dependability & maintainability theory and methods part 1: introduction and definitions
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Andrea Bobbio Dipartimento di Informatica Universit à del Piemonte Orientale, “ A. Avogadro ” 15100 Alessandria (Italy) [email protected] - http://www.mfn.unipmn.it/~bobbio/IFOA. Dependability & Maintainability Theory and Methods Part 1: Introduction and definitions. - PowerPoint PPT PresentationTRANSCRIPT
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Dependability & Maintainability Theory and Methods
Part 1: Introduction and definitions
Andrea BobbioDipartimento di Informatica
Università del Piemonte Orientale, “A. Avogadro”15100 Alessandria (Italy)
[email protected] - http://www.mfn.unipmn.it/~bobbio/IFOA
IFOA, Reggio Emilia, June 17-18, 2003
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Dependability: DefinitionDependability: Definition
Dependability is the property of a system to be dependable in time, i.e. such that reliance can justifiably be placed on the service it delivers.
Dependability extends the interest on the system from the design and construction phase to the operational phase (life cycle).
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What dependability theory and practicewants to avoid
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dependability
measures
reliabilityavailabilitymaintainabilitysafetysecurity
means fault forecastingfault tolerancefault removalfault prevention
threats faults errorsfailures
Dependability: TaxonomyDependability: Taxonomy
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Quantitative analysisQuantitative analysis
The quantitative analysis aims at numerically evaluating measures to characterize the dependability of an item:
Risk assessment and safety
Design specifications
Technical assistance and maintenance
Life cycle cost
Market competition
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Risk assessment and safetyThe risk associated to an activity is given proportional to the probability of occurrence of the activity and to the magnitute of the consequences.
A safety critical system is a system whose incorrect behavior may cause a risk to occur, causing undesirable consequences to the item, to the operators, to the population, to the environment.
R = P M
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Design specifications
Technological items must be dependable.
Some times, dependability requirements (both qualitative and quantitative) are part of the design specifications:
Mean time between failures
Total down time
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Technical assistance and maintenance
The planning of all the activity related to the technical assistance and maintenance is linked to the system dependability (expected number of failure in time).
planning spare parts and maintenance crews;
cost of the technical assistance (warranty period);
preventive vs reactive maintenance.
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Market competition
The choice of the consumers is strongly influenced by the perceived dependability.
advertisement messages stress the dependability;
the image of a product or of a brand may depend on the dependability.
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Purpose of evaluation
Understanding a system– Observation– Operational environment– Reasoning
Predicting the behavior of a system– Need a model– A model is a convenient abstraction– Accuracy based on degree of extrapolation
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Methods of evaluation
Measurement-Based Most believable, most expensive Not always possible or cost effective during system design
Model-Based Less believable, Less expensive Analytic vs Discrete-Event Simulation Combinatorial vs State-Space Methods
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Measurement-BasedMost believable, most expensive;
Data are obtained observing the behavior of physical objects.
field observations; measurements on prototypes; measurements on components (accelerated tests).
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Closed-formAnswers
NumericalSolution
Analytic
Simulation
All models are wrong; some models are useful
Models
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Methods of evaluation
Measurements + Models data bank
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The probabilistic approachThe probabilistic approachThe mechanisms that lead to failure a technological object are very complex and depend on many physical, chemical, technical, human, environmental … factors.
The time to failure cannot be expressed by a determin-istic law.
We are forced to assume the time to failure as a random variable.
The quantitative dependability analysis is based on a probabilistic approach.
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ReliabilityReliability
The reliability is a measurable attribute of the dependability and it is defined as:
The reliability R(t) of an item at time t is the probability that the item performs the required function in the interval (0 – t) given the stress and environmental conditions in which it operates.
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Basic Definitions: cdfLet X be the random variable representing the time to failure of an item.
The cumulative distribution function (cdf) F(t) of the r.v. X is given by:
F(t) = Pr { X t }
F(t) represents the probability that the item is already failed at time t (unreliability) .
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Basic Definitions: cdf
Equivalent terminoloy for F(t) :
CDF (cumulative distribution function)
Probability distribution function
Distribution function
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Basic Definitions: cdf
1
0
F(t)
ta
F(b)
F(a)
b
F(0) = 0lim F(t) = 1t
F(t) = non-decreasing
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Basic Definitions: ReliabilityLet X be the random variable representing the time to failure of an item.
The survivor function (sf) R(t) of the r.v. X is given by:
R (t) = Pr { X > t } = 1 - F(t)
R(t) represents the probability that the item is correctly working at time t and gives the reliability function .
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Basic Definitions
Equivalent terminology for R(t) = 1 -F(t) :
Reliability
Complementary distribution function
Survivor function
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Basic Definitions: Reliability
1
0
R(t)
ta b
R(0) = 1lim R(t) = 0t
R(t) = non-increasing
R(a)
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Basic Definitions: density
Let X be the random variable representing the time to failure of an item and let F(t) be a derivable cdf:
The density function f(t) is defined as:
d F(t)f (t) = ——— dt
f (t) dt = Pr { t X < t + dt }
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Basic Definitions: Density
0
f (t)
ta b
f(x) dx = Pr { a < X b } = F(b) – F(a) a
b
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Basic Definitions: Density
1
0
f (t)
t
00
dttRdtttfXEMTTF
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Basic Definitions
Equivalent terminology: pdf
probability density function
density function
density
f(t) = dtdF ,)(
)()(
0
t
t
dxxf
dxxftF
For a non-negativerandom variable
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Quiz 1:The higher the MTTF is, the higher the
item reliability is.1. Correct2. Wrong
The correct answer is wrong !!!
