Statistics for Maintenance Modelling: Renewal Processes

# Other Types of Renewal Process¤

## Asynchronous Renewal Process

In an asynchronous renewal process, where the first time to renewal has pdf. S/. In general, a modified renewal process has an arbitrary first time to renewal, with pdf. (say) fa, which is different from the distribution of the remaining times. The renewal equation is modified:

M(t) = Fa(t) + M(t - u)f(u) du.

Such processes are sometimes used to model situations where a component is supposed to have a different time to failure after its first repair.

## General Alternating Renewal Process

Alternating renewal processes generalise to the case where there are n possible states, not just two, and there is a probability Pji of jumping to j state from state i. Times spent in each state have pdfs fi(t).

## Reward-Renewal Process

Renewal-reward processes are also sometimes useful. Here we may think of a system where failure or operation has an associated cost Y, where Y is a random variable that may be correlated with X. We may want to find the expected cost to time t.