Statistics for Maintenance Modelling: Repairable Systems

# Homogeneous Poisson Process Models (HPP)¤

A stochastic point process may be regarded as a sequence of events distributed in a continuum according to some probability mechanism. If we take the continuum to be time and the events as failures, it is a frame for repairable systems.

There are a variety of stochastic point process models for the failures of repairable systems, but we will focus on the Poisson process (see The Poisson Process screen in the Continuous Lifetime Distributions session). This process is conceptually simple and it can model happy and sad systems as well as systems with constant ROCOF. The relevant statistical methodology is well developed and easy to apply.

For an HPP, natural assumptions, which may or may not be satisfied in any particular example, are:

- failures occurring in disjoint time intervals are independent
- the ROCOF is constant.

If both assumptions are satisfied, then it is called a homogeneous Poisson process. Two important properties of a HPP are:

1. the number of failures N in a time interval of length t has a Poisson distribution with mean t, so that:
   P(N = k) = ( t)k / k!
2. the times between successive failures are independent random variables each having an exponential density with scale parameter , so that:
   P(t > x) =