Question

Buses arrive at a certain stop according to a Poisson process with rate λ. If you take the bus from that stop then it takes a time R, measured from the time at which you enter the bus, to arrive home. If you walk from the bus stop then it takes a time W to arrive home. Suppose your policy when arriving at the bus stop is to wait up to time s, and if a bus has not yet arrived by that time then you walk home.

Find the value of s which maximizes the expected time from when you arrive at the bus stop until you reach home.

Answer 1

From the relation above we see that if is positive we maximize the expectation by letting , i.e. . If is negative we maximize the expectation by letting i.e. . When , then the expectation is independent of .