Statistics for Maintenance Modelling: Renewal Processes

# Why Renewal Theory?¤

The main interest in renewal theory is the calculation of the renewal function, because this is what is needed for applications.

1. We are interested in calculating it analytically where we can, i.e. in getting a mathematical expression in closed form in terms of elementary functions. Later on we shall obtain such a result, and shall be at pains to replace an infinite series expansion with an expression involving sines and cosines. This exemplifies what is meant by 'closed form' and 'elementary functions'. Such calculations give insight into renewal theory and no mathematician can resist them!
2. We are also interested in calculating the renewal function numerically. After all, this is what is needed in practice.
3. Finally, its asymptotic behaviour for large times t is of interest. It is of course also possible to calculate higher moments than the renewal function (the mean number of failures). This is touched on here only very briefly. See Cox (1962) for more details.