1. 01-02-2012
Reliability Reliability Function
• A relative frequency definition of reliability can • Let us put some N items under life test under the
be stated as follows: same environmental conditions and observe the
If a large number of independent systems/ number of failures at a predetermined interval of
subsystems/components are operated at the same
y p p time, regularly. Obviously the number of failures
time, the R(t) can be estimated to be the ratio of would go on i
ld increasing until all th N it
i til ll the items putt
systems/subsystems/ components still running under test have failed. The test is terminated only
over time t and the initial number of when all the items have failed. Let us consider that
systems/subsystems/components put to operation. we are considering catastrophic failures only. Let us
assume that the number of failures over a time t are:
nf (t). Naturally, the survivals shall be:
Systems/subsystems/components can be called, in ns(t) = N- nf (t).
general, as items.
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Reliability Function Reliability Function
• From the definition of probability, we can • The item unreliability likewise can be
define reliability as: defined as:
• Obviously this definition requires that the • In fact Q(t) shall also be the cumulative
test is conducted for a large value of N distribution (cdf) or F(t) if the random
under identical conditions of tests and the variable, t , is taken as time-to-failure for
tests are conducted independently. the observation, in this test.
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2. 01-02-2012
Reliability and Unreliability Reliability Function
• Cumulative distribution function can also be
defined as: • The cumulative distribution function
increases from 0 to 1 as the random
variable, t increases from zero to its highest
, g
value towards infinity. Since at t = 0, the
item was operating, Q(t)=F(t)=0, but as
• Reliability (Survival Function) is given by: t, F(t) 1. Also, at t = 0, R(t)= 1 and
R(t) 0 as t. We also know that the
Density Function would be defined by: derivative of cdf of a continuous rv provides
probability density function or f(t).
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Reliability Function Reliability Function
• By dividing both sides of equation (1), we can
define what is known as hazard rate, viz.,
• Realizing that,
we obtain an important relationship:
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3. 01-02-2012
Hazard Rate
Reliability Function
In other words, the hazard rate is the
• It is important to highlight at this stage the
difference between rate of failing and hazard conditional probability of an
rate. At two different points of time, say at t1 and
p , y instantaneous failure given that the
f g
t2, we can have the same rate of failing but as the component is surviving upto time t,
number of survivals would go on reducing the
hazard rate would not remain the same even if the
divided by the length of the short time
rate of failing is the same. Hazard rate is a better interval involved.
indicator of how hazardous a situation is at a given
point of time.
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Reliability Function Reliability Function
• It is obvious from the foregoing expressions that
• Now reverting to an expression derived earlier, we
f(t) is the rate of failing normalized to the
have:
original population put to test whereas h(t) is the
rate of failing normalized to the number of
survivals. The difference forms of f(t) and h(t) • We can solve this first order differential equation
that can be used to obtain them from the histogram by separable variable technique with initial
of life-test data, is given by: condition that at t = 0, R(0) = 1. We then obtain a
very important relationship in reliability studies:
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4. 01-02-2012
Reliability Function Characterizations
Reliability Characteristics of a Unit
A probability distribution is completely
characterized by each of the following
entities:
Hazard Rate • Density function (f(t))
f(t), h(t) h(t)=f(t)/R(t) f(t1) • Cumulative distribution function (F(t))
• Reliability (R(t))
• Hazard rate (h(t))
R(t1) Area
Q(t1) or
F(t1) Area
In other words: if one of these entities is known, we
t1
can compute the other entities from it.
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Fig. 3.1: Reliability Characteristic of a Unit
Case I (Density Function is known) Case II (Distribution Function is known)
t f (t )  F ' (t )
F (t )   f ( x )dx
0 R(t )  1  F ( t )

R(t )   f ( x )dx F ' (t )
t h( t ) 
f (t ) 1  F (t )
h(t ) 
R( t ) 15 16
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5. 01-02-2012
Case III (Reliability is known) Reliability and MTTF
t
  h(u)du
f ( t )   R' ( t ) R( t )  e 0
t
F ( t )  1  R( t ) Th function
The f i (t ) 
cumulative hazard rate.
 h( u)du
0
is called the
i ll d h
 R' ( t ) The expected lifetime is often called Mean Time To
h(t )  Failure (MTTF).
R( t ) It is dangerous to make decisions based on MTTF
only; never cross the river based on average depth.
Consider the variance as well!
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MTTF Reliability Expressions
The mean-time-to-failure in case of a continuous • Using the expression:
random variable such as time- to- failure, t ,is given • We can obtain the reliability of a unit, which
by: follows a given failure distribution or has a
  t    i h d F
given hazard rate. For example:l
 
