An Introduction to Reliability and
Life Distributions
Dr Jane Marshall
Product Excellence using 6 Sigma
Module
PEUSS 2011/2012
Reliability and Life distributions
Page 1
Objectives of the session
• Probability distribution functions
• Life time distributions
• Fitting Reliability distributions using Hazard
Plotting
• Interpretation
PEUSS 2011/2012
Reliability and Life distributions
Page 2
1
Data types of interest
• Sample data from a population of items
• For example:
– 100 ipods put on test, 12 fail, analyse the times to
failure
– 1000 aircraft engine controllers operating in-service,
collect all the times to failure data and analyse
• Not only times but distance or cycles etc.
PEUSS 2011/2012
Reliability and Life distributions
Page 3
Histogram
Histogram of hours to failure
35
120.00%
30
100.00%
25
Frequency
80.00%
20
60.00%
15
40.00%
10
20.00%
5
0
.00%
9
5
37
6.
23
75
3.
46
5
12
1.
69
5
8.
91
75
.8
45
11
5
25
.2
.6
73
00
13
16
e
or
M
Hours to failure
Frequency
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Cumulative %
Reliability and Life distributions
Page 4
2
Probability distribution
Hours to failure
35
Frequency
30
25
20
15
10
5
-500
0
-5 0
500
1000
1500
2000
2500
Hours to failure
• The area under the curve is equal to 1
• The area under the curve between two values is the
probability
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Reliability and Life distributions
Page 5
Failure Time distributions
• PDF (Probability density function)
• The CDF (Cumulative Distribution Function)
– The CDF gives the probability that a unit will fail before time t
or alternatively the proportion of units in the population that
will fail before time t.
• The Survival Function (sometimes known as
reliability function)
– Complement of the CDF.
• The Hazard Function
– Conditional probability of failing in the next small interval
given survival up to time t.
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Reliability and Life distributions
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3
Probability density Function:
• PDF - Probability of falling between two values
Frequency
PDF, f(t) (%)
1.2
1
0.8
P(t1<t<t2)=
0.6

t2
t1
f(t) dt
0.4
0.2
0
1
1
2
3
3
4
5
5
6
7
7
9
Value
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Reliability and Life distributions
Page 7
Probability distributions
Hours to failure
Relative frequency
0.5
0.4
0.3
0.2
0.1
0
-500
-0.1 0
500
1000
1500
2000
2500
hours to failure
Probability of failure between 500 and 1000 hours is given by the area
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Reliability and Life distributions
Page 8
4
Standard Normal distribution
m
-1s
-2s
+1s
68.27%
-3s
+2s
+3s
95.45%
99.73%
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Reliability and Life distributions
Page 9
Cumulative distribution function
• The CDF known as F(t)
Frequency
CDF, F(t) (%)
1
1.2
t
F( t) =
1
-f ( t ) dt
0.8
0.6
0.4
0.2
0
1
1
2
3
3
4
5t 5
Value
6
7
7
9
F(t) gives the
probability that a
measured value will fall
between -  and t
Failure Function, F(t)
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Reliability and Life distributions
Page 10
5
Cumulative distribution
Cumulative probabilty
cumulative probability
1.2
1
0.8
0.6
0.4
0.2
0
-500
-0.2 0
500
1000
1500
2000
2500
hours to failure
The probability of failure before 500 hours is 0.8
or 80% will have failed by 500hrs
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Reliability and Life distributions
Page 11
Survival function
• The survival function or reliability function R(t)
Frequency
(%)
R(t)
1.2
1
0.8
R(t) = 1 - F(t) and
F(t) = 1 - R(t)
0.6
0.4
0.2
0
1
1
2
3
3
4
5t 5
6
7
7
9
Value
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Reliability and Life distributions
Page 12
6
Survival Function
Survival Function
Probability of survival
1
0.8
0.6
0.4
0.2
0
-500
0
500
1000
1500
2000
2500
Hours to failure
The probability of surviving up to 500 hrs is 0.2
Or 20% have survived up to 500 hrs
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Reliability and Life distributions
Page 13
Hazard function
1 - F (t )
R (t )
• Figure shows
increasing hazard
function
Frequency
(%)
h(t)
• The Hazard function is defined as probability of
failure in next time interval given survival to time
Reliability Function R(t)
1
t
1.2
1
• h(t) = f ( t ) = f ( t )
0.8
0.6
0.4
0.2
0
1
1
2Hazard
3 3 Function
4 5 5h(t)6
7
7
9
Value
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Reliability and Life distributions
Page 14
7
Hazard function
Bath-tub curve
Useful Life
Time
Infant
Mortality
PEUSS 2011/2012
Wear Out
Reliability and Life distributions
Page 15
Probability distributions
• Exponential distribution
• Weibull distribution
• Normal distribution
• Lognormal distribution
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Reliability and Life distributions
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8
Exponential distribution
•
•
•
•
•
•
•
Simplest of all life models
One parameter, 
PDF, f(t) = e- t
CDF, F(t) = 1- e- t and R(t) = e- t
Hazard function, h(t) =  i.e. constant
MTBF = 1/  and failure rate = 
1/ is the 63rd percentile i.e. time at which 63%
of population will have failed
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Reliability and Life distributions
Page 17
Exponential distribution
Probability
1.0
Hazard Function
0.160
0.0
0.155
0
10
20
30
40
Survival Function
50
Rate
0.5
0.150
0
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Reliability and Life distributions
10
20
30
40
50
Page 18
9
Failure rate - example
•
•
•
•
•
•
10 components of a particular type in each PCB
5 PCBS in each unit
200 units in the field
Total operating time to date for all units is 10,000 hours
There have been 30 confirmed failures of this component
The failure rate is given by:
– 30/5*200*10*10,000 = 0.000003 = 3 fpmh (failures per million hours)
• The MTTF is 1/0.000003 = 333,333
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Reliability and Life distributions
Page 19
Example
•
•
•
•
•
100 units in the field
Total operating hours is 30,000
Number of confirmed failures is 60
MTBF = 30,000*100/60 = 50,000
Removal rate includes all units removed
regardless of whether they have failed
• Use 200 removals
• MTBR = 15000
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Reliability and Life distributions
Page 20
10
Weibull distribution
• Most useful lifetime in reliability analysis
• 2 parameter Weibull
– Shape parameter - 
– Scale parameter - 
•
•
•
•
When < 1 decreasing hazard function
When > 1 increasing hazard function
When =1 constant hazard function
 is the characteristic life, 63rd percentile
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Reliability and Life distributions
Page 21
Weibull distribution
PDF : f (t ) 

