Weibull Prediction of Future Failures

This is an example of a recently published in the Reliability Analytics Toolkit called Weibull Prediction of Future Failures. This tool is based on work described in references 1 and 2. For a population of N items placed on test, this tool calculates the expected number of failures for some future time interval based on the following two inputs:
1. the estimated Weibull shape parameter and
2. some number of failures (X>=1) during the initial time interval (t1).

For example, if there are 20,000 items that have started operation at “time 0” and 8 have found to be in a failed state at the 3 year point.  It is desired to estimate how many additional items will fail between the 3 and 10 year point.  The figure below shows the inputs required.  Although a single value could be entered for input #1, beta, three values separated by commas are entered for purposes of performing sensitivity analysis.  It is estimated that the Weibull shape parameter is in the range of 3.0 to 3.6, with a typical value of 3.3.

The resulting estimated number of failures is 413, assuming a Weibull shape parameter of 3.3, with a lower 90% bound of 206 failures and an upper bound of 753.

The calculation of confidence limits in the above table depends on p and q being small. If p + q is greater than 0.10, confidence limits are not shown (references provide examples with p+q=0.03). p is the probability of item failure up to t1. Given that an item survives to t1, q is the probability of item failure from t1 to t2. r is the probability of item survival to t2.
p + q + r = 1.0

 

References:

  1. Nordman, D. J., & Meeker, W. Q. (2002). Weibull Prediction Intervals for a Future Number of Failures. Technometrics. 44, 15-23. .
  2. Nelson, W. (2000). Weibull prediction of a future number of failures. Quality and Reliability Engineering International. 16, 23-26.
  3. Abernethy, Robert, The New Weibull Handbook Fifth Edition, Reliability and Statistical Analysis for Predicting Life, Safety, Supportability, Risk, Cost and Warranty Claims
  4. Nelson, Wayne, Applied Life Data Analysis (Wiley Series in Probability and Statistics)
  5. http://www.barringer1.com/wdbase.htm, database of typical Weibull shape and characteristic life parameters.