Statistics for Maintenance Modelling: Renewal Processes

# Asymptotic Formulae¤

The behaviour of the renewal function at large time t values is its asymptotic behaviour as t . Studying this gives useful insight, but it is not directly related to calculations needed for replacement modelling, and this more theoretical section can be skipped on first reading.

We start with the result we proved on screen 10:

M(t) = Fk(t)

where Fk is the k-fold convolution of the distribution function (the distribution function corresponding to the k-fold convolution of the pdf., or the distribution function of the sum of k random variables). This is the probability that at least k events have occurred (the kth event has occurred) by time t.

Let the Laplace transform of the pdf f be (s). Then the transform of Fk(t) is (s)k/s. Taking the Laplace transform of the equation for M(t), and summing the geometric series, we obtain:

(s) = (1/s) (s)k

= (1/s) (s)k = | This looks a lot simpler than the original expression, and will yield the asymptotic formula fairly easily (but by a non-rigorous argument).