t


CDF : F (t )  1  e
t
 
 1   
e
t
 
 
Re liability : R (t )  e
PEUSS 2011/2012


t
 
 
Reliability and Life distributions

Page 22
11
Weibull distribution
 t-1

• When =1, h(t)= 1/ =  therefore =1/ 
• When >3.5 the distribution approximates to a
normal distribution
• h(t) =
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Reliability and Life distributions
Page 23
Three parameter Weibull
• A three parameter distribution can be used if
failures do not start at t=0, but after a finite
time . The parameter,  is called the failurefree time or location parameter
F (t )  1  e
PEUSS 2011/2012
 (t  ) 


  
Reliability and Life distributions

24
12
Hazard function
Bath-tub curve and the Weibull
Useful Life
<1
=1
>1
Time
Infant
Mortality
PEUSS 2011/2012
Wear Out
Reliability and Life distributions
Page 25
Normal Distribution
• Not used as often in reliability work
– Can represent severe wear-out mechanism
– Rapidly Increasing hazard function
• e.g.’s, filament bulbs, IC wire bonds
• Location parameter, m , is the mean
• Scale parameter, , is the standard deviation
• Lognormal more versatile, always positive
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Reliability and Life distributions
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13
Fitting parametric distributions
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Reliability and Life distributions
Page 27
Fitting parametric distributions
•
•
•
•
Censoring
Repaired and non repaired
Probability plotting
Hazard plotting
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Reliability and Life distributions
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14
Censoring structures
• Complete data
• Single censored
– Units started together and data analysed before all units have
failed
– Right, interval and left
• Time censored
– Censoring time is fixed
• Failure censored
– Number of failures is fixed
• Multiply censored
– Different running times intermixed with failure times – field data
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Reliability and Life distributions
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Complete Data
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Reliability and Life distributions
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Right Censored data
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Reliability and Life distributions
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Interval Censored
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Reliability and Life distributions
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Left censored
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Reliability and Life distributions
Page 33
Repaired and non-repaired data
• Non-repaired data when only one failure can
occur and interested in time to failure
• Repaired data when interested in the pattern of
times between failures
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Reliability and Life distributions
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17
Probability plotting
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Reliability and Life distributions
Page 35
Areas to be covered
•
•
•
•
•
•
Introduction to probability plotting
Assumptions
How to do a Weibull plot
Estimating the parameters
Testing assumptions
Examples
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Reliability and Life distributions
Page 36
18
What is probability plotting?
• Graphical estimation method
• Based on cumulative distribution function CDF or F(t)
• Probability papers for parametric distributions, e.g.
Weibull
• Axis is transformed so that the true CDF plots as a
straight line
• If plotted data fits a straight line then the data fits the
appropriate distribution
• Parameter estimation
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Reliability and Life distributions
Page 37
Assumptions
• Data must be independently identically
distributed (iid)
– No causal relationship between data items
– No trend in the time between failures
– All having the same distribution
• Non-repaired items
• Repaired items with no trend in the time
between failures
• Time to first failure of repaired items
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Reliability and Life distributions
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19
Example of test for trend
• Machine H fails at the following running times
(hours):
– 15, 42, 74, 117, 168, 233, and 410
• Machine S fails at the following running times
(hours):
– 177, 242, 293, 336, 368, 395, and 410
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Reliability and Life distributions
Page 39
Trend Analysis
machine S running times to failure
8
7
6
5
4
3
2
1
0
order number
order number
machine H running times to failure
0
100
200
300
400
500
8
7
6
5
4
3
2
1
0
0
machine time to failure
100
200
300
400
500
machine time to failure
This system is getting better with
time, the failure times are getting
further and further apart
This system is getting worse with time,
the failure times are getting closer and
closer together.
In neither case can Weibull analysis be used as there is
trend in the data.
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Reliability and Life distributions
Page 40
20
Making a Weibull plot
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Reliability and Life distributions
