Statistics for Maintenance Modelling: Renewal Processes

# The Integral Equation¤

The equation:

M(t) = Fk(t)

can be re-expressed as an integral equation:

M(t) = F(t) + f(u) M(t - u) du

and this can also be Laplace-transformed to yield:

(s) = (which was derived earlier by a different route)

In fact, there are many starting points for deriving these three formulae. An illuminating one is to derive
M(t) = F(t) + f(u) M(t - u) du directly. We start by assuming that the first renewal occurs at time u. The renewal function M(t) is then 1 + M(t - u), the renewal at time u plus the expected number of renewals after that. This occurs with pdf. f(u), so relaxing the assumption (the first renewal can occur at any time from 0 to t), we have:

M(t) = f(u) du + f(u) M(t - u) du | It is possible to use this integral equation to derive the asymptotic formula for M(t) directly without taking Laplace transforms.