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

8.4.5.3.

Exponential law model

Estimates of the parameters of the Exponential Law model can be obtained from either a graphical procedure or maximum likelihood estimation

Recall from section 1 that the Exponential Law refers to a NHPP process with repair rate $$ M'(t) = m(t) = \mbox{ exp } (\alpha + \beta t) \, . $$ This model has not been used nearly as much in industrial applications as the Power Law model, and it is more difficult to analyze. Only a brief description will be given here.

Since the expected number of failures is given by
$$ M(t) = (1/\beta) \cdot \mbox{ exp } (\alpha + \beta t) $$ and $$ \mbox{ln } M(t) = -\alpha \mbox{ ln } \beta + \beta t \, , $$ a plot of the cumulative failures versus time of failure on a log-linear scale should roughly follow a straight line with slope \(\beta\). Doing a regression fit of \(y = \mbox{ ln (cumulative failures)}\) versus \(x = \mbox{ time of failure}\) will provide estimates of the slope \(\beta\) and the intercept \(-\alpha \mbox{ ln } \beta\).

Alternatively, maximum likelihood estimates can be obtained from the following pair of equations: $$ \sum_{i=1}^r t_i + \frac{r}{\beta} - \frac{rT}{1 - e^{-\beta T}} = 0 $$ $$ \alpha = \mbox { ln } \left( \frac{r\beta}{e^{-\beta T} -1} \right) \, . $$

The first equation is non-linear and must be solved iteratively to obtain the maximum likelihood estimate for \(\beta\). Then, this estimate is substituted into the second equation to solve for the maximum likelihood estimate for \(\alpha\).