Reliability Model for Air
Compressor Failure
SMRE Term Project
Paul Zamjohn
August 2008
Paul Zamjohn
Page 1
Summer 2008
Data on “large air compressors” for a military base
near the seacoast will be analyzed to determine the
probabilistic failure structure. Air compressors require
“bleeding” prior to operation to function properly; the
data below represents failure due to binding in the bleed
system. Salt air due to proximity to the ocean is believed
to be a major contributor, nothing is known about other
variables and their impact to reliability.
Analysis will include:
 Generating the descriptive statistics
 Selecting the distribution that best describes the
data and the distribution parameters
 Calculating the failure probability density function
(f)
 Calculate the cumulative distribution function (F)
 Calculating the survival probability function (R)
 Calculating the hazard function (z)
 Determining the MTTF
 Calculating the MRL
 Perform Monte Carlo simulation to model and assess
reliability
Compressor Failure Data:
Operating time for 202 compressors (failed and unfailed units)
operating hours
frequency
0-200
0
201-300
2
301-400
0
401-500
0
501-600
2
601-700
2
701-800
10
801-900
26
901-1000
27
1001-1100
22
1101-1200
24
1201-1300
24
1301-1400
11
1401-1500
11
1501-1600
20
1601-1700
8
1701-1800
4
1801-1900
2
1901-2000
3
2001-2100
3
2101-2200
1
Paul Zamjohn
Page 2
Summer 2008
Minitab was used to find a distribution model that matched
the data under review.The data used is interval censored
therefore the arbitrary censoring option wass selected.
The Weibull distribution provides the best approximation
for the compressor failure data. The correlation
coefficients for the four distributions compared are:
Probability Plot for Start
LSXY Estimates-Arbitrary Censoring
C orrelation C oefficient
Weibull
0.971
Lognormal
0.952
E xponential
*
Loglogistic
0.954
Lognormal
99.9
99.9
90
99
50
90
P er cent
P er cent
Weibull
10
50
1
10
0.1
0.1
1
500
1000
2000
Star t
E xponential
2000
99.9
90
99
P er cent
50
P er cent
1000
Star t
Loglogistic
99.9
10
1
0.1
500
90
50
10
1
1
10
100
Star t
1000
Weibull
Lognormal
Exponentiol
Loglogistic
Paul Zamjohn
10000
0.1
1000
Star t
10000
0.971
0.952
*
0.954
Page 3
Summer 2008
Minitab provides a shape parameter (λ) of 3.97329 and a
scale (α) of 1335.04 for the Weibull distribution. When α
is greater than 1 the failure rate function is increasing.
In this case it will be rapidly increasing.
Distribution Overview Plot for Start
LSXY Estimates-Arbitrary Censoring
P robability Density F unction
Table of S tatistics
S hape
3.97329
S cale
1335.04
M ean
1209.63
S tDev
341.417
M edian
1217.40
IQ R
473.737
A D*
0.242
C orrelation
0.971
Weibull
99.9
0.0012
90
50
P DF
P er cent
0.0008
0.0004
0.0000
10
1
500
1000
1500
Star t
0.1
2000
500
1000
2000
Star t
S urv iv al F unction
H azard F unction
100
Rate
P er cent
0.0075
50
0.0050
0.0025
0
0.0000
500
1000
1500
Star t
2000
500
1000
1500
Star t
2000
The failure rate function for this distribution (Fw) is
defined as:
The reliability (survivor) function (Rw) is defined as:
Paul Zamjohn
Page 4
Summer 2008
>
The probability density function (fw) is defined as:
fw 
d
F (t )
dt
Paul Zamjohn
Page 5
Summer 2008
>
Paul Zamjohn
Page 6
Summer 2008
The failure rate function (zw) is defined as:
>
The mean time to failure function (MTTFw) is defined as:

