Statistics for Maintenance Modelling: Renewal Processes

# Calculating the Renewal Function Analytically: Erlang (2) Distribution (2)¤

Note that M(t):

- increases initially as M(t) = 1/2(t)2 for small t (quadratically)
- and then increases linearly as t
- equals t/2 - 1/4 for large t.

Since for the interfailure time distribution, the mean = 2/, and the variance 2 = 2/2, for large t when 0,

M(t) t/ + (2 - 2)/(22)

This formula also holds for the exponential distribution, where = . We shall see that this asymptotic formula is true generally for 'non-arithmetic' distributions with finite variance.

Some harder exact calculations are deferred until after this asymptotic formula has been discussed in more detail.