1. EMIS 7300 SYSTEMS ANALYSIS METHODS FALL 2005
Dr. John Lipp
Copyright © 2005 Dr. John Lipp

2. 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

3. 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

4. Reliability Definitions (cont.)
The mean time to failure (MTTF) is the preferred term when a device is not repairable
The term mean time between failure (MTBF) is preferred when a device can be repaired
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

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Hazard Rate
The hazard rate is the instantaneous probability that a part that has survived to time t will suddenly fail
Typically, h(t) is high during infancy and old age
“Burn-in”
“Old age” 1
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

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Hazard Rate (cont.)
The instantaneous failure rate, h(t), has the following properties:
h(t)  0 , t  0
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

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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 3: r3
Device N: rN …
Device: r =
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

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Parallel Reliability
Device 1: q1
Device 2: q2
Device 3: q3
Device N: qN … =
Device: q
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

9. 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

10. 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

11. Exponential Parameter Estimation
The parameter l can be estimated with
or confidence intervals computed via
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

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Half-life of radioactive material  exponential
Print / make transparency
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

13. Series Exponential Reliability
A series connection of N identical, exponentially reliable components has an overall exponential reliability
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

14. 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

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Weibull Reliability
When the failure rate isn’t constant with respect to time, the distribution that fits the data is usually the Weibull
Reliability
Hazard Rate
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
Gamma
Function
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

16. 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).
 is the shape parameter
l is the scale parameter
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

17. Copyright  2005 Dr. John Lipp
Weibull PDF
f(t) t
t is in multiples of 
1.8
1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
β=0.5
β=5.0
β=3.44
β=2.5
β=1.0
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp

18. Weibull Probability Paper
Probability paper for the Weibull is based on the logarithm of the logarithm of the inverse reliability
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

19. 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

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Homework
Probability and Statistics for Reliability: An Introduction
EMIS 7300 Fall 2005
Copyright  2005 Dr. John Lipp