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SEMI E10-0304 E © SEMI 1986, 2004 14 Table A1-3 1 -Sided Lower Confidence Bound Fac tors for the MTBF p (Failure Censored Data) Use for failure censored data to multiply the MTBF p estimate to obtain a lower bound at the…

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SEMI E10-0304
E
© SEMI 1986, 2004 13
Table A1-1 1-Sided Lower Confidence Bound Factors for the MTBF
p
(Time or Cycle Censored Data or Fixed
Length Test)
Use for time or cycle censored data to multiply the MTBF
p
or MCBF estimate to obtain a lower bound at the given confidence level.
For 0 failures, multiply the operating hours or cycles by the factor corresponding to the desired confidence level.
CONFIDENCE LEVEL
# FAILS
r
60% 70% 80% 85% 90% 95% 97.5%
0 1.091 0.831 0.621 0.527 0.434 0.334 0.271
1 0.494 0.410 0.334 0.297 0.257 0.211 0.179
2 0.644 0.553 0.467 0.423 0.376 0.318 0.277
3 0.718 0.630 0.544 0.499 0.449 0.387 0.342
4 0.763 0.679 0.595 0.550 0.500 0.437 0.391
5 0.795 0.714 0.632 0.589 0.539 0.476 0.429
6 0.817 0.740 0.661 0.618 0.570 0.507 0.459
7 0.834 0.760 0.684 0.642 0.595 0.532 0.485
8 0.848 0.777 0.703 0.662 0.616 0.554 0.508
9 0.859 0.790 0.719 0.679 0.634 0.573 0.527
10 0.868 0.802 0.733 0.694 0.649 0.590 0.544
12 0.883 0.821 0.755 0.718 0.675 0.617 0.572
15 0.899 0.841 0.780 0.745 0.704 0.649 0.606
20 0.916 0.864 0.809 0.777 0.739 0.688 0.647
30 0.935 0.892 0.844 0.816 0.783 0.737 0.700
50 0.953 0.918 0.879 0.856 0.829 0.790 0.759
100 0.969 0.943 0.915 0.897 0.877 0.847 0.822
500 0.987 0.976 0.962 0.954 0.944 0.929 0.916
Table A1-2 1-Sided Upper Confidence Bound Factors for the MTBF
p
Use to multiply the MTBF
p
estimate to obtain an upper bound at the given confidence level (time censored or failure censored data).
CONFIDENCE LEVEL
# FAILS
r
60% 70% 80% 85% 90% 95% 97.5%
1 1.958 2.804 4.481 6.153 9.491 19.496 39.498
2 1.453 1.823 2.426 2.927 3.761 5.628 8.257
3 1.313 1.568 1.954 2.255 2.722 3.669 4.849
4 1.246 1.447 1.742 1.962 2.293 2.928 3.670
5 1.205 1.376 1.618 1.795 2.055 2.538 3.080
6 1.179 1.328 1.537 1.687 1.904 2.296 2.725
7 1.159 1.294 1.479 1.610 1.797 2.131 2.487
8 1.144 1.267 1.435 1.552 1.718 2.010 2.316
9 1.133 1.247 1.400 1.507 1.657 1.917 2.187
10 1.123 1.230 1.372 1.470 1.607 1.843 2.085
12 1.108 1.203 1.329 1.414 1.533 1.733 1.935
15 1.093 1.176 1.284 1.357 1.456 1.622 1.787
20 1.077 1.147 1.237 1.296 1.377 1.509 1.637
30 1.060 1.115 1.185 1.231 1.291 1.389 1.482
50 1.044 1.085 1.137 1.170 1.214 1.283 1.347
100 1.029 1.058 1.093 1.115 1.144 1.189 1.229
500 1.012 1.025 1.039 1.049 1.060 1.078 1.094
SEMI E10-0304
E
© SEMI 1986, 2004 14
Table A1-3 1-Sided Lower Confidence Bound Factors for the MTBF
p
(Failure Censored Data)
Use for failure censored data to multiply the MTBF
p
estimate to obtain a lower bound at the given confidence level. Failure censored data means
the test or observation period lasts as long as needed to obtain a preset number of failures.
CONFIDENCE LEVEL
# FAILS
r
60% 70% 80% 85% 90% 95% 97.5%
1 1.091 0.831 0.621 0.527 0.434 0.334 0.271
2 0.989 0.820 0.668 0.593 0.514 0.422 0.359
3 0.966 0.830 0.701 0.635 0.564 0.477 0.415
4 0.958 0.840 0.725 0.665 0.599 0.516 0.456
5 0.955 0.849 0.744 0.688 0.626 0.546 0.488
6 0.954 0.856 0.759 0.706 0.647 0.571 0.514
7 0.953 0.863 0.771 0.721 0.665 0.591 0.536
8 0.954 0.869 0.782 0.734 0.680 0.608 0.555
9 0.954 0.874 0.791 0.745 0.693 0.623 0.571
10 0.955 0.878 0.799 0.755 0.704 0.637 0.585
12 0.956 0.886 0.812 0.771 0.723 0.659 0.610
15 0.958 0.895 0.828 0.790 0.745 0.685 0.639
20 0.961 0.906 0.846 0.812 0.772 0.717 0.674
30 0.966 0.920 0.870 0.841 0.806 0.759 0.720
50 0.971 0.935 0.896 0.872 0.844 0.804 0.772
100 0.978 0.952 0.923 0.906 0.885 0.855 0.830
500 0.989 0.978 0.964 0.956 0.945 0.930 0.918
Table A1-4 Test Length Guide
Use to determine the test time needed to demonstrate a desired MTBF
p
at a given confidence level if r failures occur. Multiply the desired MTBF
p
by the k factor corresponding to r and the confidence level.
