8.
Assessing Product Reliability
8.1. Introduction 8.1.9. How can you model reliability growth?


The Exponential Law is another useful reliability growth model to try when the Power law is not fitting well 
When the data points in a Duane plot
show obvious curvature, a model that might fit better is the
NHPP Exponential Law.
For this model, if \(\beta\) < 0, the repair rate improves over time according to $$ m(t) = e^{\alpha + \beta t} \, . $$ The corresponding cumulative expected failures model is $$ M(t) = A \left( 1e^{\beta t} \right) \, . $$ This approaches the maximum value of \(A\) expected failures as \(t\) goes to infinity, so the cumulative failures plot should clearly be bending over and asymptotically approaching a value $$A = \frac{e^\alpha}{\beta} \, . $$ Rule of thumb: First try a Duane plot and the Power law model. If that shows obvious lack of fit, try the Exponential Law model, estimating parameters using the formulas in the Analysis Section for the Exponential law. A plot of cum fails versus time, along with the estimated \(M(t)\) curve, can be used to judge goodness of fit. 