Page 41
Rank the data
• Probability graph papers are based on plots of
the variable against cumulative probability
• For n< 50 the cumulative percentage probability
is estimated using median ranks tables
• For n< 100 use benard’s approximation for the
median rank ri
ri = i - 0.3
n+0.4
Where i is the ith order value and n is the sample size
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Reliability and Life distributions
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21
Example
Failure number (i)
Ranked hrs at failure (ti)
Median Rank from tables
Cumulative % Failed at ti - F(t)
1
300
6.7
2
410
16.2
3
500
25.9
4
600
35.5
5
660
45.2
6
750
54.8
7
825
64.5
8
900
74.1
9
1050
83.8
10
1200
93.3
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Reliability and Life distributions
Page 43
•
•
•
•
Plot times on x-axis
Plot CDF on y-axis
Fit line through the data
Draw perpendicular line from
estimation point to the
fitted line.
• Read off the estimate of β
• η is the value given on
from the intersection line
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Reliability and Life distributions
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22
Interpreting the plot
• If the data produced a straight line then:
– The data can be modelled by the Weibull distribution.
• If β<1 then data shows a decreasing hazard function
– e.g. Infant mortality, weak components, low quality
• If β=1 then data shows a constant hazard function
– e.g. useful life of product
• If β>1 then data shows a increasing hazard function
– e.g. wear-out, product reaching end of life
• η is the value in time by which 63.2% of all failures will
have occurred and is termed the characteristic life
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Reliability and Life distributions
Page 45
Hazard function
Bath-tub curve and the Weibull
Useful Life
<1
=1
>1
Time
Infant
Mortality
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Wear Out
Reliability and Life distributions
Page 46
23
Interpreting the plot
• If the data did not produce a straight light then:
– There may be an amount of failure-free time
• This may appear concave when viewed from the bottom
right hand corner of the sheet
– There may be more than one failure mode present
• This may appear convex shape or cranked shape (also
known as dog-leg shape)
• In this case the data needs to separated into failures
associated with each failure mode using expert judgement
and analysed separately
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Reliability and Life distributions
Page 47
Example: Poor fit due to 3 months
offset
Weibull Plot of Time-in-Service
Weibull Plot of Time in service (Months)
2-Parameter Weibull - 95% CI
Censoring Column in Censoring - ML Estimates
3-Parameter Weibull - 95% CI
Censoring Column in Censoring - ML Estimates
99
Table of Statistics
Shape
1.87010
Scale
57.6561
Mean
51.1897
10
10
5
3
2
5
3
2
1
0.01
0.1
1.0
10.0
Time in service (Months)
100.0
Table of Statistics
Shape
0.873095
Scale
297.337
Thres
2.9997
90
80
70
60
50
40
30
20
Pe r ce nt
Pe r c e n t
99
90
80
70
60
50
40
30
20
1
0.01
0.0001
0.0010
0.0100
0.1000
1.0000 10.0000 100.00001000.000010000.0000
Time in service (Months) - Threshold
The same data plotted with a three-Parameter Weibull distribution shows a good
fit with 3 months offset (location – 2.99 months)
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Reliability and Life distributions
Page 48
24
Example of two failure modes
Weibull CDF
Mode 2 Beta
= 11.9
Mode 1 Beta
= 0.75
Time to failure ( hours)
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Reliability and Life distributions
Page 49
Adjusted rank for censored data
Rank
1
2
3
4
5
6
7
8
Time
10
30
45
49
82
90
96
100
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Status
Suspension
Failure
Suspension
Failure
Failure
Failure
Failure
Suspension
Reverse
rank
8
7
6
5
4
3
2
1
Adjusted rank
Suspended...
[7 X 0
+(8+1)]/ (7+1) = 1,125
Suspended…
[5 X 1,125 +(8+1)]/ (5+1) = 2,438
[4 X 2,438 +(8+1)]/ (4+1) = 3,750
[3 X 3,750 +(8+1)]/ (3+1) = 5,063
[2 X 5,063 +(8+1)]/ (2+1) = 6,375
Suspended...
Reliability and Life distributions
Median
rank
9,8 %
25,5
41,1
56,7
72,3
%
%
%
%
Page 50
25
Weibull Analysis using
software tools
• Number of software packages that can do Weibull
plotting (and other distributions), these include:
–
–
–
–
Minitab
Relex
WinSMITH
Reliasoft
• Concentrate on getting good quality data, correct
assumptions and correct interpretation from the
software
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Reliability and Life distributions
Page 51
Hazard Plotting
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Reliability and Life distributions
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26
Contents
•
•
•
•
Assumptions
Fitting parametric distributions
Estimating parameters
Using results for decision making
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Reliability and Life distributions
Page 53
Assumptions
• Non-repaired items
• Repaired items with no trend in the time
between failures
• Time to first failure of repaired items
• Individual failure modes from non-repaired items
• Can deal with censored data
in particular multiply censored data
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Reliability and Life distributions
Page 54
27
Hazard Plotting
• Cumulative hazard function
H(t)=