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Hazard (failure) rate
h(t) t = Conditional Prob. system will fail in (t, t + t) given that it is survived until time t
f(t) t = Unconditional Prob. System will fail in (t, t + t)
)(1)(
)()()(
tFtf
tRtfth
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is the conditional probability that the unit will fail in the interval given that it is functioning at time t.
is the unconditional probability that the unit will fail in the interval
Difference between the two sentences:– probability that someone will die between 90 and 91, given that he
lives to 90– probability that someone will die between 90 and 91
The Failure Rate of a Distribution
tΔth),( ttt
ttf ),( ttt
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DFR IFR
Decreasing failure rate Increasing fail. rate
h(t)
t
CFRConstant fail. rate
(useful life)
(infant mortality – burn in) (wear-out-phase)
Bathtub curve
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Infant mortality (dfr)Also called infant mortality phase or reliability growth phase. The failure rate decreases with time.
Caused by undetected hardware/software defects; Can cause significant prediction errors if steady-state failure rates are used;Weibull Model can be used;
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Useful life (cfr)The failure rate remains constant in time (age independent) .
Failure rate much lower than in early-life period.
Failure caused by random effects (as environmental shocks).
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Wear-out phase (ifr)The failure rate increases with age.
It is characteristic of irreversible aging phenomena (deterioration, wear-out, fatigue, corrosion etc…)
Applicable for mechanical and other systems.
(Properly qualified electronic parts do not exhibit wear-out failure during its intended service life)
Weibull Failure Model can be used
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Cumul. distribution function:
Reliability :
Density Function :
Failure Rate (CFR):
Mean Time to Failure:
0 1 tetF t
0 t tetf
0 ttR e t
tRtfth
1MTTF
Exponential DistributionFailure rate is age-independent (constant).
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2.50
The Cumulative Distribution Function of an Exponentially Distributed Random
Variable With Parameter = 1
F(t)1.0
0.5
0 1.25 3.75 5.00 t
F(t) = 1 - e - t
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2.50
The Reliability Function of an Exponentially Distributed Random
Variable With Parameter = 1
R(t)1.0
0.5
0 1.25 3.75 5.00 t
R(t) = e - t
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Exponential Density Function (pdf)
f(t)
MTTF = 1/
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Memoryless Property of the Exponential Distribution
Assume X > t. We have observed that the
component has not failed until time t
Let Y = X - t , the remaining (residual) lifetime
y
t
etXPtyXtPtXtyXP
tXyYPyG
1)(
)()|(
)|()(
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Memoryless Property of the Exponential Distribution (cont.)
Thus Gt(y) is independent of t and is identical to the original exponential distribution of X
The distribution of the remaining life does not depend on how long the component has been operating
An observed failure is the result of some suddenly appearing failure, not due to gradual deterioration
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Quiz 3: If two components (say, A and B) have independent
identical exponentially distributed times to failure, by the “memoryless” property, which of the following is
true? 1. They will always fail at the same time2. They have the same probability of failing at time
‘t’ during operation3. When these two components are operating
simultaneously, the component which has been operational for a shorter duration of time will survive longer
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0
0
0 1
1
tetR
tettf
tetF
t
t
t
Weibull Distribution
Distribution Function:
Density Function:
Reliability:
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1
1
0 1
)(
)( ttth
tR
tf
Weibull Distribution : shape parameter;
: scale parameter.
Failure Rate:
1 DfrCfr
Ifr
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Failure Rate of the Weibull Distribution with Various Values of
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Weibull Distribution for Various Values of
Cdf density
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We use a truncated Weibull Model
Infant mortality phase modeled by DFR Weibull and the steady-state phase by the exponential
0 2,190 4,380 6,570 8,760 10,950 13,140 15,330 17,520Operating Times (hrs)
Failu
re-R
ate
Mul
tiplie
r
76543210
Figure 2.34 Weibull Failure-Rate Model
Failure Rate Models
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Failure Rate Models (cont.)
This model has the form:
where:steady-state failure rate
is Weibull shape parameter
Failure rate multiplier =
SS
W tCt
1)(760,8760,81
tt
SSWC ,11
SSW t)(
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Failure Rate Models (cont.)
There are several ways to incorporate time dependent failure rates in availability modelsThe easiest way is to approximate a continuous function by a piecewise constant step function
2,190 4,380 6,570 10,950 13,140 15,330 17,520Operating Times (hrs)
Failu
re-R
ate
Mul
tiplie
r
76543210
Discrete Failure-Rate Model
8,7600
1
2 SS
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Failure Rate Models (cont.)
Here the discrete failure-rate model is defined by:
ss
W t
2
1)(
760,8760,8380,4380,40
ttt
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A lifetime experimentA lifetime experiment
N i.i.d components are put in a life test experiment.
1
2
3
4
N
t = 0
X 1
X 2X 3
X 4
X N
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A lifetime experimentA lifetime experiment1234
N
X 1X 2
X 3X 4
X N