10.7 Buses arrive to a certain stop according to a Poisson process with rate . If you...
Question:
10.7 Buses arrive to a certain stop according to a Poisson process with rate λ. If you take the bus from that stop, then it takes a time S measured from the time you enter the bus to reach home. If you walk from that bus stop, then it takes a time T to reach home. Suppose that the rule you decide to follow is: wait a deterministic amount of time s, and then if the bus has not arrived yet, you walk home.
a) Compute the expected time to reach home from the moment you arrive at the bus stop.
b) Show that if T < 1/λ + S, then the expectation in part
a) is minimized by letting s = 0; and if T > 1/λ + S, then the expectation in part
a) is minimized by letting s = ∞. Also show that, when T = 1/λ + S, all s values give the same expected time.
c) Give an intuitive explanation why it is enough to consider 0 and ∞ as possible values for s when minimizing the expected time.
You may assume that S and T are deterministic in this problem. For extra points, repeat the reasoning with S and T random variables. How do the conditions in part
b) change?
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