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68-540: The definition of MTBF depends on the definition of what is considered a failure. For complex, repairable systems, failures are considered to be those out of design conditions which place the system out of service and into a state for repair. Failures which occur that can be left or maintained in an unrepaired condition, and do not place the system out of service, are not considered failures under this definition. In addition, units that are taken down for routine scheduled maintenance or inventory control are not considered within

136-464: A constant failure rate λ {\displaystyle \lambda } implies that T {\displaystyle T} has an exponential distribution with parameter λ {\displaystyle \lambda } . Since the MTBF is the expected value of T {\displaystyle T} , it is given by the reciprocal of the failure rate of the system, Once

204-488: A dangerous condition. It can be calculated as follows: where B 10 is the number of operations that a device will operate prior to 10% of a sample of those devices would fail and n op is number of operations. B 10d is the same calculation, but where 10% of the sample would fail to danger. n op is the number of operations/cycle in one year. In fact the MTBF counting only failures with at least some systems still operating that have not yet failed underestimates

272-609: 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

340-417: A process in which events occur continuously and independently at a constant average rate; the distance parameter could be any meaningful mono-dimensional measure of the process, such as time between production errors, or length along a roll of fabric in the weaving manufacturing process. It is a particular case of the gamma distribution . It is the continuous analogue of the geometric distribution , and it has

408-420: 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,

476-431: 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

544-514: A sample size greater than two, with a correction factor to the MLE: λ ^ = ( n − 2 n ) ( 1 x ¯ ) = n − 2 ∑ i x i {\displaystyle {\widehat {\lambda }}=\left({\frac {n-2}{n}}\right)\left({\frac {1}{\bar {x}}}\right)={\frac {n-2}{\sum _{i}x_{i}}}} This

612-470: A sum of two independent random variables is the convolution of their individual PDFs . If X 1 {\displaystyle X_{1}} and X 2 {\displaystyle X_{2}} are independent exponential random variables with respective rate parameters λ 1 {\displaystyle \lambda _{1}} and λ 2 , {\displaystyle \lambda _{2},} then

680-497: A system out of two serial components can be calculated as: and for a system out of two parallel components MDT can be calculated as: Through successive application of these four formulae, the MTBF and MDT of any network of repairable components can be computed, provided that the MTBF and MDT is known for each component. In a special but all-important case of several serial components, MTBF calculation can be easily generalised into which can be shown by induction, and likewise since

748-460: Is F − 1 ( p ; λ ) = − ln ⁡ ( 1 − p ) λ , 0 ≤ p < 1 {\displaystyle F^{-1}(p;\lambda )={\frac {-\ln(1-p)}{\lambda }},\qquad 0\leq p<1} The quartiles are therefore: And as a consequence the interquartile range is ln(3)/ λ . The conditional value at risk (CVaR) also known as

SECTION 10

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816-662: Is an unbiased MLE estimator of 1 / λ {\displaystyle 1/\lambda } and the distribution mean. The bias of λ ^ mle {\displaystyle {\widehat {\lambda }}_{\text{mle}}} is equal to B ≡ E ⁡ [ ( λ ^ mle − λ ) ] = λ n − 1 {\displaystyle B\equiv \operatorname {E} \left[\left({\widehat {\lambda }}_{\text{mle}}-\lambda \right)\right]={\frac {\lambda }{n-1}}} which yields

884-450: Is Gamma(n, λ) distributed. Other related distributions: Below, suppose random variable X is exponentially distributed with rate parameter λ, and x 1 , … , x n {\displaystyle x_{1},\dotsc ,x_{n}} are n independent samples from X , with sample mean x ¯ {\displaystyle {\bar {x}}} . The maximum likelihood estimator for λ