MTTFw   Rw(t )dt
0
>
Paul Zamjohn
Page 7
Summer 2008
The mean life remaining at 500, 1000, and 1500 hours is:
>
>
>
Monte Carlo Simulation:
A Monte Carlo analysis was performed to simulate and assess
the reliability of this model. The following steps were
taken:
1. Define the cumulative failure function
2. Invert the function and solve for (t)
3. Use excel’s (RAND) function for F to compute time
4. Repeat 1000 times
The equations used are:
.
Fw  1  e
 t

 ( )3.97329
 

e 3.8150905E 13*t
3.97329
 1  Fw
 3.8150905E  13 * t 3.97329  ln( 1  Fw)
1
1
t (
) ln( 1  Fw) 3.97329
 3.8150905E  13
1
1
t (
) ln( 1  RAND ()) 3.97329
 3.8150905E  13
Rw  e
Paul Zamjohn
3.8150905E 13*t 3.97329
Page 8
Summer 2008
Failure rates for the following times were simulated:
t=250, 500, 750, 1000, 1250, 1500, 1750, 2000 (see excel
document)
Simulation results:
Random #
1989
1186
1215
1068
1519
1031
944
1045
1293
t=250
1
1
1
1
1
1
1
1
1
t=500
1
1
1
1
1
1
1
1
1
t=750
1
1
1
1
1
1
1
1
1
t=1000
1
1
1
1
1
1
0
1
1
t=1250
1
0
0
0
1
0
0
0
1
t=1500
1
0
0
0
1
0
0
0
0
t=1750
1
0
0
0
0
0
0
0
0
t=2000
0
0
0
0
0
0
0
0
0
250
500
750
1000
1250
1500
1750
2000
MC
0.9990
0.9740
0.9030
0.7230
0.4780
0.1970
0.0630
0.0040
MTTF=
1209.7069
EQ
0.9987
0.9800
0.9038
0.7282
0.4631
0.2042
0.0533
0.0069
The Monte Carlo simulation closely matched the expected
value for Rw.
Comparison of Monte Carlo vs.Equation
1.2000
1.0000
0.8000
MC
EQ
0.6000
0.4000
0.2000
0.0000
250
500
750
1000
1250
1500
1750
2000
(hours)
Alternate distributions:
Paul Zamjohn
Page 9
Summer 2008
The data was analyzed to determine if there were other
distributions that would provide a better model for the
compressor data. Mini-tab was used to identify alternate
distribution functions with high correlation to the data.
Since the data used is interval censored the arbitrary
censored plot was used.
Probability Plot for Start
LSXY Estimates-Arbitrary Censoring
C orrelation C oefficient
3-P arameter Weibull
0.978
3-P arameter Lognormal
0.991
2-P arameter E xponential
*
3-P arameter Loglogistic
0.991
3-Parameter Lognormal
99.9
99.9
90
99
50
90
Percent
Percent
3-Parameter Weibull
10
50
1
10
0.1
500
0.1
1
1000
Start - Threshold
2000
8400
2-Parameter Exponential
10200
3-Parameter Loglogistic
99.9
99.9
90
99
Percent
50
Percent
9000
9600
Start - Threshold
10
1
90
50
10
1
0.1
1
10
100
1000
Start - Threshold
10000
0.1
12000
13000
14000
Start - Threshold
The 3-parameter lognormal distribution provides the best
correlation (0.991) The probability density function was
taken from mini-tab help, although this was found to be in
error. Sigma and mu had to be switched to provide the
proper distribution function. The cumulative distribution,
reliability, and hazard functions for the 3-parameter
lognormal distribution were determined and plotted against
the distribution functions for the weibull distributions
for comparison.
Paul Zamjohn
Page 10
Summer 2008
3-Parameter Lognormal
>
>
>
>
>
>
>
>
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Paul Zamjohn
Page 11
Summer 2008
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Paul Zamjohn
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Summer 2008
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The probability density, cumulative distribution, and
reliability plots are very close. The hazard rate or
failure plots were close up to ~1500 hours then became
divergent. The fact that all 202 compressors in the study
failed by 2200 hours leads me to believe the weibull
distribution is a better fit. I will use the results
obtained with the Weibull model.
Paul Zamjohn
Page 13
Summer 2008