「利用者:YasuakiH/Weibull distribution」の版間の差分
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故障分布のモデル化では、多くの確率分布を用いることができる（{{仮リンク|確率分布のリスト|en|List of probability distributions}}を参照）。一般的なモデルは、[[指数分布|指数密度関数]]に基づく'''指数故障分布'''、
:<math>F(t)=\int_{0}^{t} \lambda e^{-\lambda \tau}\, d\tau = 1 - e^{-\lambda t} \!</math> である。
これに対するハザード率関数は、
:<math>h(t) = \frac{f(t)}{R(t)} = \frac{\lambda e^{-\lambda t}}{e^{-\lambda t}} = \lambda </math> となる。
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Mixtures of DFR variables are 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 [[:en:exponential distribution|exponentially distributed]] random variables are [[:en:Hyperexponential distribution|hyperexponentially distributed]].
main: 故障率曲線
故障率減少（decreasing failure rate、DFR）とは、ある事象が将来の一定の時間間隔で発生する確率が、時間の経過とともに減少していく現象を表す。故障率減少は、初期に起こる故障が解消または修正される「初期故障」の期間を表すことができ、λ(t) が[[減少|減少関数]]である状況に対応する。
DFR変数の混合<!-- mixtures -->はDFRである。[[指数分布]]確率変数の混合は、{{仮リンク|超指数分布|en|Hyperexponential distribution}}である。
===Renewal processes===
For a [[:en:renewal process|renewal process]] with DFR renewal function, inter-renewal times are concave.<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 }}</ref>
=== 再生過程 ===
DFR再生関数を持った{{仮リンク|再生理論|en|Renewal theory|label=再生過程}}では、再生間時間<!--inter-renewal times-->は凹になる{{訳語疑問点|date=2021年10}}。Brownは逆に、再生間時間が凹になるためにはDFRが必要であると推測したが、この推測は離散的な場合にも連続的な場合にも成り立たないことが示されている。
===Applications===
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Increasing failure rate is an intuitive concept caused by components wearing out. Decreasing failure rate describes a system which improves with age.<ref name="proschan" />
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 reliability of aircraft air conditioning systems were individually found to have an [[:en:exponential distribution|exponential distribution]], and thus in the pooled population a DFR.<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>
===Coefficient of variation===
When the failure rate is decreasing the [[:en:coefficient of variation|coefficient of variation]] is ⩾ 1, and when the failure rate is increasing the coefficient of variation is ⩽ 1.<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===
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The relationship of FIT to MTBF may be expressed as: MTBF = 1,000,000,000 x 1/FIT.
===Additivity===
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[https://www.quanterion.com/mission-reliability-and-logistics-reliability-a-design-paradox/ "Mission Reliability and Logistics Reliability: A Design Paradox"].
</ref>
===Example===
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or 799.8 failures for every million hours of operation.
==See also==
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*[[:en:Weibull distribution|Weibull distribution]]
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* [[故障率曲線]]
==References==
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