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# Imprecise Reliability (tutorial)

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Imprecise Reliability
(tutorial)
Lev Utkin
ISIPTA 09, July 2009
Lev Utkin Imprecise Reliability (tutorial) Reliability is the measurable capability of a system to perform its
intended function in the required time under speci ed conditions.
Reliability index is a quantitative measure of reliability properties.
PrfX > Yg, H(t) = PrfX > tg, EX, ...
Reliability is the probability that a system will perform
satisfactorily for at least a given period of time when used under
stated conditions.
2 Reliability as the property (capability):
De nitions of reliability
Standard de nitions of reliability
1 Reliability as the probability:
Lev Utkin Imprecise Reliability (tutorial) Reliability is the measurable capability of a system to perform its
intended function in the required time under speci ed conditions.
Reliability index is a quantitative measure of reliability properties.
PrfX > Yg, H(t) = PrfX > tg, EX, ...
2 Reliability as the property (capability):
De nitions of reliability
Standard de nitions of reliability
1 Reliability as the probability:
Reliability is the probability that a system will perform
satisfactorily for at least a given period of time when used under
stated conditions.
Lev Utkin Imprecise Reliability (tutorial) Reliability is the measurable capability of a system to perform its
intended function in the required time under speci ed conditions.
Reliability index is a quantitative measure of reliability properties.
PrfX > Yg, H(t) = PrfX > tg, EX, ...
De nitions ...

Subjects

##### Independence (Kansas)

Informations Imprecise Reliability (tutorial)
Lev Utkin
ISIPTA09, July
Lev Utkin
2009
Imprecise Reliability
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Reliability
(tutorial)
Lev Utkin
Imprecise
probability:
Denitions of reliability
of reliability
Standard denitions
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Reliability
as
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Standard denitions of reliability
1Reliability as the probability:
Denitions of reliability
Reliabilityis theprobabilitythat a system will perform satisfactorily for at least a given period of time when used stated conditions.
under
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2Reliability as the property (capability):
Reliabilityis theprobabilitythat a system will perform satisfactorily for at least a given period of time when used under stated conditions.
1Reliability as the probability:
Denitions of reliability
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Standard denitions of reliability
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2Reliability as the property (capability):
Reliabilitycapability of a system to perform itsis the measurable intended function in the required time under specied conditions.
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1Reliability as the probability:
Reliabilityis theprobabilitythat a system will perform satisfactorily for at least a given period of time when used under stated conditions.
Standard denitions of reliability
Denitions of reliability
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Standard denitions of reliability
1Reliability as the probability:
Reliabilityis theprobabilitythat a system will perform satisfactorily for at least a given period of time when used under stated conditions.
2Reliability as the property (capability):
Reliabilitycapability of a system to perform itsis the measurable intended function in the required time under specied conditions.
Reliability indexis a quantitative measure of reliability properties.
PrfX>Yg,H(t) =PrfX>tg,EX, ...
LevUtkinImpreciseReliability(tutorial) Denitions of reliability
Two main tasks in reliability analysis
Two di¤erent aspects (problems, parts) of reliability analysis can be selected:
1
2
3
Statistical inference of the component (system) reliability measures
Standard methods of statistical inference, regression analysis, etc. for computing reliability measures from statistical data and expert judgments
System reliability analysis
Probabilities (expectations) of a function of random times to failure.
Some specic problems of reliability and risk analyses
LevUtkinImpreciseReliabiliyt(tutorial) Denitions of reliability
System reliability analysis
e If there is a vector ofnrandom variablesX= (X1, ...,Xn):
unit times to failure for a system ofnunits, load or stress factors for a structural systems, switch times, times to repair, etc.
e and a system reliability is dened as a functionY=g(X):
system time to failure, e stress minus strength (g(X) =X1X2), etc.
then our goal is to compute reliability indices
e e Prfg(X)>tg,Eg(X), ...
under two assumptions.
LevUtkinImpreciseReliabiliyt(tutorial) Denitions of reliability
Two main assumptions
1all probabilities of events or probability distributions of r.v. X1, ...,Xnare known or perfectly determinable; 2the system units or r.v.X1, ...,Xnare statistically independent or their dependence is precisely known.
The assumptions are usually not fullled. As a result, thereliability may be toounreliableand risky.
LevUtkinImpreciseReliability(tutorial) 3
Alternative approaches
How to deal with the large imprecision by analyzing large systems? How to interpret the possibilistic reliability measures? How to take into account conditions of independence? How how how ...?
Reliability in the framework of random sets and evidence theories (Hall and Lawry 2001,Tonon, Bernardini, Elishako¤ 1999, Oberguggenberger, Fetz, Pittschmann 2000, etc.)
1 2
Interval reliability (interval probabilities in the framework of standard interval calculation) /not interesting/. Fuzzy (possibilistic) reliability as an extension of interval models (Cai et al. 1996, de Cooman 1996, Utkin-Gurov 1996, etc.). The models have many open questions:
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An attempt to consider sets of distributions:
ageingaspects of lifetime distributions, in particular, IFRA (increasing failure rate average) and DFRA (decreasing failure rate average) distributions (Barlow and Proschan 1975); various nonparametric or semi-parametric classes of probability distributions (Barzilovich and Kashtanov 1971).
1
Frechet bounds for series systems (Y=min(X1, ...,Xn)) (Barlow and Proschan 1975).
An attempt to use some models of joint probability distributions for taking into account the lack of independence:
3
An attempt to use bounds for system reliability (Lindqvist and Langseth 1998).
)
2
Elements of imprecise reliability in the classical approach
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