8. Assessing Product Reliability
8.4. Reliability Data Analysis
8.4.5. How do you fit system repair rate models?

8.4.5.2.

Power law (Duane) model

The Power Law (Duane) model has been very successful in modeling industrial reliability improvement data

Brief Review of Power Law Model and Duane Plots

Recall that the Power Law is a NHPP with the expected number of fails, M(t), and the repair rate, M'(t) = m(t), given by: 

The parameter  = 1-b is called the Reliability Growth Slope and typical industry values for growth slopes during reliability improvement tests are in the .3 to .6 range. 

If a system is observed for a fixed time of T hours and failures occur at times t₁, t₂, t₃, ..., t_r (with the start of the test or observation period being time 0), a Duane plot is a plot of (t_i / i) versus t_i on log-log graph paper. If the data are consistent with a Power Law model, the points in a Duane Plot will roughly follow a straight line with slope  and intercept (where t = 1 on the log-log paper) of -log₁₀a.

MLE's for the Power Law model are given

Estimates for the Power Law Model

Computer aided graphical estimates can easily be obtained by doing a regression fit of Y = ln (t_i / i) vs X = ln t_i. The slope is the  estimate and e^-intercept is the a estimate. The estimate of b is 1-. The Dataplot command for the regression fit is FIT Y X. 

However, better estimates can easily be calculated. These are modified maximum likelihood estimates (corrected to eliminate bias). The formulas are given below for a fixed time of T hours, and r failures occurring at times t₁, t₂, t₃, ..., t_r. 

The estimated MTBF at the end of the test (or observation) period is 

Approximate confidence bounds for the MTBF at end of test are given

Approximate Confidence Bounds for the MTBF at End of Test

We give an approximate 100×(1-)% confidence interval (M_L, M_U) for the MTBF at the end of the test. Note that M_L is a 100×(1-/2)% lower bound and M_U is a 100×(1-/2)% upper bound. The formulas are: 

with  is the upper 100×(1-/2) percentile point of the standard normal distribution. 
 

Dataplot calculations for the Power Law (Duane) Model 

Dataplot Estimates And Confidence Bounds For the Power Law Model 

Dataplot will calculate , a, and the MTBF at the end of test, along with a 100x(1-)% confidence interval for the true MTBF at the end of test (assuming, of course, that the Power Law model holds). The user needs to pull down the Reliability menu and select "Test" and "Power Law Model". The times of failure can be entered on the Dataplot spread sheet. A Dataplot example is shown next. 

Case Study 1: Reliability Improvement Test Data Continued

Dataplot results fitting the Power Law model to Case Study 1 failure data

This case study was introduced in section 2, where we did various plots of the data, including a Duane Plot. The case study was continued when we discussed trend tests and verified that significant improvement had taken place. Now we will use Dataplot to complete the case study data analysis. 

The observed failure times were: 5, 40, 43, 175, 389, 712, 747, 795, 1299 and 1478 hours, with the test ending at 1500 hours. After entering this information into the "Reliability/Test/Power Law Model" screen and the Dataplot spreadsheet and selecting a significance level of .2 (for an 80% confidence level), Dataplot gives the following output:

THE RELIABILITY GROWTH SLOPE BETA IS 0.516495 

THE A PARAMETER IS 0.2913 

THE MTBF AT END OF TEST IS 310.234 

THE DESIRED 80 PERCENT CONFIDENCE INTERVAL IS: 
(157.7139 , 548.5565) 
AND 157.7139 IS A (ONE-SIDED) 90 PERCENT 
LOWER LIMIT 

Note: The downloadable package of statistical programs, SEMSTAT, will also calculate Power Law model statistics and construct Duane plots. The routines are reached by selecting "Reliability" from the main menu then the "Exponential Distribution" and finally "Duane Analysis".