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```


How do you know how long your design
is going to last?
Is there any way we can predict how long
it will work?
Why do Reliability Engineers get paid so
much?
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3
Definitions

Random Experiment

Event or Outcomes

Event Space
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
What is an axiom?

2 of the 3 axioms
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
What is a random variable (RV)?
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What is a PDF?

Math Definition
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Normal Density
pX(x)
1
fX (x ) 
e
2 


1 x  
 

2  

2
x
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Uniform Density
1
fX (x ) 
,
b a
a x b
pX(x)
1/(b-a)
x
a
b
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Data for lifelengths of batteries (in hundreds of hours)
0.406
0.685
4.778
1.725
8.223
2.343
1.401
1.507
0.294
2.230
0.538
0.234
4.025
3.323
2.920
5.088
1.458
1.064
0.774
0.761
5.587
0.517
3.246
2.330
1.064
2.563
0.511
2.782
6.426
0.836
0.023
0.225
1.514
3.214
3.810
3.334
2.325
0.333
7.514
0.968
3.491
2.921
1.624
0.334
4.490
1.267
1.702
2.634
1.849
0.186
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Exponential Density
fX (x )  e
 x
,
x,  0
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Example: Based on the data for lifelengths of batteries
previously given, the random variable X representing
the lifelength has associated with it an exponential
density function with  = 0.5.
(a) Find the probability that the lifelength of a particular
battery is less than 200 or greater than 400 hours.
(b) Find the probability that a battery lasts more than
300 hours given that it has already been in use for
more than 200 hours.
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Square Roots of the lifelengths
of batteries
0.637
1.531
0.733
2.256
2.364
1.601
0.152
1.826
1.868
1.126
0.828
1.184
0.484
1.207
0.719
0.715
0.474
1.525
1.709
1.305
2.186
1.228
2.006
1.032
1.802
1.668
1.230
0.577
1.274
1.623
1.313
0.542
1.823
0.880
1.526
2.535
1.793
2.741
0.578
1.360
2.868
1.493
1.709
0.872
1.032
0.914
1.952
0.984
2.119
0.431
WeiBull Distribution
fX (x )  x
e
 1
 x 
,
x 0
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Example: The length of service time during which a
certain type of thermistors produces resistances within its
specifications has been observed to follow a Weibull
Distribution with  = 1/50 and  = 2 (measurements in
thousand of hours).
(a) Find the probability that one of these thermistors, to
be installed in a system today, will function properly
for over 10,000 hours.
(b) Find the expected lifelength for the thermistor of this
type.
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Reliability (defn)

Failure Rate
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(t)
burn-in
service lifetime / constant failure rate
wear-out
mechanical
devices
electrical devices
t
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

See the book for derivation of R(t).
If the failure rate is constant, then R(t) = ?
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Example: Consider a transistor with a constant failure rate of
 = 1/106 hours.
(a) What is the probability that the resistor will be operable in 5 years?
(b) Determine the MTTF and the reliability at the MTTF.
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Def’n (Series System) =

We model this as
S1
S2
...
Sn
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Definition: Redundancy
Definition: Parallel System
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S1
S2
.
.
.
Sn
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Example: Redundant Array of Independent disks (RAID)
In a RAID, multiple hard drives are used to store the same data,
thus achieving redundancy and increased reliability. One or more
of the disks in the system can fail and the data can still be recovered.
However, if all disks fail, then the data is lost.
Assume that each disk drive has a failure rate of  = 10 failures/106
hrs. How many disks must the system have to achieve a reliability of
98% in 10 years?
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S2
S1
S3
S4
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Quiz: Redundant Array of Independent disks (RAID)
Your company intends to design, manufacture, and market a new
RAID for network servers. The system must be able to store a total of
500 GB of user data and must have a reliability of at least 95% in 10
years. In order to develop the RAID system, 20-GB drives will be
designed and utilized. To meet the requirement, you have decided to
use a bank of 25 disks (25x20 GB = 500 GB) and utilize a system
redundancy of 4 (each of the 25 disks has a redundancy of 4). What
must the reliability of the 20 GB drive be in 10 years in order to meet
the overall system reliability requirement?
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What factors influence the failure rate?
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Low Frequency FET, Appendix C.
  b T A Q E failures 10 hours
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How would you find each of these?
25
Vcc = 5V
IC
RC = 100
VO = 5V or 0V
=
RB = 10k
VI
VO
2N3904
VI = 0 or 5V
IB
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CA
Ambient, TA
Junction, TJ
JC
Case, TC
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TJ
JC
PD
TC
CA
TA
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SA
Ambient, TA
Heat Sink, TS
CS
thermal
conductive
paste
Junction, TJ
JC
Case, TC
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TJ
JC
TC
PD
CS
TS
SA
TA
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PD,max, mW
with heat sink
without heat sink
slopes = -1JA mW/ oC
TA, oC
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So far, we have only looked at a single device.
We are interested in collection of devices into
a system!
For example
Vcc = 5V
IC
RC = 100
VO = 5V or 0V
=
RB = 10k
VI
VO
2N3904
VI = 0 or 5V
IB
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