A maintenance company services elevators in a premium high-rise office building in London.

From historical data the elevator breaks down on average once every 100 days. Such failures

cause chaos and so the building owners want the maintenance company to improve the

situation. Given the high rental rates the building owners attract for such a premium office

space, the owners demand that the most likely number of failures in any given year should be

zero. The office building is open 250 days per year.

The maintenance company therefore install a predictive maintenance system. Through a

series of networked sensors the system monitors the ongoing condition of the elevators. A

machine learning algorithm trained on the monitoring data can often detect a fault before

the elevator breaks down, raising an alarm and giving the company time to conduct

maintenance on it before it fails. The machine learning algorithm has a true positive rate of

75% and a false positive rate of 5%.

a) What is the probability per year of at least one elevator failure without the predictive

maintenance system in place?

b) Given the predictive maintenance system is put in place, draw a probability tree

diagram showing all four possible outcomes of the algorithm. Calculate the

probability of each possible outcome.

c) What is the probability per year of at least one elevator failure with the predictive

maintenance system in place? Will the introduction of the predictive maintenance

system meet the building owners' demands?

d) Assuming the maintenance company only maintains the elevator when the predictive

maintenance system reports a problem, what percentage of maintenance activities

are unnecessary?

Analysis of past failures identify that a single component, a sensor monitoring the stress

on the wires that lift the elevator, is responsible for many of the elevator failures.

Engineering data from the manufacturer shows that the sensor failed 10 times over

25,000 hours of operation.

e) What is the Mean Time Between Failure (MTBF)?

f) What is the random variable equation for the time to wait for 1 failure of the sensor?

State the 5th, 50th and 95th percentiles of the distribution.

g) Sketch the distribution to indicate its shape and comment on what issues that might

therefore occur in simply using the mean time between failure (that is the MTBF).