Statistics for Maintenance Modelling: Renewal Processes

# Renewal Process Replacement Costs¤

The expected cost of operating a system for time t, that is replaced on failure at cost c and has no running costs is cM(t). The cost of issuing a warranty is an example. Block replacement of components is another. Here components (such as light bulbs that are hard to access) are replaced on failure, and all bulbs are replaced as planned at time t.

If:

- the cost of replacing a failed bulb is cf
- the cost of a block replacement per bulb is cb

then:

- the expected accumulated cost for a bulb under this strategy up to time t is cf M(t) + cb
- the cost per unit time is (cf M(t) + cb)/ t.

## Minimising Costs

The obvious best policy is to choose the replacement age t to minimise total cost per unit time.

## A Non-Time Based Application

Inventory is another (rather strange!) application. Here demand for a product per week (say) is the random variate (rather than a time), and rather than mean number of failures by time we seek the mean number of weeks for total demand to exceed stock level.