Statistics for Maintenance Modelling: Renewal Processes

# Discrete Renewal Processes¤

When time is discrete, it is possible to solve the renewal equation very simply. It becomes:

Mn = Fn + Mn - k pk | where pk is the probability that failure occurs at time k Fn is the distribution function of the probabilities pk at time n Mn and is the renewal function at time n.

Then and we solve recursively for M0, M1, ..., Mn.

Usually, p0 = 0, when we have the slight simplification:

Mn = Fn + Mn - k pk

In discrete-time block replacement, failures are replaced at cost cf at times 1..n - 1, and a planned replacement is done at time n at cost cb whether a failure has occurred at time n or not. The cost per unit time is therefore:

(cb + cf Mn - 1)/n.

It is simple to find the time n that minimises cost per unit time.