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EMIS 7300
SYSTEMS ANALYSIS METHODS
FALL 2005
Dr. John Lipp
Copyright © 2005 Dr. John Lipp
Reliability Definitions
• Reliability is the probability that an item will successfully
perform its intended function
– In general, reliability is a function of time, R(t).
– Un-reliability is denoted Q(t) = 1 – R(t).
• Reliability / un-reliability are related to the CDF
– R(t) = P(working up to time t) = 1 – F(t).
– Q(t) = P(failure before time t) = F(t).
• Note that R(t) + Q(t) = 1.
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-2
Reliability Definitions (cont.)
• The probability of failure in a given time interval, t1 to t2, can
be expressed in terms of either reliability or unreliability
functions, i.e.,
P(t1 < T < t2) = R(t1) - R(t2)
= Q(t2) - Q(t1)
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-3
Reliability Definitions (cont.)
• The mean time to failure (MTTF) is the preferred term when a
device is not repairable

MTTF   R(t )dt
0
• The term mean time between failure (MTBF) is preferred
when a device can be repaired

MTBF   R (t )dt
0
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-4
Hazard Rate
• The hazard rate is the instantaneous probability that a part that
has survived to time t will suddenly fail
dQ(t ) / dt
f (t )
h(t ) 

R(t )
1  F (t )
• Typically, h(t) is high during infancy and old age
“Burn-in”
“Old age”
1
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-5
Hazard Rate (cont.)
•
The instantaneous failure rate, h(t), has the following
properties:
h(t)  0 , t  0
t
lim  h( )d  
t 
0
t
R(t )  e
EMIS 7300 Fall 2005

 h ( ) d
0
Copyright  2005 Dr. John Lipp
S4P1-6
Series Reliability
• Independent, series connected devices with reliabilities r1, r2,
r3, etc. have an overall reliability of r = r1 r2 r3 … rN.
Device 1: r1
Device 2: r2
Device: r
…
Device 3: r3
=
Device N: rN
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-7
Parallel Reliability
Device 1: q1
Device 2: q2
Device: q
…
Device 3: q3
=
Device N: qN
• Independent, parallel connected devices with un-reliabilities
q1, q2, q3, etc. have overall un-reliability q = q1 q2 q3 … qN.
• Equivalent to r = 1 – (1 – r1)(1 – r2)(1 – r3) … (1 – rN).
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-8
Exponential Reliability
• The most basic model of reliability is the situation where the
failure rate is constant over time.
• The result is an exponential model,
R(t) = e-lt
Q(t) = 1 – e-lt
h(t) = l
MTTF / MTBF = 1/l
• Applies when the failure mechanism is simple .
• Recall that the exponential distribution is memoryless
• The Exponential Model is most often associated with
electronic equipment.
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-9
Exponential Probability Paper
• -log R(t) = lt, that is, a logarithm paper should show a straight
line with exponential reliability.
Sample correlation coefficient
r quality of linear fit
ln 0.01
-log R(t)
slope = l
ln 0.1
ln 1.0
time
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-10
Exponential Parameter Estimation
• The parameter l can be estimated with
lˆ 
n
1

t1  t2  t3  tn t
or confidence intervals computed via
12 / 2, 2 n
2nt
EMIS 7300 Fall 2005
l 
2 / 2, 2 n
2nt
Copyright  2005 Dr. John Lipp
S4P1-11
• Half-life of radioactive material  exponential
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-12
Series Exponential Reliability
• A series connection of N identical, exponentially reliable
components has an overall exponential reliability
R(t )  R1 (t ) R2 (t ) R3 (t )  RN (t )
 e l1t e l2t e l3t  e lN t
 lt
e  i
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-13
Non-constant Failure Rates
Increasing
Failure Rate
ln 0.01
-log R(t)
ln 0.1
Decreasing
Failure Rate
ln 1.0
time
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-14
Weibull Reliability
• When the failure rate isn’t constant with respect to time, the
distribution that fits the data is usually the Weibull
– Reliability
R(t )  e
– Hazard Rate
h(t )  blb t b 1
  lt b
• When b < 1 the failure rate is increasing.
• When b = 1 the failure rate is constant.
• When b > 1 the failure rate is increasing.
– MTTF/MTBF
 1
1  
 b
Gamma
Function
l
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-15
Weibull Reliability (cont.)
• Weibull is the first choice reliability model
– Models “weakest link in the chain” (series).
– 20-30 points usually required to discredit.
– Note: includes exponential (b = 1).
•
b is the shape parameter
•
l is the scale parameter
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-16
Weibull PDF
f(t)
1.8
β=5.0
1.6
1.4
1.2
β=0.5
β=3.44
β=1.0
β=2.5
1.0
0.8
0.6
0.4
0.2
0
0
0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4
t
t is in multiples of 
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-17
Weibull Probability Paper
• Probability paper for the Weibull is based on the logarithm of
the logarithm of the inverse reliability
1
log log
 b log l  b log t
R(t )
• That is,
– Y = log(failure time)
– X = log(log(1 / (1 – Median Rank (Y))))
– Fit Y = mX + b using the least squares method
(AKA simple linear regression).
– Estimate of l = e-b and b = m-1.
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-18
The Log-Normal Distribution
• A log-normal tends to be a model of reliability when
– A large number of failures have to occur, that is, an
effective model is that of a large, parallel system, the
failure tends to have a log-normal distribution.
– Non-linear increases in the failure rate of the components.
– The log-normal appears concave down on Weibull paper.
• The Lognormal Model is often used as the failure distribution
for mechanical items and for the distribution of repair times.
• If T ~ LN(,), then Y = lnT ~ N(,).
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-19
Homework
• Probability and Statistics for Reliability: An Introduction
http://quanterion.com/ReliabilityQues/V4N2.html
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp
S4P1-20