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


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: $$ M(t) = a t^b, \,\,\,\,\, M'(t) = abt^{b1} = \alpha t ^{\beta} \, . $$
The parameter \(\beta = 1b\) 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 loglog 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 loglog 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 \(t_1, \, t_2, \, t_3, \, \ldots, \, t_r\). $$ \begin{eqnarray} \hat{\beta} & = & 1  \frac{r1}{\sum_{i=1}^r \mbox{ ln } \left( \frac{T}{t_i} \right)} , \,\,\, \hat{\alpha} = \frac{r}{T^{1\hat{\beta}}} \\ & & \\ \hat{b} & = &1  \hat{\beta} = \frac{r1}{\sum_{i=1}^r \mbox{ ln } \left( \frac{T}{t_i} \right)} \, . \end{eqnarray} $$ 

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) % onesided lower confidence bound and \(M_U\) is a 100(1\(\alpha\)/2) % onesided upper confidence bound. The formulas are: $$ \begin{eqnarray} M_L & = & \mbox{ MTBF } \,\, \frac{r(r1)}{\left[ r + \frac{z^2_{1\alpha/2}}{4} + \sqrt{r \, \frac{z^2_{1\alpha/2}}{2} + \frac{z^4_{1\alpha/2}}{16}} \right]^2} \\ & & \\ & & \\ M_U & = & \mbox{ MTBF } \,\, \frac{r(r1)}{\left( r  z_{1\alpha/2} \sqrt{\frac{r}{2}} \right)^2} \end{eqnarray} $$ where \(z_{1\alpha/2}\) is the 100(1\(\alpha\)/2) percentile point of the standard normal distribution. 

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 \(\beta\) = 0.5165The analyses in this section can can be implemented using R code. 