Statistics for Maintenance Modelling: Continuous Lifetime Distributions 1

# Using the Poisson Distribution as a Model to Replace Lightbulbs¤

The manufacturing of large specialised electrical/electronic products is often carried out in large project halls. Due to the ceiling height in these halls, changing failed light bulbs requires some effort so there is a tendency to wait for a number of bulbs to have failed before carrying out the replacement. So as to avoid affecting production, the replacement task is carried out at weekends. However for safety reasons there is a limit on how many bulbs can have failed before immediate remedial action is needed. Hence a balance needs to be struck between replacing bulbs when only a small number have failed and waiting until larger numbers have failed and running the risk that the safety limit will be exceeded and a more costly maintenance action will be needed.

A natural model for the number of bulbs failing in a week is a Poisson distribution. Suppose that = 1.1 then the probability that there are:

- 0 failures in a week is 0.33
- 1 failure in a week is 0.37
- 2 failures in a week is 0.20
- 3 failures in a week is 0.07

Hence if the safety limit is 5 failures and at the weekend when the decision is made not to maintain this weekend, there is then 1 failure, the probability that an emergency maintenance is needed in the forthcoming week is:

1 - (0.33 + 0.37 + 0.20 + 0.07) = 0.03 (i.e. 1 - the probability of upto 4 failures in total)