952-1354: Is also exponentially distributed, with parameter λ = λ 1 + ⋯ + λ n . {\displaystyle \lambda =\lambda _{1}+\dotsb +\lambda _{n}.} This can be seen by considering the complementary cumulative distribution function : Pr ( min { X 1 , … , X n } > x ) = Pr ( X 1 > x , … , X n > x ) = ∏ i = 1 n Pr ( X i > x ) = ∏ i = 1 n exp ⁡ ( − x λ i ) = exp ⁡ ( − x ∑ i = 1 n λ i ) . {\displaystyle {\begin{aligned}&\Pr \left(\min\{X_{1},\dotsc ,X_{n}\}>x\right)\\={}&\Pr \left(X_{1}>x,\dotsc ,X_{n}>x\right)\\={}&\prod _{i=1}^{n}\Pr \left(X_{i}>x\right)\\={}&\prod _{i=1}^{n}\exp \left(-x\lambda _{i}\right)=\exp \left(-x\sum _{i=1}^{n}\lambda _{i}\right).\end{aligned}}} The index of

1020-877: Is available in closed form: assuming λ 1 > λ 2 {\displaystyle \lambda _{1}>\lambda _{2}} (without loss of generality), then H ( Z ) = 1 + γ + ln ⁡ ( λ 1 − λ 2 λ 1 λ 2 ) + ψ ( λ 1 λ 1 − λ 2 ) , {\displaystyle {\begin{aligned}H(Z)&=1+\gamma +\ln \left({\frac {\lambda _{1}-\lambda _{2}}{\lambda _{1}\lambda _{2}}}\right)+\psi \left({\frac {\lambda _{1}}{\lambda _{1}-\lambda _{2}}}\right),\end{aligned}}} where γ {\displaystyle \gamma }

1088-567: Is because for items that are inexpensive to procure , it is often more cost-effective not to maintain (repair) them. Repair costs can be expensive, including costs for the labor for the removal the broken or worn out part (described as unserviceable), cost of replacement with a working (serviceable) from inventory, and also the cost of the actual repair, including possible shipping costs to a repair vendor. At maintenance facilities, such as might be found at Main Operating Bases , inventory

1156-1050: Is constructed as follows. The likelihood function for λ, given an independent and identically distributed sample x = ( x 1 , ..., x n ) drawn from the variable, is: L ( λ ) = ∏ i = 1 n λ exp ⁡ ( − λ x i ) = λ n exp ⁡ ( − λ ∑ i = 1 n x i ) = λ n exp ⁡ ( − λ n x ¯ ) , {\displaystyle L(\lambda )=\prod _{i=1}^{n}\lambda \exp(-\lambda x_{i})=\lambda ^{n}\exp \left(-\lambda \sum _{i=1}^{n}x_{i}\right)=\lambda ^{n}\exp \left(-\lambda n{\overline {x}}\right),} where: x ¯ = 1 n ∑ i = 1 n x i {\displaystyle {\overline {x}}={\frac {1}{n}}\sum _{i=1}^{n}x_{i}}

1224-687: Is controlled by site personnel. Maintenance personnel will formally "turn-in" unserviceable items for repair, receiving a funding credit in the process. These "turn-ins" will be fixed, reconditioned, or replaced. Maintenance personnel can also be issued repaired or new items back from inventory. These processes are assisted by automated logistics management systems. In the Navy/Marine Corps supply system repairable items are identified with certain two character cognizance symbols (COGs) and one character Material Control Codes (MCCs). In United States Marine Corps Aviation, repairables are managed by

1292-453: Is equal to the unconditional probability of observing the event more than 10 seconds after the initial time. The exponential distribution and the geometric distribution are the only memoryless probability distributions . The exponential distribution is consequently also necessarily the only continuous probability distribution that has a constant failure rate . The quantile function (inverse cumulative distribution function) for Exp( λ )