H(t)=

t
0
h(t) dt
t
0
f(t) /1-F(t) dt
H(t)= -ln[1-F(t)]
• Relationship allows derivation of cumulative
hazard plotting paper
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Reliability and Life distributions
Page 55
Weibull Hazard Plotting
• h(t) =
 t-1 and H(t) = t



()
• If H is the cumulative hazard value then
Log t = 1 log H + log 

• Weibull hazard paper is log-log paper
• The slope is 1/  and when H=1, t= 
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Reliability and Life distributions
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Hazard plotting procedure
• Tabulate times in order and rank
• Reverse rank
• For each failure, calculate the hazard interval
– hi = 1/ no of items remaining after previous
failure/censoring (i.e. 1/reverse rank)
• For each failure, calculate the cumulative hazard
function H
n
– H = h1 + h2 + ………. +
• Plot the cumulative hazard against life value
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Reliability and Life distributions
Page 57
Example 1 – vehicle shock
absorbers
Distance to failure for
Shock absorbers
F denotes failure
PEUSS 2011/2012
Distance (km)
6700 F
17520 F
6950
17540
7820
17890
8790
18450
9120 F
18960
9660
18980
9820
19410
11310
20100 F
11690
20100
11850
20150
11880
20320
12140
20900 F
12200 F 22700 F
12870
23490
13150 F 26510 F
13330
27410
13470
27490 F
14040
27890
14300 F 28100
Reliability and Life distributions
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29
Example 1 – vehicle shock
absorbers
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
Reverse
rank
38
37
36
35
34
33
32
31
30
29
28
27
26
25
24
23
22
21
20
Distance
(km)
6700 F
6950
7820
8790
9120 F
9660
9820
11310
11690
11850
11880
12140
12200 F
12870
13150 F
13330
13470
14040
14300 F
Hazard
(1/rank)
1/38
Cumulative
hazard
0.0263
1/34
0.0557
1/26
0.0942
1/24
0.1359
1/20
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20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
17520 F
17540
17890
18450
18960
18980
19410
20100 F
20100
20150
20320
20900 F
22700 F
23490
26510 F
27410
27490 F
27890
28100
1/19
0.2385
1/12
0.3218
1/8
1/7
0.4468
0.5896
1/5
0.7896
1/3
1.1229
0.1859
Reliability and Life distributions
Page 59
Example 1
• Plot the data on log 2 cycle paper x log 2 cycle
paper
• Estimate Weibull shape parameter
• Estimate Weibull scale parameter
• Interpret results
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30
Example 1
distance
Cumulative hazard plot for shock absorbers on linear
paper
30000
25000
20000
15000
10000
5000
0
0
0.2
0.4
0.6
0.8
1
1.2
Cumulative hazard
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Reliability and Life distributions
Page 61
Example 1
Log cumulative hazard for shock absorbers
= 2.6
= 28500km
R2 = 0.98
log distance
100000
10000
1000
0.01
0.1
1
10
log hazard
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Example 1
Since looking at one known failure mode use the estimated
parameters to fit to the distribution
Probability of survival
Survival plot for vehicle shock absorbers with
Beta =2.6 and Eta=29000km
1.2
1
0.8
0.6
0.4
0.2
0
0
5000
10000
15000
20000 25000 30000 35000 40000
kilometers
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Reliability and Life distributions
Page 63
Example 2:Hazard plot on
linear paper
Cumulative hazard plot for O ring failures
2000
h o u rs
1500
1000
500
0
0
1
2
3
4
5
6
cumulative hazard
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32
Example 2: Hazard plot on log
paper
log cumulative hazard for O ring failures
= 1.01
