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

## 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 $$\alpha$$.