1360-1571: Is given by Δ ( λ 0 ∥ λ ) = E λ 0 ( log ⁡ p λ 0 ( x ) p λ ( x ) ) = E λ 0 ( log ⁡ λ 0 e λ 0 x λ e λ x ) = log ⁡ ( λ 0 ) − log ⁡ ( λ ) − ( λ 0 − λ ) E λ 0 ( x ) = log ⁡ ( λ 0 ) − log ⁡ ( λ ) + λ λ 0 − 1. {\displaystyle {\begin{aligned}\Delta (\lambda _{0}\parallel \lambda )&=\mathbb {E} _{\lambda _{0}}\left(\log {\frac {p_{\lambda _{0}}(x)}{p_{\lambda }(x)}}\right)\\&=\mathbb {E} _{\lambda _{0}}\left(\log {\frac {\lambda _{0}e^{\lambda _{0}x}}{\lambda e^{\lambda x}}}\right)\\&=\log(\lambda _{0})-\log(\lambda )-(\lambda _{0}-\lambda )E_{\lambda _{0}}(x)\\&=\log(\lambda _{0})-\log(\lambda )+{\frac {\lambda }{\lambda _{0}}}-1.\end{aligned}}} Among all continuous probability distributions with support [0, ∞) and mean μ ,

1428-1465: Is given by E ⁡ [ X ( i ) X ( j ) ] = ∑ k = 0 j − 1 1 ( n − k ) λ E ⁡ [ X ( i ) ] + E ⁡ [ X ( i ) 2 ] = ∑ k = 0 j − 1 1 ( n − k ) λ ∑ k = 0 i − 1 1 ( n − k ) λ + ∑ k = 0 i − 1 1 ( ( n − k ) λ ) 2 + ( ∑ k = 0 i − 1 1 ( n − k ) λ ) 2 . {\displaystyle {\begin{aligned}\operatorname {E} \left[X_{(i)}X_{(j)}\right]&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\operatorname {E} \left[X_{(i)}\right]+\operatorname {E} \left[X_{(i)}^{2}\right]\\&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\sum _{k=0}^{i-1}{\frac {1}{(n-k)\lambda }}+\sum _{k=0}^{i-1}{\frac {1}{((n-k)\lambda )^{2}}}+\left(\sum _{k=0}^{i-1}{\frac {1}{(n-k)\lambda }}\right)^{2}.\end{aligned}}} This can be seen by invoking

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1496-584: Is given by E ⁡ [ X ] = 1 λ . {\displaystyle \operatorname {E} [X]={\frac {1}{\lambda }}.} In light of the examples given below , this makes sense; a person who receives an average of two telephone calls per hour can expect that the time between consecutive calls will be 0.5 hour, or 30 minutes. The variance of X is given by Var ⁡ [ X ] = 1 λ 2 , {\displaystyle \operatorname {Var} [X]={\frac {1}{\lambda ^{2}}},} so

1564-913: Is given by The exponential distribution is sometimes parametrized in terms of the scale parameter β = 1/ λ , which is also the mean: f ( x ; β ) = { 1 β e − x / β x ≥ 0 , 0 x < 0. F ( x ; β ) = { 1 − e − x / β x ≥ 0 , 0 x < 0. {\displaystyle f(x;\beta )={\begin{cases}{\frac {1}{\beta }}e^{-x/\beta }&x\geq 0,\\0&x<0.\end{cases}}\qquad \qquad F(x;\beta )={\begin{cases}1-e^{-x/\beta }&x\geq 0,\\0&x<0.\end{cases}}} The mean or expected value of an exponentially distributed random variable X with rate parameter λ

1632-453: Is interpreted as the waiting time for an event to occur relative to some initial time, this relation implies that, if T is conditioned on a failure to observe the event over some initial period of time s , the distribution of the remaining waiting time is the same as the original unconditional distribution. For example, if an event has not occurred after 30 seconds, the conditional probability that occurrence will take at least 10 more seconds

1700-399: 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

1768-514: Is not exponentially distributed, if X 1 , ..., X n do not all have parameter 0. Let X 1 , … , X n {\displaystyle X_{1},\dotsc ,X_{n}} be n {\displaystyle n} independent and identically distributed exponential random variables with rate parameter λ . Let X ( 1 ) , … , X ( n ) {\displaystyle X_{(1)},\dotsc ,X_{(n)}} denote

