Integrated circuit failure times in hours during stress test David Swanick DSES-6070 HV5 Statistical...
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Integrated circuit failure times in hours during stress test David Swanick DSES-6070 HV5 Statistical Methods for Reliability Engineering Summer 2008 Professor Ernesto Gutierrez-Miravete August 14, 2008
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Transcript of Integrated circuit failure times in hours during stress test David Swanick DSES-6070 HV5 Statistical...
Integrated circuit failure times in hours during stress test
David Swanick
DSES-6070 HV5
Statistical Methods for Reliability Engineering
Summer 2008
Professor Ernesto Gutierrez-Miravete
August 14, 2008
Integrated circuit failure times in hours during stress testFailure data on integrated circuits during stress testing; data includes:total number of units tested, number and times of failures, number of units surviving, and length of test time. Objective of the study is to apply fundamental concepts of reliability engineering and produce an appropriate reliability model. This will involve analysis of the provided failure data.Analysis will include:a) Computing descriptive statistics for datab) Constructing histogramsc) Selecting best fitting distribution for reliability model.d) Determining distribution parameters and confidence intervals.e) Determine analytical expressions for failure probability distributionfunction, survival probability function, hazard function, MTTF, and MRL by evaluating with Maple.Data: IC Data (Meeker, 1987)
Integrated circuit failure times in hours –stress test
Test ended at 1,370 hours –28 failed; 4,128 units were still running at end of test –(right) censored
0.1 0.1 0.15 0.6
0.8 0.8 1.2 2.5
3 4 4 6
10 10 12.5 20
20 43 43 48
48 54 74 84
94 168 263 593
Time of failure No. Failed Survivors R z F Z
0 0 0 4156 1 0.000192493 0 0.0020460.1 0 <t< 20 16 4140 0.99615 1.20773E-05 0.00385 0.0007250.1 20 <t< 40 1 4139 0.99591 6.04011E-05 0.00409 0.000725
0.15 40 <t< 60 5 4134 0.994706 1.20948E-05 0.005294 0.0003630.6 60 <t< 80 1 4133 0.994466 2.41955E-05 0.005534 0.0002420.8 80 <t< 100 2 4131 0.993985 0 0.006015 00.8 100 <t< 120 0 4131 0.993985 0 0.006015 01.2 120 <t< 140 0 4131 0.993985 0 0.006015 0.0001212.5 140 <t< 160 0 4131 0.993985 1.21036E-05 0.006015 0.000121
3 160 <t< 180 1 4130 0.993744 0 0.006256 04 180 <t< 200 0 4130 0.993744 0 0.006256 04 200 <t< 220 0 4130 0.993744 0 0.006256 06 220 <t< 240 0 4130 0.993744 0 0.006256 0.000121
10 240 <t< 260 0 4130 0.993744 1.21065E-05 0.006256 0.00012110 260 <t< 280 1 4129 0.993503 0 0.006497 0
12.5 280 <t< 300 0 4129 0.993503 0 0.006497 020 300 <t< 320 0 4129 0.993503 0 0.006497 020 320 <t< 340 0 4129 0.993503 0 0.006497 043 340 <t< 360 0 4129 0.993503 0 0.006497 043 360 <t< 380 0 4129 0.993503 0 0.006497 048 380 <t< 400 0 4129 0.993503 0 0.006497 048 400 <t< 420 0 4129 0.993503 0 0.006497 054 420 <t< 440 0 4129 0.993503 0 0.006497 074 440 <t< 460 0 4129 0.993503 0 0.006497 084 460 <t< 480 0 4129 0.993503 0 0.006497 094 480 <t< 500 0 4129 0.993503 0 0.006497 0
168 500 <t< 520 0 4129 0.993503 0 0.006497 0263 520 <t< 540 0 4129 0.993503 0 0.006497 0593 540 <t< 560 0 4129 0.993503 0 0.006497 0.000121
560 <t< 580 0 4129 0.993503 1.21095E-05 0.006497 -0.003512580 <t< 600 1 4128 0.993263 0 0.006737 0
Time (hours)Interval
28 failure times of the circuit boards were recorded. The number of failures were put into 31 bins, 20 hour intervals, and the number of survivors was calculated. From that data R was calculated by dividing the number of survivors by the total number tested. F was obtained by subtracting R from 1. z was found by dividing the number failed by the time interval, then dividing by the number of survivors. Z was computed by averaging the number of failures and multiplying it by the time interval.