k FACTOR FOR GIVEN CONFIDENCE LEVELS
# FAILS
r
50% 60% 75% 80% 90% 95%
0 0.693 0.916 1.39 1.61 2.30 3.00
1 1.68 2.02 2.69 2.99 3.89 4.74
2 2.67 3.11 3.92 4.28 5.32 6.30
3 3.67 4.18 5.11 5.52 6.68 7.75
4 4.67 5.24 6.27 6.72 7.99 9.15
5 5.67 6.29 7.42 7.90 9.28 10.51
6 6.67 7.35 8.56 9.07 10.53 11.84
7 7.67 8.38 9.68 10.23 11.77 13.15
8 8.67 9.43 10.80 11.38 13.00 14.43
9 9.67 10.48 11.91 12.52 14.21 15.70
10 10.67 11.52 13.02 13.65 15.40 16.96
15 15.67 16.69 18.48 19.23 21.29 23.10
20 20.68 21.84 23.88 24.73 29.06 30.89
SEMI E10-0304
E
© SEMI 1986, 2004 15
APPENDIX 2
RELIABILITY GROWTH OR DEGRADATION MODELS
NOTICE: This appendix was approved as an official part of SEMI E10 by full letter ballot procedure. It offers detailed
information related to Section 8.
A2-1 Introduction
A2-1.1 E-MTBF
p
may be substituted for MTBF
p
in all
calculations in this section.
A2-1.2 If the times between failures (known as
“interarrival times”) of a repairable system or piece of
equipment are independent random times sampled from
the same exponential distribution, then the (theoretical)
rate of occurrence of failures (“ROCOF”) is a constant
λ and the MTBF
p
is just 1/λ. This situation is known in
the reliability literature as a homogeneous poisson
process (HPP). An HPP assumption underlies the
definition of MTBF
p
given in Section 6, and the
confidence limit factors described in Section 7 and
Appendix 1. These concepts are described in detail in
Ascher and Feingold [1] and Tobias and Trindade [2].
A2-1.3 If reliability is either improving or degrading
with time, then the ROCOF is no longer a constant and
a MTBF
p
calculation will be misleading.
A2-1.4 This appendix contains a simple test for trend
that may be applied if a time-varying ROCOF is
suspected, as well as a description of a well known and
powerful model that may be used when reliability
improvement trends are evident in the equipment
failure time data.
A2-2 Testing for Trends
A2-2.1 A non-parametric reverse arrangement test
(RAT) devised by Kendall [3] and further developed
into a table by Mann [4] will be described. Begin by
writing the interarrival times in the order they occurred.
For a period with r failures, these might be X
1
, X
2
, …,
X
r
. Starting from left to right, define a reversal as any
instance in which a lesser value occurs before any
subsequent greater value in the sequence. In other
words, any time we have X
i
< X
j
and i < j, we count it
as a reversal. For example, suppose a piece of
equipment has r = 4 failures at 30, 160, 220, and 360
hours of productive time. The interarrival times are 30,
130, 60, and 140. The total number of reversals is 3 + 1
+ 1 = 5.
A2-2.2 A larger than expected number of reversals
indicates an improving trend; a smaller number of
reversals than expected indicates a degradation trend.
A2-2.3 For r up to 12, use Table A2-1 below (adapted
from [2]) to determine whether a given number of
reversals, R, is statistically significant at the 100 × (1 –
α) confidence level.
A2-2.4 For r greater than 12, approximate critical
values for the number of reversals (based on Kendall’s
normal approximation) can be calculated from:
R
(r; 1-
α
)
= z
critical
(2r
+
5)(r − 1)r
72
+
r(r
−
1)
4
−
1
2
Table A2-1 Critical Values R
r;1-
α
the Number of Reversals for the Reverse Arrangement Test at a Given
Confidence Level
Sample Size Single-Sided Lower Critical Value (Too Few
Reversals Provide Evidence of Degradation)
Single-Sided Upper Critical Value (Too Many
Reversals Provide Evidence of Improvement)
r 99% 95% 90% 90% 95% 99%
4 0 0 6 6
5 0 1 1 9 9 10
6 1 2 3 12 13 14
7 2 4 5 16 17 19
8 4 6 8 20 22 24
9 6 9 11 25 27 30
10 9 12 14 31 33 36
11 12 16 18 37 39 43
12 16 20 23 43 46 50