= 360hrs
R2 = 0.98
10000
log hours
1000
100
10
1
0.01
0.1
1
10
log cumulative hazard
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Reliability and Life distributions
Page 65
Example 2: Interpretation of
results
•  = 1.01 is approximately an exponential
distribution and constant failure rate
•  = 360 hrs = 1/ = Mean Time to Failure
• Calculating the MTTF from the data gives:
– Total hours/number of failures
– 26839/73 = 367 hrs
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Example 2: Survival function
R(t) - Survival Function for O ring failures
Probability of survival
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
2000
Hours to failure
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Reliability and Life distributions
Page 67
Example 2: Failure Distribution
F(t) for O ring failures
Probability of failure
1.2
1
0.8
0.6
0.4
0.2
0
0
500
1000
1500
2000
Hours to failure
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34
Example 3 : pumps
pump no
• Two dominant failure
modes
– Impeller failure (I)
– Motor failure (m)
age at failure
1
1180 m
2
6320 m
3
1030 i
4
120 m
5
2800 i
6
970 i
7
2150 i
8
700 m
9
640 i
10
1600 i
11
520 m
12
PEUSS 2011/2012
failure mode
1090 i
Reliability and Life distributions
Page 69
Example 3: ignoring failure
modes
log cumulative hazard for all failures
10000
age
1000
100
10
1
0.01
0.1
1
10
cum hazard
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Reliability and Life distributions
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Example 3: impeller failure
log cumulative hazard plot for impeller failures
= 1.95
= 1900hrs
R2 = 0.93
10000
a ge a t fa ilur e
1000
100
10
1
0.1
1
10
cumulative hazard
PEUSS 2011/2012
Reliability and Life distributions
Page 71
Example 3: motor failure
log cumulative hazard plot for motor failures
= 0.76
= 3647hrs
R2 = 0.978
a ge a t fa ilure
10000
1000
100
10
1
0.01
0.1
1
10
cumulative hazard
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Reliability and Life distributions
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36
Advantages of Cum Hazard
Plotting
• It is much easier to calculate plotting positions
for multiply censored data using cum hazard
plotting techniques.
• Linear graph paper can be used for exponential
data and log-log paper can be used for Weibull
data.
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Reliability and Life distributions
Page 73
Disadvantages of Cum
Hazard Plotting
• It is less intuitively clear just what is being plotted.
– Cum percent failed (i.e., probability plots) is meaningful and
the resulting straight-line fit can be used to read off times
when desired percents of the population will have failed.
– Percent cumulative hazard increases beyond 100% and is
harder to interpret.
• Normal cum hazard plotting techniques require exact
times of failure and running times.
• With computer software for probability plotting, the
main advantage of cum hazard plotting goes away
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Reliability and Life distributions
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37
Summary
• Important lifetime distributions
– Failure distribution (CDF), Survival function R(t) and
the hazard function h(t)
• Some parametric distributions
– Exponential, Weibull and Normal
• Weibull probability plotting
• Distribution fitting using hazard plotting
techniques
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Reliability and Life distributions
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38