1836-1803: Is one minus the probability level at which the CVaR equals the threshold x {\displaystyle x} . It is derived as follows: p ¯ x ( X ) = { 1 − α | q ¯ α ( X ) = x } = { 1 − α | − ln ⁡ ( 1 − α ) + 1 λ = x } = { 1 − α | ln ⁡ ( 1 − α ) = 1 − λ x } = { 1 − α | e ln ⁡ ( 1 − α ) = e 1 − λ x } = { 1 − α | 1 − α = e 1 − λ x } = e 1 − λ x {\displaystyle {\begin{aligned}{\bar {p}}_{x}(X)&=\{1-\alpha |{\bar {q}}_{\alpha }(X)=x\}\\&=\{1-\alpha |{\frac {-\ln(1-\alpha )+1}{\lambda }}=x\}\\&=\{1-\alpha |\ln(1-\alpha )=1-\lambda x\}\\&=\{1-\alpha |e^{\ln(1-\alpha )}=e^{1-\lambda x}\}=\{1-\alpha |1-\alpha =e^{1-\lambda x}\}=e^{1-\lambda x}\end{aligned}}} The directed Kullback–Leibler divergence in nats of e λ {\displaystyle e^{\lambda }} ("approximating" distribution) from e λ 0 {\displaystyle e^{\lambda _{0}}} ('true' distribution)

1904-430: 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

1972-517: Is the Euler-Mascheroni constant , and ψ ( ⋅ ) {\displaystyle \psi (\cdot )} is the digamma function . In the case of equal rate parameters, the result is an Erlang distribution with shape 2 and parameter λ , {\displaystyle \lambda ,} which in turn is a special case of gamma distribution . The sum of n independent Exp( λ) exponential random variables

2040-478: 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. MTBF serves as a crucial metric for managing machinery and equipment reliability. Its application

2108-404: Is the subfactorial of n The median of X is given by m ⁡ [ X ] = ln ⁡ ( 2 ) λ < E ⁡ [ X ] , {\displaystyle \operatorname {m} [X]={\frac {\ln(2)}{\lambda }}<\operatorname {E} [X],} where ln refers to the natural logarithm . Thus the absolute difference between

Mean time between failures - Misplaced Pages Continue

2176-447: Is the network in which the components are arranged in parallel, and P F ( c , t ) {\displaystyle PF(c,t)} is the probability of failure of component c {\displaystyle c} during "vulnerability window" t {\displaystyle t} . Intuitively, both these formulae can be explained from the point of view of failure probabilities. First of all, let's note that

2244-2184: 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: mtbf ( c 1 ∥ c 2 ) = 1 1 mtbf ( c 1 ) × PF ( c 2 , mdt ( c 1 ) ) + 1 mtbf ( c 2 ) × PF ( c 1 , mdt ( c 2 ) ) = 1 1 mtbf ( c 1 ) × mdt ( c 1 ) mtbf ( c 2 ) + 1 mtbf ( c 2 ) × mdt ( c 2 ) mtbf ( c 1 ) = mtbf ( c 1 ) × mtbf ( c 2 ) mdt ( c 1 ) + mdt ( c 2 ) , {\displaystyle {\begin{aligned}{\text{mtbf}}(c_{1}\parallel c_{2})&={\frac {1}{{\frac {1}{{\text{mtbf}}(c_{1})}}\times {\text{PF}}(c_{2},{\text{mdt}}(c_{1}))+{\frac {1}{{\text{mtbf}}(c_{2})}}\times {\text{PF}}(c_{1},{\text{mdt}}(c_{2}))}}\\[1em]&={\frac {1}{{\frac {1}{{\text{mtbf}}(c_{1})}}\times {\frac {{\text{mdt}}(c_{1})}{{\text{mtbf}}(c_{2})}}+{\frac {1}{{\text{mtbf}}(c_{2})}}\times {\frac {{\text{mdt}}(c_{2})}{{\text{mtbf}}(c_{1})}}}}\\[1em]&={\frac {{\text{mtbf}}(c_{1})\times {\text{mtbf}}(c_{2})}{{\text{mdt}}(c_{1})+{\text{mdt}}(c_{2})}}\;,\end{aligned}}} where c 1 ∥ c 2 {\displaystyle c_{1}\parallel c_{2}}

2312-479: 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. Repairable A repairable component is a component of a finished good that can be designated for repair. Repairable components tend to be more expensive than non-repairable components ( consumables ). This

