Statistics for Maintenance Modelling: Renewal Processes

# Evaluating Pie(t)¤

We could use this equation to evaluate (t) numerically or analytically, after calculating v. Taking Laplace transforms:

(s) = (1 - (s)a)(1 + (s))/s

where = a b/(1 - a b), and so:

(s) =

We can find (t) by inverting the Laplace transform. Asymptotically, expanding a = 1 - a s, b = 1 - b s,

(s) = a/(s(a + b))
and so (t) a/(a + b), which is clearly the long-run proportion of time that the system is 'up'.