1 Stracener_EMIS 7305/5305_Spr08_01.31.08 Reliability Models & Applications (continued) Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering EMIS 7305/5305 Systems Reliability, Supportability and Availability Analysis Systems Engineering Program Department of Engineering Management, Information and Systems
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Stracener_EMIS 7305/5305_Spr08_01.31.08 1 Reliability Models & Applications (continued) Dr. Jerrell T. Stracener, SAE Fellow Leadership in Engineering.
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Stracener_EMIS 7305/5305_Spr08_01.31.08
Reliability Models & Applications (continued)
Dr. Jerrell T. Stracener, SAE Fellow
Leadership in Engineering
EMIS 7305/5305Systems Reliability, Supportability and Availability Analysis
Systems Engineering ProgramDepartment of Engineering Management, Information and Systems
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Stracener_EMIS 7305/5305_Spr08_01.31.08
The Normal or Gaussian Model:
• DefinitionA random variable T is said to have the Normal (Gaussian) Distribution with parameters and , where > 0, if the density function of T is:
, for - < t <
• DefinitionIf T ~ N(,) and if , then Z ~ N(0,1)
the Standard Normal Distribution and Cumulative Probability is tabulated for various values of z.
22
t2
1
e2
1 )t(f
-T
Z
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Properties of the Normal Model:
• Probability Distribution Function
Where (Z) is the Cumulative Probability DistributionFunction of the Standard Normal Distribution.
• Reliability Function
provided that P(T < 0) 0
t
)t(F
t
-1 )t(R
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Properties of the Normal Model:
• MTBF (Mean Time Between Failure)
• Standard Deviation of Time to Failure =
• Failure Rate
MTBF
)t(R
)t(f )t(h
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Properties of the Normal Model - Failure Densities:
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The Normal Model - Example
Example=1,000=100
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0
z
z
Normal Distribution:
Standard Normal Distribution ~ X ~ N (, )
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Properties of the Normal Model - Standard Normal Distribution:
-t
Z
Table of Probabilities p
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Standard Normal Distribution0 0.01 0.02 0.03 0.04 0.09 0.06 0.07 0.08 0.09
• DefinitionA random variable T is said to have the Lognormal Distribution with parameters and , where - < < and > 0, if the density function of T is:
, for t > 0
, for t 0
• RemarkThe Lognormal Model is often used as the failure distribution for mechanical items and for the distribution of repair times.
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tln2
1
e2t
1 )t(f
0
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Properties of the Lognormal Model:
• Failure Distribution
where (z) is the cumulative distribution function
• Reliability Function
• If T ~ LN(,), then Y = lnT ~ N(,)
tln
)t(F
tln
-1 )t(R
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Properties of the Lognormal Model:
• MTBF (Mean Time Between Failures)
•Variance of Time to Failure
• Failure Rate
2
2
1
eMTBF
tln
1
)t(f )t(h
1ee)T(Var222
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The Lognormal Model:
Failure rate functions for various values of and
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The Lognormal Model:
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A theoretical justification based on a certain materialfailure mechanism underlies the assumption that ductile strength X of a material has a lognormal distribution. Suppose the parameters are = 5 and = 0.1
(a) Compute E(X) and Var(X)
(b) Compute P(X > 120)
(c) Compute P(110 X 130)
(d) What is the value of median ductile strength?
(e) If ten different samples of an alloy steel of this type were subjected to a strength test, how many would you expect to have strength at least 120?
(f) If the smallest 5% of strength values were unacceptable, what would the minimum acceptable strength be?
Example - Ductile Strength
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Example - Solution
59.223
01005.0*83.22247
)1()(
is X of variance theand
,16.149
)(
is X ofmean The
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2
2
2
2
)1.0(5
2
eeXVar
e
eXE
(a)
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9834.0
0166.01
)13.2(1
)1.0
579.4(1
)1.0
5120ln(1
)120(1)120(
ZP
ZP
ZP
XPXP(b)
Example - Solution
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0921.0
0013.00934.0
)00.3()32.1(
)32.100.3(
)1.0
5130ln
1.0
5110ln()130110(
ZP
ZPXP(c)
41.1485.0 ex(d)
Example - Solution
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83.9
)9834.0(10
)(
and
4)B(10,0.983~Y Then,
120least at ofstrength with testsofnumber Y
(b)part from 9834.0)120(
np
YE
XPp
(e)
Example - Solution
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0.05,-1.645)P(Z Since
,05.0)XP(Xwhich
for,say X, of value thefind toneed We
0.05
05.0
x
(f)
Example - Solution
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9015.125
Finally,
8355.4
),645.1)(1.0(5ln
,645.1ln
05.0)ln
P(Z
and
8355.40.05
0.05
0.05
0.05
ex
x
x
x
Example - Solution
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The Binomial Model:
Definition
- If X is the number of successes in n trials, where:
(1) The trials are identical and independent,
(2) Each trial results in one of two possible outcomes
success or failure,
(3) The probability of success on a single trial is p, and is constant from trial to trial, then X has the binomial Distribution with Probability Mass Function given by:
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The Binomial Model
Probability Mass Function
, for x = 0, 1, 2, ... , n
, otherwise
where
x)(XPp)n,|b(x)x(b
0
!xn!x
!n
x
n
xnx ppx
n
1
=
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Binomial Distribution
Rule:
)p,n;x(bq
p
1x
xn )p,n;1x(b
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Binomial Distribution
• Mean or Expected Value
= np
• Standard Deviation
= (npq)1/2 ,
where q=1-p
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The Binomial Model - Example Application 1:
Four Engine Aircraft
Engine Unreliability Q(t) = p = 0.1
Mission success: At least two engines survive
Find RS(t)
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The Binomial Model - Example Application 1- Solution
X = number of engines failing in time t
RS(t) = P(x 2) = b(0) + b(1) + b(2)
= 0.6561 + 0.2916 + 0.00486 = 0.9963
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Number of Failures Model:
• Definition- If T ~ E() and if X is the number of failures occurringin an interval of time, t, then X ~ P(t/ ), the Poisson Distribution with Probability Mass Function given by:
An item has a failure rate of = 0.002 failures per hourif the item is being put into service for a period of 1000hours. What is the probability that 4 spares in stock willbe sufficient?
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Expected number of failures (spares required) = t = 2
P(enough spares) = P(X 4)
= p(0) + p(1) + p(2) + p(3) + p(4) = 0.945
or about a 5% chance of not having enough spares!
The Poisson Model - Example Application - Solution