2380-1095: Is the sample mean. The derivative of the likelihood function's logarithm is: d d λ ln ⁡ L ( λ ) = d d λ ( n ln ⁡ λ − λ n x ¯ ) = n λ − n x ¯   { > 0 , 0 < λ < 1 x ¯ , = 0 , λ = 1 x ¯ , < 0 , λ > 1 x ¯ . {\displaystyle {\frac {d}{d\lambda }}\ln L(\lambda )={\frac {d}{d\lambda }}\left(n\ln \lambda -\lambda n{\overline {x}}\right)={\frac {n}{\lambda }}-n{\overline {x}}\ {\begin{cases}>0,&0<\lambda <{\frac {1}{\overline {x}}},\\[8pt]=0,&\lambda ={\frac {1}{\overline {x}}},\\[8pt]<0,&\lambda >{\frac {1}{\overline {x}}}.\end{cases}}} Consequently,

2448-670: The Repairables Management Division of the Aviation Supply Department. In the United States Air Force , repairables can be identified by their ERRC designation or SMR code. Exponential distribution In probability theory and statistics , the exponential distribution or negative exponential distribution is the probability distribution of the distance between events in a Poisson point process , i.e.,

2516-426: The bias-corrected maximum likelihood estimator λ ^ mle ∗ = λ ^ mle − B . {\displaystyle {\widehat {\lambda }}_{\text{mle}}^{*}={\widehat {\lambda }}_{\text{mle}}-B.} An approximate minimizer of mean squared error (see also: bias–variance tradeoff ) can be found, assuming

2584-1065: The complementary cumulative distribution function : Pr ( T > s + t ∣ T > s ) = Pr ( T > s + t ∩ T > s ) Pr ( T > s ) = Pr ( T > s + t ) Pr ( T > s ) = e − λ ( s + t ) e − λ s = e − λ t = Pr ( T > t ) . {\displaystyle {\begin{aligned}\Pr \left(T>s+t\mid T>s\right)&={\frac {\Pr \left(T>s+t\cap T>s\right)}{\Pr \left(T>s\right)}}\\[4pt]&={\frac {\Pr \left(T>s+t\right)}{\Pr \left(T>s\right)}}\\[4pt]&={\frac {e^{-\lambda (s+t)}}{e^{-\lambda s}}}\\[4pt]&=e^{-\lambda t}\\[4pt]&=\Pr(T>t).\end{aligned}}} When T

2652-1906: The expected shortfall or superquantile for Exp( λ ) is derived as follows: q ¯ α ( X ) = 1 1 − α ∫ α 1 q p ( X ) d p = 1 ( 1 − α ) ∫ α 1 − ln ⁡ ( 1 − p ) λ d p = − 1 λ ( 1 − α ) ∫ 1 − α 0 − ln ⁡ ( y ) d y = − 1 λ ( 1 − α ) ∫ 0 1 − α ln ⁡ ( y ) d y = − 1 λ ( 1 − α ) [ ( 1 − α ) ln ⁡ ( 1 − α ) − ( 1 − α ) ] = − ln ⁡ ( 1 − α ) + 1 λ {\displaystyle {\begin{aligned}{\bar {q}}_{\alpha }(X)&={\frac {1}{1-\alpha }}\int _{\alpha }^{1}q_{p}(X)dp\\&={\frac {1}{(1-\alpha )}}\int _{\alpha }^{1}{\frac {-\ln(1-p)}{\lambda }}dp\\&={\frac {-1}{\lambda (1-\alpha )}}\int _{1-\alpha }^{0}-\ln(y)dy\\&={\frac {-1}{\lambda (1-\alpha )}}\int _{0}^{1-\alpha }\ln(y)dy\\&={\frac {-1}{\lambda (1-\alpha )}}[(1-\alpha )\ln(1-\alpha )-(1-\alpha )]\\&={\frac {-\ln(1-\alpha )+1}{\lambda }}\\\end{aligned}}} The buffered probability of exceedance

