Buses arrive at a certain stop according to a Poisson process with rate A. 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 that your policy when arriving at the bus stop is to wait up to a time s, and if a bus has not yet arrived by that time then you walk home.

(a) Compute the expected time from when you arrive at the bus stop until you reach home.

(b) Show that if W < 1/A + R then the expected time of part

(a) is minimized by letting s = 0; if W> 1/A + R then it is minimized by lettings (that is, you continue to wait for the bus), and when W1/AR all values of s give the same expected time.

(c) Give an intuitive explanation of why we need only consider the cases s = 0 and so when minimizing the expected time.