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SEMI E10-0304 E © SEMI 1986, 2004 17 p Figure A2-1 Duane Plot of CUM MTB F p vs. Time Tren d N o Trend Section s 8.1-8.3/A ppendix 2 Estimate the achieved MTBF p at the end of the period Sections 6.2.1-6.2.2 Use standard…

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A2-2.5 In this equation z
critical
comes from the critical
values of the standard normal distribution (for 90%
significance, z
critical
= 1.282, for 95% significance, z
critical
= 1.645, and for 99% significance, z
critical
= 2.33). The
formula calculates the critical value for detecting an
improvement trend. For degradation trends (a small
number of reversals) use (r)(r − 1)/2 minus R
r;1-α
as the
critical value. Note that (r)(r − 1)/2 just the total
possible number of reversals when there are r failures.
A2-2.6 For example, with 17 failures, the formula for
R
r;1-α,
using 95% significance, gives a critical number of
reversals of R
17,95
= 88. The maximum number of
reversals is 17 × 16/2 = 136. That means that observing
88 or more reversals signals a likely improvement
trend, while observing 136 − 88 = 48 or less reversals
signals a likely degradation trend.
A2-2.7 The example given in the next section shows an
application for the reverse arrangement test using Table
A2-1.
A2-2.8 The AMSAA Reliability Growth Model:
Assume the sequence of interarrival time indicates an
improvement trend. This will typically be the case
during reliability improvement testing, where failures
are analyzed down to root causes and actions are taken
to improve the equipment’s reliability. Duane [5]
observed that a plot of t
k
/k versus t
k
, where t
k
is the
system age at the time of kth failure, typically appears
linear on log versus log graph paper. The slope β of
this line measures the rate of reliability growth. Typical
empirical values of β lie between 0.3 and 0.6. Crow [6]
developed this empirical observation into the power
relationship model used by the U.S. Army Materials
Systems Analysis Activity (AMSAA model). This
model has proved successful in a wide range of
applications.
A2-2.9 The AMSAA model assumes that during
reliability improvement testing the MTBF
p
is improving
with time and has an instantaneous value denoted by
MTBF
I
(t). When the test ends at time T, the MTBF
p
becomes a constant with the value MTBF
I
(T). An
estimate of the MTBF
p
after a test of T hours with r
failures is given by:
M
TBF
I
(T )
=
T
r× (1-
β
)
(1)
A2-2.10 In this equation, β is the reliability
improvement (Duane) slope, β is estimated by
β
=
1-
r
−
1
∑
i=1
r
1n
T
t
i
(2)
using the modified maximum likelihood estimates given
by Crow [6]. Crow developed confidence limits for
MTBF
I
(T ) that are described in [2] and [6].
A2-2.11 Example: During a calendar quarter a tool has
550 hours of productive time. Eleven failures were
recorded at the following points of productive time: 18,
20, 35, 41, 67, 180, 252, 287, 390, 410, and 511 hours.
Determine whether there appears to be an improvement
trend and use the AMSAA model to estimate the
achieved MTBF
I
at the end of the quarter.
A2-2.12
Solution: The interarrival times are: 18, 2, 15,
6, 26, 113, 72, 35, 103, 20, and 101. The number of
reversals is 7 + 9 + 7 + 7 + 5 + 0 + 2 + 2 + 0 + 1 = 40.
Using Table A2-1, this is significant at greater than the
95% confidence level, indicating an improvement trend
is likely. Figure A2-1 shows the Duane plot, which
appears to show a linear improvement trend on log-log
paper. The AMSAA model equations give an
improvement slope estimate of 0.43 and an
instantaneous MTBF
P
estimate at 550 hours of 87.2.
Note that a standard calculation ignoring the
improvement trend
would yield an MTBF
P
estimate of
550/11 = 50, which is a 43% underestimate.
A2-2.13 Figure A2-2 summarizes the recommended
procedure to follow when analyzing system or
equipment reliability data, with appropriate references
to SEMI E10 sections or appendices.
A2-3 References
1. Ascher, H. and H. Feingold, Repairable Systems
Reliability, Marcel Dekkar, Inc., New York, 1984
2. Tobias, P.A. and D.C. Trindade, Applied
Reliability, Second Edition, Van Nostrand
Reinhold, Inc., New York, 1995
3. Kendall, M.G., “A New Measure of Rank
Correlation”, Biometrika, 1938, volume 30, pages
81-93
4. Mann, H.B., “Nonparametric Test Against Trend”
Econometrica, 1945, volume 13, pages 245-25
5. Duane, J.T., “Learning Curve Approach to
Reliability Monitoring,” IEEE Transactions on
Aerospace, 1964, volume 2, pages 563-566
6. Crow, L.H., “Reliability Analysis for Complex
Repairable Systems,” Reliability and Biometry, F.
