Statistics for Maintenance Modelling: Renewal Processes

# Alternating Renewal Processes¤

Here for example a machine might be alternately working or being repaired after failure. The successive times to failure and repair times form a renewal process. Successive idle and busy periods of the server of a queue form an alternating renewal process.

The main result we want to derive is the probability that the system is working at time (t), and its asymptotic value.

Let the pdfs of time to failure and repair be fa(t), fb(t) respectively. The time from insertion of a component in a socket to its renewal (end of its repair period) has pdf. f(t) = fa(u) fb(t - u) du, and these periods constitute an ordinary renewal process. In terms of Laplace transforms, = a b and (s) = (s)/(s(1 - (s))).

The probability satisfies the equation:

(t) = Sa(t) + v(u) Sa(t - u) du,

where Sa is the survival function of time to failure, and v is the renewal density. Recall that v(t) dt is the probability that any failure (1st, 2nd, 3rd, etc) occurs in time dt, as v(t) = . The equation for (t) says that the probability that the system is working at time t is the probability that it never failed, or that it last failed at some time u and has not failed since.