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

## 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: $$M(t) = a t^b, \,\,\,\,\, M'(t) = abt^{b-1} = \alpha t ^{-\beta} \, .$$

The parameter $$\beta = 1-b$$ is called the Reliability Growth Slope and typical industry values for growth slopes during reliability improvement tests are in the 0.3 to 0.6 range.

If a system is observed for a fixed time of $$T$$ hours and failures occur at times $$t_1, \, t_2, \, t_3, \, \ldots, \, t_r$$ (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 $$\beta$$ and intercept (where $$t$$ = 1 on the log-log paper) of $$\mbox{-log}_{10} \alpha$$.

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 $$Y = \mbox{ ln } \left( t_i / i \right)$$ versus $$X = \mbox{ ln } t_i$$. The slope is the $$\beta$$ estimate and $$e^{\mbox{-intercept}}$$ is the $$a$$ estimate. The estimate of $$b$$ is 1-$$\beta$$.

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 $$t_1, \, t_2, \, t_3, \, \ldots, \, t_r$$. $$\begin{eqnarray} \hat{\beta} & = & 1 - \frac{r-1}{\sum_{i=1}^r \mbox{ ln } \left( \frac{T}{t_i} \right)} , \,\,\, \hat{\alpha} = \frac{r}{T^{1-\hat{\beta}}} \\ & & \\ \hat{b} & = &1 - \hat{\beta} = \frac{r-1}{\sum_{i=1}^r \mbox{ ln } \left( \frac{T}{t_i} \right)} \, . \end{eqnarray}$$

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-$$\alpha$$) % confidence interval $$(M_L, \, M_U)$$ for the MTBF at the end of the test. Note that $$M_L$$ is a 100(1-$$\alpha$$/2) % one-sided lower confidence bound and $$M_U$$ is a 100(1-$$\alpha$$/2) % one-sided upper confidence bound. The formulas are: $$\begin{eqnarray} M_L & = & \mbox{ MTBF } \,\, \frac{r(r-1)}{\left[ r + \frac{z^2_{1-\alpha/2}}{4} + \sqrt{r \, \frac{z^2_{1-\alpha/2}}{2} + \frac{z^4_{1-\alpha/2}}{16}} \right]^2} \\ & & \\ & & \\ M_U & = & \mbox{ MTBF } \,\, \frac{r(r-1)}{\left( r - z_{1-\alpha/2} \sqrt{\frac{r}{2}} \right)^2} \end{eqnarray}$$ where $$z_{1-\alpha/2}$$ is the 100(1-$$\alpha$$/2) percentile point of the standard normal distribution.

Case Study 1: Reliability Improvement Test Data Continued

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 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. We estimate $$\beta$$, $$a$$, and the MTBF at the end of test, along with a 100(1-$$\alpha$$) % confidence interval for the true MTBF at the end of test (assuming, of course, that the Power Law model holds). The parameters and confidence intervals for the power law model were estimated to be the following.

Estimate of $$\beta$$ = 0.5165
Estimate of $$a$$ = 0.2913
Estimate of MTBF at the end of the test =
310.234
80 % two-sided confidence interval:
(157.7139 , 548.5565)
90 % one-sided lower confidence limit =
157.7139
The analyses in this section can can be implemented using R code.