Statistics for Maintenance Modelling: Repairable Systems

# Parametric Forms of Lambda(t) for NHPP Models¤

Two particular forms for (t) are:

1. 1(t) = - gives a simple model to describe a happy system (1 < 0) or a sad system (1 > 0). If 1 = 0 we have a HPP.
2. 2(t) = - referred to as the Weibull or power law process (with A = ). The name Weibull process is a misnomer here since only the time to the first failure has a Weibull distribution. Once again this model may represent a sad or happy system as discussed above.

The parameters within (t) can be estimated using the maximum likelihood method. Suppose we observe a system over the time interval (0, t1] with failures observed at t1, t2, ... tn. The likelihood may be obtained as follows. The probability of observing no failures in (0, t1), one failure in (t1, t1 + t1), no failures in (t1 + t1, t2), one failure in (t2, t2 + t2) and so on up to no failures in (tn + tn, t0) is, for small t1, t2,..., tn:

L = (t1)t1 (t2)t2 ...

Dividing through by t1, t2,...,tn gives:

L = ti

and the log-likelihood is:
ln L = ln (ti) - (t) dt. Maximising this with respect to the parameters within (t) estimates their values.