Statistics for Maintenance Modelling: Renewal Processes

# Numerical Calculation¤

Often we can not get an analytic expression for M(t). For small t, M(t) = F(t) (when the probability of 2 or more replacements is small). For large t, we have the asymptotic formula. Finding M(t) in between is the problem!

The renewal integral equation is the basis of the most popular method of calculating M(t). In Xie's method, it is discretized, and solved recursively for M(), M(2), etc.

It is also possible to use the Laplace transform of this equation. If we can find the transform of the pdf., we can invert the transform numerically, using some method such as Talbot's method. Talbot's method is described by Murli and Rizzardi (1990), and compared favourably with other methods by Duffy (1993).

If we cannot find the transform of f(t) analytically (as we cannot for the Weibull distribution), we can perhaps approximate f(t) so that we can find an expression for its transform.