2720-1107: The law of total expectation and the memoryless property: E ⁡ [ X ( i ) X ( j ) ] = ∫ 0 ∞ E ⁡ [ X ( i ) X ( j ) ∣ X ( i ) = x ] f X ( i ) ( x ) d x = ∫ x = 0 ∞ x E ⁡ [ X ( j ) ∣ X ( j ) ≥ x ] f X ( i ) ( x ) d x ( since   X ( i ) = x ⟹ X ( j ) ≥ x ) = ∫ x = 0 ∞ x [ E ⁡ [ X ( j ) ] + x ] f X ( i ) ( x ) d x ( by

2788-541: The maximum likelihood estimate for the rate parameter is: λ ^ mle = 1 x ¯ = n ∑ i x i {\displaystyle {\widehat {\lambda }}_{\text{mle}}={\frac {1}{\overline {x}}}={\frac {n}{\sum _{i}x_{i}}}} This is not an unbiased estimator of λ , {\displaystyle \lambda ,} although x ¯ {\displaystyle {\overline {x}}}

Mean time between failures - Misplaced Pages Continue

2856-422: The median-mean inequality . An exponentially distributed random variable T obeys the relation Pr ( T > s + t ∣ T > s ) = Pr ( T > t ) , ∀ s , t ≥ 0. {\displaystyle \Pr \left(T>s+t\mid T>s\right)=\Pr(T>t),\qquad \forall s,t\geq 0.} This can be seen by considering

2924-484: The normal , binomial , gamma , and Poisson distributions. The probability density function (pdf) of an exponential distribution is Here λ > 0 is the parameter of the distribution, often called the rate parameter . The distribution is supported on the interval  [0, ∞) . If a random variable X has this distribution, we write  X ~ Exp( λ ) . The exponential distribution exhibits infinite divisibility . The cumulative distribution function

2992-918: The standard deviation is equal to the mean. The moments of X , for n ∈ N {\displaystyle n\in \mathbb {N} } are given by E ⁡ [ X n ] = n ! λ n . {\displaystyle \operatorname {E} \left[X^{n}\right]={\frac {n!}{\lambda ^{n}}}.} The central moments of X , for n ∈ N {\displaystyle n\in \mathbb {N} } are given by μ n = ! n λ n = n ! λ n ∑ k = 0 n ( − 1 ) k k ! . {\displaystyle \mu _{n}={\frac {!n}{\lambda ^{n}}}={\frac {n!}{\lambda ^{n}}}\sum _{k=0}^{n}{\frac {(-1)^{k}}{k!}}.} where ! n

3060-416: 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. In other words, it is assumed that the system has survived initial setup stresses and has not yet approached its expected end of life, both of which often increase the failure rate. Assuming

3128-415: 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 : where For a constant exponential distribution ,

3196-402: The MTBF of a system is known, and assuming a constant failure rate, the probability that any one particular system will be operational for a given duration can be inferred from the reliability function of the exponential distribution , R T ( t ) = e − λ t {\displaystyle R_{T}(t)=e^{-\lambda t}} . In particular,

3264-411: The components. With parallel components the situation is a bit more complicated: the whole system will fail if and only if after one of the components fails, the other component fails while the first component is being repaired; this is where MDT comes into play: the faster the first component is repaired, the less is the "vulnerability window" for the other component to fail. Using similar logic, MDT for

3332-423: The continuous improvement of manufacturing processes. Two components c 1 , c 2 {\displaystyle 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

3400-471: The corresponding order statistics . For i < j {\displaystyle i<j} , the joint moment E ⁡ [ X ( i ) X ( j ) ] {\displaystyle \operatorname {E} \left[X_{(i)}X_{(j)}\right]} of the order statistics X ( i ) {\displaystyle X_{(i)}} and X ( j ) {\displaystyle X_{(j)}}

3468-434: The definition of failure. The higher the MTBF, the longer a system is likely to work before failing. 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

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3536-550: The exponential distribution with λ = 1/ μ has the largest differential entropy . In other words, it is the maximum entropy probability distribution for a random variate X which is greater than or equal to zero and for which E[ X ] is fixed. Let X 1 , ..., X n be independent exponentially distributed random variables with rate parameters λ 1 , ..., λ n . Then min { X 1 , … , X n } {\displaystyle \min \left\{X_{1},\dotsc ,X_{n}\right\}}

