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SEMI E10-0304 E © SEMI 1986, 2004 10 7 Uncertainty Measuremen t 7.1 Th e measures of equipment reliability, availability, and maintainability defin ed in Section 6 are single value estimates. They do not indicate the unc…

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Table 1 RAM Measurement Metric Summary

EQUIPMENT RELIABILITY

Metric How It Is Measured Ref #

MTBF

: Mean (productive) time between failures

productive time/

# of failures that occur during productive time

6.2.1

E-MTBF

: Mean (productive) time between

equipment-related failures

productive time/

# of equipment-related failures

that occur during productive time

6.2.2

MCBF: Mean cycles between failures total equipment cycles/

# of failures

6.2.3

E-MCBF: Mean cycles between equipment-related

failures

total equipment cycles/

# of equipment-related failures

6.2.4

EQUIPMENT AVAILABILITY

Metric How It Is Measured Ref #

equipment dependent uptime (%) equipment uptime × 100/(oper-time –

(all maint-delay DT + out-of-spec input DT + fac-rel DT))

6.3.1

supplier dependent uptime (%) equipment uptime × 100/(oper-time –

(users maint-delay DT + out-of-spec input DT + fac-rel DT))

6.3.2

operational uptime (%) equipment uptime × 100/

operations time

6.3.3

EQUIPMENT MAINTAINABILITY

Metric How It Is Measured Ref #

MTTR: Mean time to repair total repair time/

# of failures

6.4.1

E-MTTR: Mean time to repair for equipment-related

failures

total repair time for equipment-related failures/

# of equipment-related failures

6.4.2

MTOL: Mean time off-line total equipment downtime/

# of DT events

6.4.3

equipment dependent scheduled downtime (%) equipment scheduled downtime × 100/(oper-time –

(all maint-delay DT + out-of-spec input DT + fac-rel DT))

6.4.4

supplier dependent scheduled downtime (%) equipment scheduled downtime × 100/(oper-time –

(user maint-delay DT + out-of-spec input DT + fac-rel DT))

6.4.5

EQUIPMENT UTILIZATION

Metric How It Is Measured Ref #

operational utilization (%) productive time × 100/

operations time

6.5.1

total utilization (%) productive time × 100/

total time

6.5.2

NOTE: oper-time = operational time, DT = Downtime, fac-rel = facilities related, maint-delay = maintenance delay

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7 Uncertainty Measurement

7.1 The measures of equipment reliability, availability,

and maintainability defined in Section 6 are single

value estimates. They do not indicate the uncertainty or

precision of the estimate. Precision varies depending

upon the number of failures observed and the amount of

productive time contained within the observation

period.

7.2 Precision is described by calculating a lower and

upper confidence limit for the MTBF

and presenting

this interval along with the MTBF

point estimate.

7.3 These procedures assume that the failure rate is

constant and the times between failures are

independently distributed according to the exponential

distribution. Therefore, there are no improvement or

degradation trends and it is meaningful to calculate

MTBF

. Section 8 applies when the failure times

indicate that a non-constant failure rate is present (for

example, when there is reliability growth or

degradation). Section 8 would typically apply during

prototype reliability improvement testing.

7.4 Since MTTR distributions are unlikely to follow an

exponential distribution assumption, applying these

procedures to put confidence limits on MTTR would be

inappropriate.

7.5 Note that all procedures and tables referred to in

this section apply equally well to measuring the

precision of estimates for similar metrics, where hours

are replaced by cycles or units, for example. These

procedures apply to E-MTBF

or E-MCBF in the same

way. It is also appropriate to combine data from

identical tools being used the same way, in order to

improve the precision of MTBF

estimates.

7.6 Calculation of Lower and Upper Confidence Limits

— To obtain lower and upper MTBF

limits, multiply

the MTBF

estimate by factors obtained by table look-

up (Tables A1-1 and A1-2 in Appendix 1). For the case

when there are zero failures during the measurement

period, lower confidence limit factors for the MTBF

are given in the first row of Table A1-1 (they multiply

the amount of productive time that had no failures to

obtain the desired MTBF

lower limit). There is no

upper limit estimate for performance when there are

zero failures.

7.6.1 Calculation of the MTBF

Lower Limit — Use

Table A1-1 in Appendix 1 to obtain a k

r;conf

factor,

where r is the number of failures observed during the

measurement period and conf is the confidence level

desired. The rows of Table A1-1 correspond to

different values of r and the columns correspond to

different values of conf. Confidence levels ranging

from 80 percent to 95 percent are typical choices.

7.6.1.1 Since the equipment being measured has

demonstrated (at a given confidence level) that it is at

least as good as the MTBF

lower limit, this lower limit

is an important and useful performance statistic, and is

often used contractually.

7.6.1.2 Note that the factors in Table A1-1 for 90%

confidence are less than 0.5 until the number of failures

equals or exceeds 4. This means that when the number

of failures is under 4, the MTBF

lower limit will be

less than half the MTBF

estimate, and confidence

intervals will be wide. From the point of view of

precision, it is advantageous to have had 4 or more

failures.

7.6.1.3 Example: During a given calendar quarter, a

tool was productive for 1200 hours and had 6 failures.

The MTBF

estimate is 1200/6 = 200 hours. A 90

percent lower limit factor from Table A1-1

(corresponding to r = 6 failures) is 0.570. That means

that 200 × 0.570 = 114.0 hours is a 90 percent lower

confidence limit for the true tool MTBF

7.6.2 Calculation of the MTBF

Upper Limit — Use

Table A1-2 in Appendix 1 to obtain a k

r;conf

factor,

where r is the number of failures observed during the

measurement period and conf is the confidence level

desired. The rows of Table A1-2 correspond to

different values of r and the columns correspond to

different values of conf. Confidence levels ranging

from 80 percent to 95 percent are typical choices.

7.6.2.1 Example: During a given calendar quarter, a

tool was productive for 1200 hours and had 6 failures.

The MTBF

estimate is 1200/6 = 200 hours. A 90

percent upper limit factor from Table A1-2

(corresponding to r = 6 failures) is 1.904. That means

that 200 × 1.904 = 380.8 hours is a 90 percent upper

confidence limit for the true tool MTBF

7.6.3 Calculation of a Confidence Interval for the

MTBF

— Lower and upper 100 × (1 – α/2) confidence

limits for the MTBF

can be combined to give a 100 ×

(1 – α) confidence interval. Here α/2 is the chance of

missing on either end of the interval. A 90 percent

lower limit has an α/2 = 0.1 chance of not being low

enough to capture the true MTBF

, and the same is true

for a 90 percent upper limit. Therefore, a 90 percent

lower limit and a 90 percent upper limit combine to

give an 80 percent confidence interval. Similarly, a 95

percent lower limit and a 95 percent upper limit would

combine to give a 90 percent confidence interval.

7.6.3.1 Example: During a calendar quarter, a tool was

productive for 1200 hours and had 6 failures. The

MTBF

estimate is 1200/6 = 200 hours. The 90 percent

lower and upper limits are 114 and 380.8 respectively

(see Sections 7.6.1 and 7.6.2). The interval (114,

SEMI E10-0304

380.8) is then an 80 percent confidence interval for the

true tool MTBF

7.6.4 Calculation of the MTBF

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

7.6.5 Choosing a test length in order to be able to

demonstrate a required MTBF

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

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

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

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

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

(or MCBF) and E-MTBF

(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

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).