8. (15 points) Walk vs. Wait: 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 R seconds (constant), measured from the time at which you enter the bus, to arrive home. If you walk from the bus-stop then it takes W seconds (constant) to arrive home. Once you decide to walk, you can't take the bus in the middle (there's no other bus stop on the way home). Suppose that your policy when arriving at the bus stop is to rst wait up to 3 seconds for a bus, and if a bus arrives within .9 seconds, you take that bus to home. If a bus has not yet arrived by 3 seconds, then you walk home. (a) (10 points) What is the expected time from when you arrive at the bus stop until you reach home? Express your answer in terms of R, W, s, A. (b) (5 points) What is your optimal policy (optimal value of 3) to minimize the expected time to home? (consider all possible cases). Give an intuitive explanation of your optimal policy for each case you consider