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Reliability Growth Check (Duane Power-Law Model, Growth Slope, MTBF at End of Test)

We have been fixing faults as they appear during a development or debugging test: is reliability actually growing, what MTBF has the equipment reached at the end of the test, and has it reached the goal?

Reliability growth analysis with the power-law (Duane, Crow–AMSAA) model from up to fifteen cumulative failure times in a test that ends at a fixed time: the growth slope, the model parameters by modified maximum likelihood, the instantaneous MTBF at the end of the test with approximate confidence bounds, the cumulative MTBF, and the test hours needed to reach the MTBF goal if growth continues at the same rate.

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How this tool works

Inputs, units and where to find them

InputUnitTypical source
Test end timehTotal test hours when the test stopped (at or after the last failure).
Number of failuresfailuresHow many failure times below to use, in order.
Confidence for the lower limit%Confidence for the one-sided lower limit on the end-of-test MTBF; the upper limit pairs with it as a two-sided interval.
MTBF goalhMTBF the equipment must reach, from a specification or plan.
Cumulative test hours at each failure
Failure 1 timehCumulative test hours when failure 1 occurred.
Failure 2 timehCumulative test hours when failure 2 occurred.
Failure 3 timehCumulative test hours when failure 3 occurred.
Failure 4 timehCumulative test hours when failure 4 occurred.
Failure 5 timehCumulative test hours when failure 5 occurred.
Failure 6 timehCumulative test hours when failure 6 occurred.
Failure 7 timehCumulative test hours when failure 7 occurred.
Failure 8 timehCumulative test hours when failure 8 occurred.
Failure 9 timehCumulative test hours when failure 9 occurred.
Failure 10 timehCumulative test hours when failure 10 occurred.
Failure 11 timehCumulative test hours when failure 11 occurred.
Failure 12 timehCumulative test hours when failure 12 occurred.
Failure 13 timehCumulative test hours when failure 13 occurred.
Failure 14 timehCumulative test hours when failure 14 occurred.
Failure 15 timehCumulative test hours when failure 15 occurred.

What each input means

Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.

What this means: Test end time
Illustration: Test end timeA timeline from a start mark to an end mark; Test end time is the time between them.startendTExample: 1500 hTest end time
What it is
The total test hours when the growth test stopped, at or after the last failure.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1500 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A longer failure-free end of test raises the growth estimate and the end-of-test MTBF.
Get it right
Use the same test clock as the failure times; the test should stop at a planned time, not at a failure.
What this means: Number of failures
Illustration: Number of failuresA row of identical items being counted; Number of failures is how many there are.rExample: 10 failuresNumber of failures
What it is
How many of the failure times listed below are used, from the first; at least two are needed.
Units and format
Enter a number in failures: failures. The form accepts 2 failures to 15 failures. Use a whole number. Required: the tool does not calculate until every input is valid.
Example
The form opens with 10 failures as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
More failures narrow the confidence limits.
Get it right
Enter every relevant failure in the test; leaving failures out overstates the MTBF.
What this means: Confidence for the lower limit
Illustration: Confidence for the lower limitA bar of 100 parts with the share marked; Confidence for the lower limit is that share out of 100. The bar follows the value you type.0 %100 %CExample: 90%Confidence for the lower limit
What it is
The confidence for the one-sided lower limit on the end-of-test MTBF; the upper limit pairs with it as a two-sided interval at twice the remainder.
Units and format
Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 50.1% to 99.9%. Required: the tool does not calculate until every input is valid.
Example
The form opens with 90% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
A higher confidence lowers the lower limit.
Get it right
NIST’s case study uses 90 %, giving an 80 % two-sided interval.
What this means: MTBF goal
Illustration: MTBF goalA timeline from a start mark to an end mark; MTBF goal is the time between them.startendMTBFExample: 150 hMTBF goal
What it is
The mean time between failures the equipment must reach, from a specification or development plan.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 150 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Only moves the pass or fail line and the projected test hours.
Get it right
Use the same failure definition as the test records.

