A pizza delivery company changed from petrol to electric vehicles (EVs) one year ago. Because the company was worried about vehicle battery failure affecting their business, they agreed a contract with the EV supplier who pays £1000 each time a vehicle battery fails to compensate for any lost pizza business.

The vehicles are at all times either out delivering pizzas or are having their battery recharged, unless either the vehicle suffers a battery failure or is involved in a crash.

(a) Draw a multi-state model diagram which could be used to analyse EV batteries and the compensation arrangement clearly labelling and defining all terms. 

(b) Over a week the fleet of EV vehicles were found to be out delivering pizza for a total of 1136 hours and having their batteries recharged for 544 hours. Vehicles were recharged 112 times in total. There were 4 battery failures of which 3 occurred whilst vehicles were being driven and one during a recharge. One vehicle was involved in a crash. Develop a formula and then find the maximum likelihood estimate for the transition intensity for battery failure whilst out delivering pizza. 

(c) How useful is this estimate in (b) for assessing value of the £1000 compensation terms?