3604-429: 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: where c 1 ; c 2 {\displaystyle c_{1};c_{2}}

3672-452: 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. MTTFd is an extension of MTTF, and is only concerned about failures which would result in

3740-466: The formula for the mdt of two components in parallel is identical to that of the mtbf for two components in series. 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,

3808-509: The hazard, λ {\displaystyle \lambda } , is constant. In this case, the MBTF is where λ ^ {\displaystyle {\hat {\lambda }}} is the maximum likelihood estimate of λ {\displaystyle \lambda } , maximizing the likelihood given above and k = ∑ σ i {\displaystyle k=\sum \sigma _{i}}

3876-420: The key property of being memoryless . In addition to being used for the analysis of Poisson point processes it is found in various other contexts. The exponential distribution is not the same as the class of exponential families of distributions. This is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes many other distributions, like

3944-519: The mean and median is | E ⁡ [ X ] − m ⁡ [ X ] | = 1 − ln ⁡ ( 2 ) λ < 1 λ = σ ⁡ [ X ] , {\displaystyle \left|\operatorname {E} \left[X\right]-\operatorname {m} \left[X\right]\right|={\frac {1-\ln(2)}{\lambda }}<{\frac {1}{\lambda }}=\operatorname {\sigma } [X],} in accordance with

4012-812: The memoryless property ) = ∑ k = 0 j − 1 1 ( n − k ) λ E ⁡ [ X ( i ) ] + E ⁡ [ X ( i ) 2 ] . {\displaystyle {\begin{aligned}\operatorname {E} \left[X_{(i)}X_{(j)}\right]&=\int _{0}^{\infty }\operatorname {E} \left[X_{(i)}X_{(j)}\mid X_{(i)}=x\right]f_{X_{(i)}}(x)\,dx\\&=\int _{x=0}^{\infty }x\operatorname {E} \left[X_{(j)}\mid X_{(j)}\geq x\right]f_{X_{(i)}}(x)\,dx&&\left({\textrm {since}}~X_{(i)}=x\implies X_{(j)}\geq x\right)\\&=\int _{x=0}^{\infty }x\left[\operatorname {E} \left[X_{(j)}\right]+x\right]f_{X_{(i)}}(x)\,dx&&\left({\text{by

4080-458: The memoryless property to replace E ⁡ [ X ( j ) ∣ X ( j ) ≥ x ] {\displaystyle \operatorname {E} \left[X_{(j)}\mid X_{(j)}\geq x\right]} with E ⁡ [ X ( j ) ] + x {\displaystyle \operatorname {E} \left[X_{(j)}\right]+x} . The probability distribution function (PDF) of

4148-555: The memoryless property}}\right)\\&=\sum _{k=0}^{j-1}{\frac {1}{(n-k)\lambda }}\operatorname {E} \left[X_{(i)}\right]+\operatorname {E} \left[X_{(i)}^{2}\right].\end{aligned}}} The first equation follows from the law of total expectation . The second equation exploits the fact that once we condition on X ( i ) = x {\displaystyle X_{(i)}=x} , it must follow that X ( j ) ≥ x {\displaystyle X_{(j)}\geq x} . The third equation relies on

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4216-402: 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: In a similar manner, mean down time (MDT) can be defined as The MTBF is the expected value of

