Dependability & Maintainability
Theory and Methods
Part 1: Introduction and definitions
Andrea Bobbio
Dipartimento di Informatica
Università del Piemonte Orientale, “A. Avogadro”
15100 Alessandria (Italy)
[email protected] - http://www.mfn.unipmn.it/~bobbio/IFOA
A. Bobbio
IFOA, Reggio
Emilia,
June 2003
17-18, 2003
Reggio Emilia,
June 17-18,
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Dependability: 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 practice
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wants to avoid
Dependability: Taxonomy
dependability
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measures
reliability
availability
maintainability
safety
security
means
fault
fault
fault
fault
threats
faults
errors
failures
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forecasting
tolerance
removal
prevention
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Quantitative 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 safety
The risk associated to an activity is given proportional
to the probability of occurrence of the activity and to
the magnitute of the consequences.
R=PM
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.
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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-Based
Most 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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Models
Closed-form
Answers
Numerical
Solution
Analytic
Simulation
All models are wrong; some models are useful
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Methods of evaluation
Measurements + Models
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data bank
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The probabilistic approach
The 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 deterministic 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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Reliability
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: cdf
Let 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
F(t)
1
F(b)
F(a)
0
a
b
t
F(0) = 0
lim F(t) = 1
t
F(t) = non-decreasing
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Basic Definitions: Reliability
Let 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
R(t)
1
R(a)
0
a
b
t
R(0) = 1
lim R(t) = 0
t
R(t) = non-increasing
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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
f (t)
0
a
b
t
b
 f(x) dx = Pr { a < X  b } = F(b) – F(a)
a
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Basic Definitions: Density
f (t)
1
0
t
MTTF  EX    tf t dt   Rt dt
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

0
0
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Basic Definitions
Equivalent terminology: pdf
 probability density function
 density function
 density
dF
 f(t) =
dt
F (t )  
t

f ( x)dx
t
  f ( x)dx ,
0
For a non-negative
random variable
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Quiz 1:
The higher the MTTF is, the higher the
item reliability is.
1. Correct
2. Wrong
The correct answer is wrong !!!
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Hazard (failure) rate
f (t )
f (t )
h(t ) 

R(t ) 1  F (t )
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)
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The Failure Rate of a Distribution
ht  Δ t is the conditional probability that the unit will
fail in the interval ( t , t   t ) given that it is
functioning at time t.
f t   t is the unconditional probability that the unit
will fail in the interval ( t , t   t )
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
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Bathtub curve
h(t)
(infant mortality – burn in)
(wear-out-phase)
CFR
Constant fail. rate
(useful life)
DFR
IFR
t
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Decreasing
failure rate
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Increasing fail. rate
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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 steadystate 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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Exponential Distribution
Failure rate is age-independent (constant).
Cumul. distribution function: F t   1  e  t
R  t   e  t
Reliability :
Density Function :
Failure Rate (CFR):
Mean Time to Failure:
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t0
f t    e   t
f t 
ht  

R t 
1
MTTF 
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t0
t0

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The Cumulative Distribution Function
of an Exponentially Distributed Random
Variable With Parameter  = 1
F(t)
1.0
F(t) = 1 - e -  t
0.5
0
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1.25
2.50
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3.75
5.00
t
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The Reliability Function of an
Exponentially Distributed Random
Variable With Parameter  = 1
R(t)
1.0
R(t) = e -  t
0.5
0
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1.25
2.50
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3.75
5.00
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
Gt ( y )  P (Y  y | X  t )
 P( X  y  t | X  t )
P (t  X  y  t )

 1  e  y
P( X  t )
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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 time
2. They have the same probability of failing at time
‘t’ during operation
3. 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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Weibull Distribution
Distribution Function:
F t   1  e
Density Function:
f t    t e
t0
Reliability:
Rt   e
t0
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  t
 1   t 
 t
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t0
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Weibull Distribution
 : shape parameter;
 : scale parameter.
Failure Rate:
 1
 1
 1
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ht  
f (t )
  t
 1
t0
R (t )
Dfr
Cfr
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
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density
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Failure Rate Models
We use a truncated Weibull Model
Figure 2.34 Weibull Failure-Rate Model
Failure-Rate Multiplier
7
6
5
4
3
2
1
0
0
2,190
4,380
6,570
8,760
10,950 13,140 15,330 17,520
Operating Times (hrs)
Infant mortality phase modeled by DFR Weibull and the
steady-state phase by the exponential
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Failure Rate Models (cont.)
This model has the form:

(
t
)

t
W
C1
  SS
1  t  8,760
t  8,760
where:
C 1  W 1,  SS  steady-state failure rate
 is Weibull shape parameter
Failure rate multiplier = W ( t)  SS
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Failure Rate Models (cont.)
There are several ways to incorporate time dependent
failure rates in availability models
The easiest way is to approximate a continuous function
by a piecewise constant step function
Discrete Failure-Rate Model
Failure-Rate Multiplier
7
6
1
5
4
2
3
2
1
0
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0
2,190
4,380
 SS
6,570 8,760 10,950 13,140 15,330 17,520
Operating Times (hrs)
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Failure Rate Models (cont.)
Here the discrete failure-rate model is
defined by:
W ( t )   1
 2
  ss
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0  t  4,380
4,380  t  8,760
t  8,760
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A lifetime experiment
X1
1
X2
2
X3
3
X4
4
XN
N
t=0
N i.i.d components are put in a life test experiment.
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A lifetime experiment
1
2
3
4
X1
X3
X2
X4
XN
N
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Normal Distribution