A single-pump petrol station is running low on petrol. The total volume of petrol remaining for sale is 100 litres.
Suppose cars arrive to the station according to a Poisson process with rate , and that each car fills independently of all other cars and of the arrival process, an amount of petrol that is distributed as a uniform random variable over - assume for example that all car tanks have a capacity of 50 litres and drivers decide "at random" when to refill. We assume that service is instantaneous so that there are no queues at the station.
(a) On average, how many cars will the petrol station fully service (sell the full amount requested) before it runs out of petrol (and before any refilling occurs)?
(b) How much time will it take on average before the station runs out of petrol (and before any refilling occurs)?
I'm not exactly sure where to start with this question part (a). Let be uniformly distributed over , then each time a car arrives at the petrol station, the total volume of petrol decreases by . So define to be the amount of petrol that the first arrival (an "arrival" here being when a car arrives at the petrol station and refills) and be that of the second arrival, and so on. Then each is identically and independently distributed as . So by the -th arrival, the station will have litres of petrol remaining. We stop once and we basically need to find ?
That's all I've got so far, if someone can provide a solution, that would be good.