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SEMI E10-0304 E © SEMI 1986, 2004 12 APPENDIX 1 CONFIDENCE BOUND FACTORS NOTICE : This ap pendix was appr oved as an official part of SEMI E10 by full let ter ballot pr ocedure. It of fers detailed information related to…
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380.8) is then an 80 percent confidence interval for the
true tool MTBF
p
.
7.6.4 Calculation of the MTBF
p
Lower Bound when
there are Zero Failures — Use the first row of Table
A1-1 (corresponding to r = 0) to obtain a k
0;conf
factor
corresponding to the desired confidence level. Multiply
the length of the measurement period by this factor to
obtain the lower limit estimate.
7.6.4.1 Example: During a calendar quarter, a tool was
productive for 1200 hours and had zero failures. From
Table A1-1, the 90% confidence level lower limit factor
is 0.434. That means that 1200 × 0.434 = 520.8 hours,
is a 90% lower confidence limit estimate for the true
tool MTBF
p
.
7.6.5 Choosing a test length in order to be able to
demonstrate a required MTBF
p
at a given confidence,
we first must pick a maximum number of failures, r,
that can occur during the test period and still allow us to
confirm a required MTBF
p
objective at a given
confidence level. Next, the length of test time needed
can be calculated using the factors in Table A1-4 in
Appendix 1. The required MTBF
p
is multiplied by a
factor based on r and the desired confidence level to
obtain the total test time needed.
7.6.5.1 Note that minimum test times are obtained by
allowing no failures. The cost, however, of using a
minimum test length is to increase the possibility of an
acceptable tool failing the test by chance. As
mentioned in the discussion in Section 6.2.1, it is
advantageous to design a test that allows up to 4
failures, whenever possible.
7.6.5.2 Example: We would like to confirm a tool
MTBF
p
of 400 hours at an 80% confidence level. We
want to be able to pass a qualification test with 4 or less
failures. We look up the appropriate factor from Table
A1-4 and find 6.72. That means the length of test time
required is 400 × 6.72 = 2688 hours. We can do this on
one tool or split the test time across several tools.
When we have accumulated 2688 hours and if 4 or less
failures have occurred, the MTBF
p
objective of 400
hours will have been confirmed at (at least) the 80%
confidence level.
8 Reliability Growth or Degradation
Measurement
8.1 The previous calculations are meaningful only
when the MTBF
p
(or MCBF) and E-MTBF
p
(or E-
MCBF) are constant over the measurement period. If
reliability is improving (typical during design
verification and debug and also early life run-in) or if
reliability is degrading (typical near the end of life for
the piece of equipment, or if certain sub-assemblies
have been over-stressed and are wearing out) then an
overall MTBF
p
calculation is inappropriate and
misleading and other methods must be used. Exact
time of failure recording is required in order to detect
reliability improvement or reliability degradation
trends, and to fit appropriate models.
8.2 Exact Time of Failure Recording — Clock times of
failure must be converted to durations of cumulative
productive time as measured from the initial productive
use of the tool (set as time 0). This is easily
accomplished if total time is continuously monitored by
duration within each of the six equipment states.
8.2.1 Example: A machine is intended for use during
first shift operation five days a week. For simplicity,
assume 100% productive utilization. After the first
three weeks of use, it fails half-way through the day,
and is not repaired until the start of the next day’s
operation. No more failures occur before the end of the
first four weeks of operation. The exact time of failure
is 124 hours (three weeks of 5 × 8 = 40 hours per week
plus half of an 8 hour day). If a second failure occurred
two hours into the third day of the fifth week, the exact
time of failure would be 174 hours.
8.3 Reliability Growth (Degradation) Models — A
useful family of reliability growth (degradation) models
was developed by the U.S. Army Materials Systems
Analysis Activity. These AMSAA models are
described in Appendix 2, along with a general test for
reliability growth (degradation) trends. Exact time of
failure data is needed to test for trends, fit an AMSAA
model, and test the fit for adequacy. The failures used
to fit the model must occur during productive time
(other failures can occur, but these are not used to fit
reliability models).
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APPENDIX 1
CONFIDENCE BOUND FACTORS
NOTICE: This appendix was approved as an official part of SEMI E10 by full letter ballot procedure. It offers
detailed information related to Section 7.
A1-1 Introduction
A1-1.1 E-MTBF
p
may be substituted for MTBF
p
in all
calculations in this section.
A1-1.2 Tables A1-1 and A1-2 contain factors that
multiply an MTBF
p
point estimate to obtain upper and
lower confidence limits. Table A1-1 applies in the
common case where the equipment is observed for a
fixed period of time and the number of failures that will
occur is unknown in advance (time censored data). The
alternative is failure censored data, where the number
of failures is specified in advance and the equipment is
observed until that many failures occur. Table A1-3
contains lower limit factors for failure censored data.
Since failure censored data rarely occurs in tool or
equipment reliability measurement, Table A1-3 is only
included for completeness. The upper limit factors
given in Table A1-2 apply to both kinds of censored
data.
A1-1.3 Table A1-4 can be used to plan equipment
assessment or qualification tests in order to be able to
demonstrate a desired MTBF
p
at a given confidence
level. In order to use Table A1-4, you must first choose
a maximum number of failures, r, you might observe
during the test period and still be able to meet the
required MTBF
p
objective.
A1-1.4 For reference, here are the formulas for the
lower and upper confidence limit factors for time
censored data found in Tables A1-1 and A1-2:
MTBF
L
OWER
=
2
r
X
2r
+
2;1
−
α
2
×
MTBF
p
where r
=
# of
f
ailures
MTBF
UPPER
=
2r
X
2r;
α
2
×
MTBF
p
A1-1.5 In both cases, the confidence level is 100 × (1 –
α) that the true MTBF
p
is above MTBF
LOWER
and below
MTBF
UPPER
and chi square distribution tables are used.
A1-1.6 For 0 fails, use:
MTBF
LOWER
=
productive time
-log e
α
A1-1.7 Factors to use when there are 0 failures based
on this formula are given in the first row of Table A1-1.
A1-1.8 For failure censored data, MTBF
UPPER
is the
same, but the lower limit factor in Table A1-3 is:
M
TB
F
L
OWER
=
2
r
X
2
r
;
1-
α
2
×
M
TB
F
p
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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