Cumulative test hours at each failure

What this means: Failure 1 time
Illustration: Failure 1 timeA timeline from a start mark to an end mark; Failure 1 time is the time between them.startendt1Example: 5 hFailure 1 time
What it is
Cumulative test hours on the test clock when failure 1 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 5 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Record the test-clock hours at each failure, not calendar dates; count every relevant failure, fixed or not.
What this means: Failure 2 time
Illustration: Failure 2 timeA timeline from a start mark to an end mark; Failure 2 time is the time between them.startendt2Example: 40 hFailure 2 time
What it is
Cumulative test hours on the test clock when failure 2 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 40 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 1’s time; use the same clock for every failure.
What this means: Failure 3 time
Illustration: Failure 3 timeA timeline from a start mark to an end mark; Failure 3 time is the time between them.startendt3Example: 43 hFailure 3 time
What it is
Cumulative test hours on the test clock when failure 3 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 43 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 2’s time; use the same clock for every failure.
What this means: Failure 4 time
Illustration: Failure 4 timeA timeline from a start mark to an end mark; Failure 4 time is the time between them.startendt4Example: 175 hFailure 4 time
What it is
Cumulative test hours on the test clock when failure 4 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 175 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 3’s time; use the same clock for every failure.
What this means: Failure 5 time
Illustration: Failure 5 timeA timeline from a start mark to an end mark; Failure 5 time is the time between them.startendt5Example: 389 hFailure 5 time
What it is
Cumulative test hours on the test clock when failure 5 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 389 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 4’s time; use the same clock for every failure.
What this means: Failure 6 time
Illustration: Failure 6 timeA timeline from a start mark to an end mark; Failure 6 time is the time between them.startendt6Example: 712 hFailure 6 time
What it is
Cumulative test hours on the test clock when failure 6 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 712 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 5’s time; use the same clock for every failure.
What this means: Failure 7 time
Illustration: Failure 7 timeA timeline from a start mark to an end mark; Failure 7 time is the time between them.startendt7Example: 747 hFailure 7 time
What it is
Cumulative test hours on the test clock when failure 7 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 747 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 6’s time; use the same clock for every failure.
What this means: Failure 8 time
Illustration: Failure 8 timeA timeline from a start mark to an end mark; Failure 8 time is the time between them.startendt8Example: 795 hFailure 8 time
What it is
Cumulative test hours on the test clock when failure 8 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 795 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 7’s time; use the same clock for every failure.
What this means: Failure 9 time
Illustration: Failure 9 timeA timeline from a start mark to an end mark; Failure 9 time is the time between them.startendt9Example: 1299 hFailure 9 time
What it is
Cumulative test hours on the test clock when failure 9 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1299 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 8’s time; use the same clock for every failure.
What this means: Failure 10 time
Illustration: Failure 10 timeA timeline from a start mark to an end mark; Failure 10 time is the time between them.startendt10Example: 1478 hFailure 10 time
What it is
Cumulative test hours on the test clock when failure 10 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1478 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 9’s time; use the same clock for every failure.
What this means: Failure 11 time
Illustration: Failure 11 timeA timeline from a start mark to an end mark; Failure 11 time is the time between them.startendt11Example: 1600 hFailure 11 time
What it is
Cumulative test hours on the test clock when failure 11 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1600 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 10’s time; use the same clock for every failure.
What this means: Failure 12 time
Illustration: Failure 12 timeA timeline from a start mark to an end mark; Failure 12 time is the time between them.startendt12Example: 1750 hFailure 12 time
What it is
Cumulative test hours on the test clock when failure 12 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1750 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 11’s time; use the same clock for every failure.
What this means: Failure 13 time
Illustration: Failure 13 timeA timeline from a start mark to an end mark; Failure 13 time is the time between them.startendt13Example: 1900 hFailure 13 time
What it is
Cumulative test hours on the test clock when failure 13 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 1900 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 12’s time; use the same clock for every failure.
What this means: Failure 14 time
Illustration: Failure 14 timeA timeline from a start mark to an end mark; Failure 14 time is the time between them.startendt14Example: 2100 hFailure 14 time
What it is
Cumulative test hours on the test clock when failure 14 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2100 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 13’s time; use the same clock for every failure.
What this means: Failure 15 time
Illustration: Failure 15 timeA timeline from a start mark to an end mark; Failure 15 time is the time between them.startendt15Example: 2300 hFailure 15 time
What it is
Cumulative test hours on the test clock when failure 15 occurred, counted from the start of the growth test.
Units and format
Enter a number in h: hours. The form accepts 0.000001 h or more. Required: the tool does not calculate until every input is valid.
Example
The form opens with 2300 h as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
Effect on the result
Earlier failures with long gaps later raise the growth slope and the MTBF at the end of the test.
Get it right
Must be at or after failure 14’s time; use the same clock for every failure.

Method and formulas

  1. Power-law model: expected failures M(t) = a t^b, repair rate m(t) = a b t^(b − 1); growth slope β = 1 − b (NIST e-Handbook 8.4.5.2).
  2. Modified maximum likelihood for a test ending at time T: b = (r − 1) ÷ Σ ln(T ÷ tᵢ), a = r ÷ T^b; MTBF at the end of test = 1 ÷ m(T) = T ÷ (r b).
  3. Approximate limits: M_L = MTBF × r(r − 1) ÷ [r + z²/4 + √(r z²/2 + z⁴/16)]², M_U = MTBF × r(r − 1) ÷ (r − z √(r/2))², with z the standard normal value for the confidence.
  4. Example: growth slope 0.51649; MTBF at end of test 310.23 h, lower limit 157.71 h.

Limitations

  • For tests that end at a fixed time with fixes made during the test; NIST notes the bounds are approximate. A test that stops at a failure needs a slightly different estimate.
  • Projecting the goal assumes growth continues at the same slope, which needs the same rate of finding and fixing faults.
  • Example values are illustrations. Replace them with values for your actual task.

Typical uses

  • Track a test-analyse-and-fix programme on a new machine
  • Estimate the MTBF reached at the end of a debugging run
  • Plan how much more testing reaches an MTBF goal

Who it is for

  • Reliability and test engineers
  • Machine builders and commissioning engineers
  • Development engineers

What you get

  • Growth slope β and model parameters
  • Instantaneous MTBF at the end of test with confidence limits
  • Cumulative MTBF
  • Test hours for the MTBF estimate to reach the goal at the current growth rate

Interpreting the result

  • A slope β near 0 means no growth (a constant failure rate); β between 0 and 1 means improvement, and a larger β faster improvement (NIST 8.1.9.1). NIST’s case study gives β = 0.5165 and an end-of-test MTBF of 310 h.

References

  • NIST/SEMATECH e-Handbook of Statistical Methods, 8.4.5.2 Power law (Duane) model — https://www.itl.nist.gov/div898/handbook/apr/section4/apr452.htm
  • NIST/SEMATECH e-Handbook of Statistical Methods, 8.1.9.1 NHPP power law — https://www.itl.nist.gov/div898/handbook/apr/section1/apr191.htm

Learn the Concepts Behind This Tool

These KEVOS® Library readings explain the method this tool uses, with worked examples.

KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.

Open the Reliability Growth Check (Duane Power-Law Model, Growth Slope, MTBF at End of Test) calculator

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