Statistics for Maintenance Modelling: Repairable Systems

# Example of Fitting a NHPP Model to Software Failures and Estimating Parameters¤

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The following are the intervals in days between successive failures of a piece of software developed as part of a large data system: 9, 12, 11, 4, 7, 2, 5, 8, 5, 7, 1, 6, 1, 9, 4, 1, 2, 2, 6, 1, 11, 33, 7, 91, 2, 1, 87, 47, 12, 9, 135, 258, 16, 35.

In the software reliability context, we want to know if there is evidence of reliability growth, i.e. is 1 < 0? Observation of the software was made for 850 days, resulting in 34 observed failures. Therefore, n = 34, t0 = 850, and ti = 6980.

## Estimating Parameters by Substituting Lambda 1(t) into the Log-Likelihood

Using Excel's solver to solve the equation on screen 14, ti + n 1-1 nt0(1 - )-1 = 0, we have 1 = -0.004 which suggests that there is evidence of reliability growth (1 modifies t in 1(t)). 0 = ln (n 1 / ( - 1)-1), which is -1.96.

## Estimating Parameters by Substituting Lambda 2(t) into the Log-Likelihood

Applying this model to the software failure data, we have = 0.0007 and = 0.1542.
This also confirms that reliability is improving since < 1.

## Which Model of Lambda (t) is Best?

To determine which model is better in this case, one has to use other criteria such as assessing the goodness of fit, (discussed throughout the Testing Model Fitness session), or simply comparing the values of ln L. The one with the larger likelihood is the better of the two - which in this case is 2(t).