Proschan and R.J., Serfling, eds., SIAM,
Philadelphia, 1974; pp. 126-134
SEMI E10-0304
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© SEMI 1986, 2004 17
p
Figure A2-1
Duane Plot of CUM MTBF
p
vs. Time
Trend
N
o Trend
Sections 8.1-8.3/Appendix 2
Estimate the achieved
MTBF
p
at the end of
the period
Sections 6.2.1-6.2.2
Use standard MTBF
p
definition
Section 7.6/Appendix 1
Compute confidence
bounds for MTBF
p
Analyze the data for trends
Duane Plots
Trend Test
Section 8.3/Appendix 2
Fit AMSAA model
Figure A2-2
Flow Chart for Reliability Data Analysis
NOTICE: SEMI makes no warranties or representations as to the suitability of the standards set forth herein for any
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Users are cautioned to refer to manufacturer’s instructions, product labels, product data sheets, and other relevant
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the risk of infringement of such rights, are entirely their own responsibility.
SEMI E10-0304
E
© SEMI 1986, 2004 18
RELATED INFORMATION 1
MULTI-PATH CLUSTER TOOL RAM METRICS
NOTICE: This related information is not an official part of SEMI E10 and was derived from work done by the
SEMI NA Cluster Tool RAM Metrics Task Force. This related information was approved for publication by full
letter ballot on October 15, 2003.
R1-1 Introduction
R1-1.1 This related information presents tracking
requirements and standard metrics for evaluating multi-
path cluster tool reliability, availability, and
maintainability (RAM) performance. The metrics in
the main body of SEMI E10 apply to non-cluster or
single-path cluster tools and individual modules in a
straightforward manner. While these entities are either
entirely “up” (i.e., in one of the SEMI E10 uptime
states) or entirely “down” (i.e., in one of the SEMI E10
downtime states), multi-path cluster tools may still be
capable of processing with some modules in an
unscheduled downtime state. Furthermore, the effect of
module unscheduled downtime on multi-path cluster
tool performance depends on the specific multi-path
cluster tool configuration and the combination of
modules in an unscheduled downtime state at each
point in time.
R1-1.2 Because module performance provides a
sufficient lowest common denominator for evaluating
multi-path cluster tool performance, all metrics in this
related information are calculated as functions of
module-level data only. Specific module tracking
requirements are presented. The metrics, Total Failure
Rate (TFR) and Cluster-Tool Mean Time to Repair
(MTTR
CT
), provide simple evaluations of aggregate
module reliability and maintainability.
R1-1.3 Other metrics are based on the specific
combinations of modules and process paths, defined
here as process flows.
R1-1.3.1 A process path is a specific set of modules
for which each module is unique and has no alternative
modules.
R1-1.3.2 A process flow is a defined set of modules
that is used to achieve a process, where any multi-path
cluster tool may have one or more such process flows.
A process flow may include alternative modules at one
or more steps of the process. A process flow may
therefore contain one or many process paths.
R1-1.4 A method of temporal mapping, defined in this
related information, is used to generate a history of
process flow and multi-path cluster tool states from
individual module states. Multi-path cluster tool
availability is then evaluated as the aggregate process
flow availability. Multi-path cluster tool reliability is
evaluated as the expected, or mean, productive time
between all process flows being in unscheduled
downtime.
R1-2 Module Tracking Requirements
R1-2.1 Multi-path cluster tool metrics require the
tracking of SEMI E10 state data from all of the modules
that impact the RAM or productivity of a multi-path
cluster tool or its process flows. This set of modules
includes processing and non-processing modules.
R1-2.1.1 processing module — an indivisible
production entity within an equipment system, e.g., a
processing chamber or station within a cluster tool.
(SEMI E79)
R1-2.1.2 non-processing module — an equipment
entity that supports the movement or conditioning of
units through the system, such as, robotic handler,
load/unload lock, pre-aligner.
R1-2.2 Multi-path cluster tool metrics require tracking
at the module level of at least productive states,
unscheduled downtime states, scheduled downtime
states, and other neutral states that are not in the first
three categories. Tracking of these states at the level of
the multi-path cluster tool as a whole is insufficient for
accurate evaluation of performance. Specific
requirements for tracking these states at the module
level, as well as requirements for handling the other
SEMI E10 states, are presented here.
R1-2.2.1 Productive state shall be tracked for each
module. For process modules, the productive time shall
include time for active loading and unloading of the
process module. Waiting times or inactive times,
including waiting for load, waiting for unload, and
process suspend times shall be specifically excluded
from productive time. Times for heating, cooling,
purging, cleaning, etc., that are specified as part of
process recipes shall be tracked as productive time.
However, similar times that are not specified as part of
process recipes shall be specifically excluded from
productive time.
NOTE 1: Productive state events may be derived from SEMI
E58 (ARAMS) state change data or the SEMI E116 (EPT)
module BUSY state events where the module or whole multi-
path cluster tool is known to be in a “manufacturing” state
and the SEMI E116 task type is either “Process” or
“Support.”