No. Failed
0
2
4
6
8
10
12
14
16
18
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
No. Failed
F
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
Time (hours)
F Series1
Reliability
0.988
0.99
0.992
0.9940.996
0.998
1
1.002
Time (hours)
R Series1
z
0
0.00005
0.0001
0.00015
0.0002
0.00025
Series1
Histogram of failure times
90
50
10
1
time of failure - Threshold
Perc
ent
1000.0100.010.01.00.1
99
90
50
10
1
time of failure - Threshold
Perc
ent
1000100101
90
50
10
1
time of failure - Threshold
Perc
ent
99
90
50
10
1
time of failure - Threshold
Perc
ent
3-Parameter Weibull0.990
3-Parameter Lognormal0.988
2-Parameter Exponential*
3-Parameter Loglogistic0.982
Correlation Coefficient
Probability Plot for time of failureLSXY Estimates-Complete Data
3-Parameter Weibull 3-Parameter Lognormal
2-Parameter Exponential 3-Parameter Loglogistic
5002500
90
50
10
1
time of failure
Perc
ent
6004002000
99
90
50
10
1
time of failure
Perc
ent
5002500
99
90
50
10
1
time of failure
Perc
ent
Smallest Extreme Value0.576
Normal0.695
Logistic0.712
Correlation Coefficient
Probability Plot for time of failureLSXY Estimates-Complete Data
Smallest Extreme Value Normal
Logistic
90
50
10
1
time of failure
Pe
rce
nt
99
90
50
10
1
time of failure
Pe
rce
nt
1000.0100.010.01.00.1
90
50
10
1
time of failure
Pe
rce
nt
99
90
50
10
1
time of failure
Pe
rce
nt
Weibull0.983
Lognormal0.986
Exponential*
Loglogistic0.981
Correlation Coefficient
Probability Plot for time of failureLSXY Estimates-Complete Data
Weibull Lognormal
Exponential Loglogistic
Goodness-of-Fit
Anderson-Darling Correlation
Distribution (adj) Coefficient
Weibull 0.720 0.983
Lognormal 0.740 0.986
Exponential 13.158 *
Loglogistic 0.825 0.981
3-Parameter Weibull 0.597 0.990
3-Parameter Lognormal 0.702 0.988
2-Parameter Exponential 3.840 *
3-Parameter Loglogistic 0.815 0.982
Smallest Extreme Value 10.065 0.576
Normal 5.625 0.695
Logistic 4.636 0.712
The three parameter Wiebull had the highest correlation coefficient, but was only slightly higher than the Wiebull, so Weibull was used for the calculations going forward.
6004002000
0.12
0.08
0.04
0.00
time of failure
PD
F
1000
.000
100.
000
10.0
001.
000
0.10
00.
010
0.00
1
90
50
10
1
time of failure
Perc
ent
6004002000
100
50
0
time of failure
Perc
ent
6004002000
0.12
0.08
0.04
0.00
time of failure
Rate
Correlation 0.983
Shape 0.511011Scale 26.9891Mean 51.8930StDev 112.577Median 13.1734IQR 48.7862Failure 28Censor 0AD* 0.720
Table of StatisticsProbability Density Function
Survival Function Hazard Function
Distribution Overview Plot for time of failureLSXY Estimates-Complete Data
Weibull
Shape and scale parameters were obtained from the Minitab output and used in the equation for F.
Once F was obtained, expressions for f, R, z, MTTF, and MRL could be generated in Maple.
F
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
0.008
20 60 100
140
180
220
260
300
340
380
420
460
500
540
580
Time (hours)
F Series1
z
0
0.00005
0.0001
0.00015
0.0002
0.00025
Series1
Above and below are comparisons of the plots for F and z compared with the charts generated in excel from the failure data.
Monte Carlo simulation performed, 1000 runs, the inverse equation used in the function cells ‘=-(1/0.01927)*LN(RAND())
Time Count Calculated0 1.000 1.000 51.81014 MTTF
20 0.682 0.65540 0.459 0.42460 0.307 0.29480 0.210 0.222100 0.148 0.175120 0.105 0.142140 0.073 0.117160 0.053 0.098180 0.040 0.083200 0.019 0.072220 0.011 0.062240 0.008 0.054260 0.003 0.047280 0.001 0.041300 0.001 0.037320 0.001 0.033340 0.001 0.029360 0.000 0.026380 0.000 0.023400 0.000 0.021420 0.000 0.019440 0.000 0.017460 0.000 0.016480 0.000 0.014500 0.000 0.013520 0.000 0.012540 0.000 0.011560 0.000 0.010580 0.000 0.009600 0.000 0.01
0.000
0.200
0.400
0.600
0.800
1.000
1.200
1.400
1.600
1.800
2.000
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Series1
Series2
Output from the Monte Carlo simulation generated an plotted against the calculated formula.
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