A simple approximation to the renewal function (reliability theory)

The authors present a simple, easy-to-understand approximation to the renewal function that is easy to implement on a personal computer. The key idea is that, for small values of time, the renewal function is almost equal to the cumulative distribution function of the interrenewal time, whereas for larger values of time an asymptotic expansion depending only on the first and second moment of the interrenewal time can be used. The relative error is typically smaller than a few percent for Weibull interrenewal times. The simple approximation methods works very well with one term if not too much accuracy is required (e.g. in the block replacement problem) or if the interrenewal (failure) distribution is not exactly known (e.g. only the first two moments are known). Although the accuracy of the simple approximation can be improved by increasing the number of terms, this strategy is not advocated since speed and simplicity are lost. If high accuracy is required, it is better to use another approximating method (e.g. power series expansion or cubic splines method).<
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