4284-2304: The probability density of Z = X 1 + X 2 {\displaystyle Z=X_{1}+X_{2}} is given by f Z ( z ) = ∫ − ∞ ∞ f X 1 ( x 1 ) f X 2 ( z − x 1 ) d x 1 = ∫ 0 z λ 1 e − λ 1 x 1 λ 2 e − λ 2 ( z − x 1 ) d x 1 = λ 1 λ 2 e − λ 2 z ∫ 0 z e ( λ 2 − λ 1 ) x 1 d x 1 = { λ 1 λ 2 λ 2 − λ 1 ( e − λ 1 z − e − λ 2 z )  if  λ 1 ≠ λ 2 λ 2 z e − λ z  if  λ 1 = λ 2 = λ . {\displaystyle {\begin{aligned}f_{Z}(z)&=\int _{-\infty }^{\infty }f_{X_{1}}(x_{1})f_{X_{2}}(z-x_{1})\,dx_{1}\\&=\int _{0}^{z}\lambda _{1}e^{-\lambda _{1}x_{1}}\lambda _{2}e^{-\lambda _{2}(z-x_{1})}\,dx_{1}\\&=\lambda _{1}\lambda _{2}e^{-\lambda _{2}z}\int _{0}^{z}e^{(\lambda _{2}-\lambda _{1})x_{1}}\,dx_{1}\\&={\begin{cases}{\dfrac {\lambda _{1}\lambda _{2}}{\lambda _{2}-\lambda _{1}}}\left(e^{-\lambda _{1}z}-e^{-\lambda _{2}z}\right)&{\text{ if }}\lambda _{1}\neq \lambda _{2}\\[4pt]\lambda ^{2}ze^{-\lambda z}&{\text{ if }}\lambda _{1}=\lambda _{2}=\lambda .\end{cases}}\end{aligned}}} The entropy of this distribution

4352-408: The probability of a system failing within a certain timeframe is the inverse of its MTBF. Then, when considering series of components, failure of any component leads to the failure of the whole system, so (assuming that failure probabilities are small, which is usually the case) probability of the failure of the whole system within a given interval can be approximated as a sum of failure probabilities of

4420-407: The probability that a particular system will survive to its MTBF is 1 / e {\displaystyle 1/e} , or about 37% (i.e., it will fail earlier with probability 63%). 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

4488-604: The random variable T {\displaystyle T} indicating the time until failure. Thus, it can be written as where f T ( t ) {\displaystyle f_{T}(t)} is the probability density function of T {\displaystyle T} . Equivalently, the MTBF can be expressed in terms of the reliability function R T ( t ) {\displaystyle R_{T}(t)} as The MTBF and T {\displaystyle T} have units of time (e.g., hours). Any practically-relevant calculation of

4556-411: 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: [REDACTED] For each observation, the "down time" is the instantaneous time it went down, which is after (i.e. greater than)

4624-2294: The variable which achieves the minimum is distributed according to the categorical distribution Pr ( X k = min { X 1 , … , X n } ) = λ k λ 1 + ⋯ + λ n . {\displaystyle \Pr \left(X_{k}=\min\{X_{1},\dotsc ,X_{n}\}\right)={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.} A proof can be seen by letting I = argmin i ∈ { 1 , ⋯ , n } ⁡ { X 1 , … , X n } {\displaystyle I=\operatorname {argmin} _{i\in \{1,\dotsb ,n\}}\{X_{1},\dotsc ,X_{n}\}} . Then, Pr ( I = k ) = ∫ 0 ∞ Pr ( X k = x ) Pr ( ∀ i ≠ k X i > x ) d x = ∫ 0 ∞ λ k e − λ k x ( ∏ i = 1 , i ≠ k n e − λ i x ) d x = λ k ∫ 0 ∞ e − ( λ 1 + ⋯ + λ n ) x d x = λ k λ 1 + ⋯ + λ n . {\displaystyle {\begin{aligned}\Pr(I=k)&=\int _{0}^{\infty }\Pr(X_{k}=x)\Pr(\forall _{i\neq k}X_{i}>x)\,dx\\&=\int _{0}^{\infty }\lambda _{k}e^{-\lambda _{k}x}\left(\prod _{i=1,i\neq k}^{n}e^{-\lambda _{i}x}\right)dx\\&=\lambda _{k}\int _{0}^{\infty }e^{-\left(\lambda _{1}+\dotsb +\lambda _{n}\right)x}dx\\&={\frac {\lambda _{k}}{\lambda _{1}+\dotsb +\lambda _{n}}}.\end{aligned}}} Note that max { X 1 , … , X n } {\displaystyle \max\{X_{1},\dotsc ,X_{n}\}}

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