MTTF 
 tf ( t )dt 


 


 du f ( t )dt 

f (t )dtdu  R(u)du 1. For h(t) = , we have :
0 00  0u 0 2. For linearly increasing hazard rate, h(t) =
This applies equally well to component MTTF, a+bt , we have:
subsystem MTTF or even to system MTTF. We
must, however, take appropriate reliability of the
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entity.
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6. 01-02-2012
Reliability Expressions Reliability Expressions
3. If the hazard rate is increasing non-linearly, i.e., • Therefore, if we know the variation of
the failure distribution is Weibull or so, with hazard rate with time,we can obtain the
hazard rate given by:
g y expression for component reliability using
the very versatile general expression:
The reliability expression would be given by:
Where T is the mission time.
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Phases of Life Typical Bathtub Curve
• There are three phases of life of any unit. These are: Bathtub curve describes the variation of hazard rate with
time, which is generally taken as life of a unit.
• Early Life or Infancy Period
• Useful Life or Prime of Life
• W
Wearout Phase or P i d
t Ph Period
• Each phase has a particular type of failure dominant
and has respectively over these three phases of life
either decreasing, constant or increasing hazard rate
characteristic. These failures result in an overall
characteristic over the life time, which apparently looks
like a bath tub . Hence the name.
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7. 01-02-2012
Early Life Useful Life
• Early life has predominantly Quality failures, which • During this period, the hazard rate is very often small
show up in early life of a unit and can be traced mainly and approximately constant. It is during this period
to the manufacturer’s carelessness and can be that a unit is put to effective use and usefully employed
attributed to defective designs, use of substandard during the entire life time. During this period, early or
material, poor workmanship or poor quality control. quality failures as well as wear out failures are
These failures result in a very high hazard rate in the negligible. O l sudden or catastrophic f il
li ibl Only dd hi failures can
beginning and keeps decreasing as the time passes. occur, which are primarily caused by sudden and step
Early failures can be eliminated through the use of increase in the stress level beyond the design strength.
debugging process which consists of operating a unit These failures occur randomly and unexpectedly.
under conditions of use for a period of time However their frequency over a long period of time is
corresponding to the preponderance of early failures. constant. One cannot eliminate these failures but their
The length of debugging period is decided by observing probability can be reduced by improving reliability at
the failure distribution and by following a specific the design stage.
debugging procedure. This is also known as burn in
period. 25 26
Wearout Life Other Hazard Models
• As the unit reaches the end of its life, parts begin to • Next slide provides a list of some of the distributions
wear out and the hazard rate of the unit begins to rise that are extensively used in reliability studies. But one
rapidly. Early or quality failures are very rare during must lose sight of the practicability and not fit a
this period and stress related failures occur with the complicated model where it is not actually necessary.
same frequency as they occur in other phases of life.
The failures that occur d i
Th f il h during this period are aptly
hi i d l
called as gradual or wearout failures and are dominant
in old age or towards the end of the life time. These
failures keep increasing slowly over the life as the
deterioration increases with age and the age at which
these become predominant depends on the
environment, a unit is operated. It is advisable that the
replacement of a unit should be done about the point tw
in time. Otherwise the sudden failure may be costly in
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consequences.
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8. 01-02-2012
Constant Hazard Rate
The useful life period of bath tub curve during which
catastrophic failures are dominant is often characterized
by constant hazard rate thus is best modeled by the
exponential failure distribution. The failure process is
memoryless and does not recognize the time already
elapsed already during the life span.
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Decreasing /Increasing Hazard rates Decreasing /Increasing Hazard rates
• Decreasing or increasing hazard rates can be modeled Decreasing/increasing hazard rates can also be modeled using
by Weibull distribution by suitably choosing the value Gamma distribution. When  < 1, decreasing hazard rates
of parameter . When  < 1, we get reliability function would be described whereas for  > 1, we can obtain
corresponding to decreasing Hazard rate. However, if  increasing hazard rates with the help of Gamma distribution.
> 1, we get reliability function corresponding to
, g y p g
increasing Hazard rate.
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9. 01-02-2012
Thanks
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