.pdf "A Realistic Evaluation of Memory Hardware Errors and Software System Susceptibility"]. 2010. p. 6. </ref> (e.g. 1,000 devices for 1,000,000 hours, or 1,000,000 devices for 1,000 hours each, or some other combination). This term is used particularly by the [[semiconductor]] industry. ===Combinations of failure types=== If a complex system consists of many parts, and the failure of any single part means the failure of the entire system, then the total failure rate is simply the sum of the individual failure rates of its parts :<math>\lambda_S = \lambda_{P1} + \lambda_{P2} + \ldots</math> however, this assumes that the failure rate <math>\lambda(t)</math> is constant, and that the units are consistent (e.g. failures per million hours), and not expressed as a ratio or as probability densities. This is useful to estimate the failure rate of a system when individual components or subsystems have already been tested.<ref> [http://www.weibull.com/hotwire/issue108/relbasics108.htm "Reliability Basics"]. 2010. </ref><ref>Vita Faraci. [http://src.alionscience.com/pdf/1Q2006.pdf "Calculating Failure Rates of Series/Parallel Networks"] {{Webarchive|url=https://web.archive.org/web/20160303224453/http://src.alionscience.com/pdf/1Q2006.pdf |date=2016-03-03 }}. 2006.</ref> Adding "redundant" components to eliminate a [[single point of failure]] may thus actually increase the failure rate, however reduces the "mission failure" rate, or the "mean time between critical failures" (MTBCF).<ref> [https://www.quanterion.com/mission-reliability-and-logistics-reliability-a-design-paradox/ "Mission Reliability and Logistics Reliability: A Design Paradox"]. </ref> Combining failure or hazard rates that are time-dependent is more complicated. For example, mixtures of Decreasing Failure Rate (DFR) variables are also DFR.<ref name="brown1980">{{Cite journal | last1 = Brown | first1 = M. | title = Bounds, Inequalities, and Monotonicity Properties for Some Specialized Renewal Processes | doi = 10.1214/aop/1176994773 | journal = The Annals of Probability | volume = 8 | issue = 2 | pages = 227–240 | jstor = 2243267| year = 1980 | doi-access = free }}</ref> Mixtures of [[exponential distribution|exponentially distributed]] failure rates are [[Hyperexponential distribution|hyperexponentially distributed]]. ===Simple example=== Suppose it is desired to estimate the failure rate of a certain component. Ten identical components are each tested until they either fail or reach 1,000 hours, at which time the test is terminated. A total of 7,502 component-hours of testing is performed, and 6 failures are recorded. The ''estimated'' failure rate is: : <math>\frac{6\text{ failures}}{7502\text{ hours}} = 0.0007998\, \frac{\text{failures}}{\text{hour}} </math> which could also be expressed as a MTBF of 1,250 hours, or approximately 800 failures for every million hours of operation.
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