Reliability math and the exponential distribution

Chet Haibel ©2013 Hobbs Engineering Corporation
Reliability
Math and the
Exponential
Distribution
0
0

General Reliability Function, R(t)
Fraction of a group surviving until a certain time.
Probability of one unit surviving until a certain time.
Monotonic downward, assuming failed things stay failed.
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
1

General Cumulative Distribution Function, F(t)
Fraction of a group failing before a certain time.
Probability of one unit failing before a certain time.
Monotonic upward, assuming failed things stay failed.
What fraction fails at 20 (arbitrary time units)?
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
2

The probability of one unit failing between two times is found by
subtracting the CDF for the two times, in this case perhaps 0.49
minus 0.41 equals 0.08 or 8%.
0
0.2
0.4
0.6
0.8
1
0 10 20 30 40 50
General Cumulative Distribution Function, F(t)
3

This is the time derivative (the slope) of F(t).
This is a histogram of when failures occur.
The height is the fraction failing per unit time.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 10 20 30 40 50
General Probability Density Function, f(t)
4

The height is the fraction failing per unit time.
The area under the pdf between two times gives the probability
of failure during this time interval, in this case perhaps 0.008
high by 10 units long for an area of 8%.
0.00
0.02
0.04
0.06
0.08
0.10
0.12
0 10 20 30 40 50
General Probability Density Function, f(t)
5

General Hazard Function, h(t)
The conditional probability density of failure, given that the
product has survived up to this point in time.
The fraction of the survivors failing per unit time.
0
5
10
15
20
0 10 20 30 40 50
6

Chet Haibel ©2013 Hobbs Engineering Corporation 7
Bathtub Curve
Operating Time (t)
HazardRate-h(t)
Useful-Life
Failures
Early-Life
Failures
aka
infant
mortalities
Wear-Out
Failures
aka
end-of-life
failures
Life to the Beginning of Wear-Out
Random-in-Time Failures

Summary of Functions Always True
R(t) = Reliability function
probability of surviving until some time
F(t) = Cumulative distribution function
probability of failure before some time
R(t) + F(t) = 1
f(t) = Probability density function failure rate
h(t) = Hazard function = f(t) / R(t)
conditional (normalized) failure rate
 
dt
F(t)d

8

Relations Among Functions Always True
All functions are not defined for t < 0
All parts, subassemblies, products, and systems are
assumed to be working at t = 0
R(t) & F(t) are probabilities and lie between 0 and 1
R(t) & F(t) are monotonic, assuming things “stay failed”
f(t) the pdf is the derivative of the CDF
where h(t)
 
dt
F(t)d

 

t
0
dh
eR(t)

R(t)
f(t)

9

Is the most widely used (and sometimes misused) failure distribution for
reliability analysis for complex electronic systems.
Is applicable when the hazard rate is constant. The hazard rate is the
surviving fraction failing per unit of time or equivalent; such as percent
per million cycles, or failures per billion (109) hours (FITs).
Requires the knowledge of only one parameter for its application.
Is used to describe steady-state failure rate conditions.
Models device performance after the Early-Life
(infant mortality) period and prior to the Wear-Out
(end-of-life) period of the Bathtub Curve.
The Exponential Failure Distribution
10

Life of 10 Constant Hazard Rate Devices
Life in Hours
11

Sort from First to Last 10 Constant Hazard Rate Devices
Life in Hours
12
How do the
devices know to
fail this way?

Fraction Surviving over Time Exponential Distribution
Exponential Distribution
 = Constant Hazard Rate
t
etR 
)(
13
Life in Hours

Chet Haibel ©2013 Hobbs Engineering Corporation 14
Reliability Function Exponential Distribution
R(t) = e-t
Extending the initial
slope intersects the
x-axis at the mean
MTTF = 1/
Life in Hours

R(t) = e-t
F(t) = 1-e-t
f(t) = e-t
h(t) = 







 θ
t
e







 θ
t
e
θ
1







 θ
t
e1
θ
1

Functions Exponential Distribution
15

Functions Exponential Distribution
pdf
Operating
Time
t
f t e t
( )  
 
CDF
Operating
Time
t
PROBABILITY DENSITY FUNCTION CULMULATIVE DISTRIBUTION
FUNCTION
F t e t
( )   
1 
t
pdf
Operating
Time
t
f t e t
( )  
 
CDF
Operating
Time
t
FUNCTION
F t e t
( )   
1 
R(t)
Operating
Time
t
THE RELIABILITY FUNCTION
R t e t
( )  
h(t)
Operating
Time
t
THE HAZARD RATE

Operating Time
t
Operating Time
t
FUNCTION
R(t)
Operating Time
t
THE RELIABILITY FUNCTION
R t e t
( )  
h(t)
Operating Time
t
THE HAZARD RATE

1 ------
 ------
h(t) = 
1 ------
16

For REPAIRABLE products, there is no limit to how many
failures one can have!
Percent of Failures Over Time
(shown for 3%per month failure rate)
0%
20%
40%
60%
80%
100%
120%
140%
160%
180%
200%
0
12
24
36
48
60
72
84
96
Months in Service
Repairable
Non-Repairable
H(t)
F(t)
Repairable & Non Repairable Systems Exponential Distribution
17

Mean is:
Standard deviation is:
To get the “point estimate” of 
λ
1
θμ 
failuresofnumbertotal
not)or(failedhoursofnumbertotal
or MTBFMTTFθ 
λ
1
θσ 
Mean & Standard Deviation Exponential Distribution
18

Reliability math and the exponential distribution

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Reliability math and the exponential distribution