Fred and Gretchen are waiting at a bus stop in Blotchville. Two bus routes, Route 1 and

Chapter 5, Problem 43

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Fred and Gretchen are waiting at a bus stop in Blotchville. Two bus routes, Route 1 and Route 2, have buses that stop at this bus stop. For Route i, buses arrive according to a Poisson process with rate i buses/minute. The Route 1 process is independent of the Route 2 process. Fred is waiting for a Route 1 bus, and Gretchen is waiting for a Route 2 bus. (a) Given that Fred has already waited for 20 minutes, on average how much longer will he have to wait for his bus? (b) Find the probability that at least n Route 1 buses will pass by before the first Route 2 bus arrives. The following result from Chapter 7 may be useful here: for independent random variables X1 Expo(1), X2 Expo(2), we have P(X1 < X2) = 1/(1+2). (c) For this part only, assume that 1 = 2 = . Find the expected time it will take until both Fred and Gretchen have caught their buses.