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A young doctor is working at night in an emergency room. Emergencies come in at times of a…

A young doctor is working at night in an emergency room. Emergencies come in at times of a Poisson process with rate 0.5 per hour. The doctor can only get to sleep when it has been 36 minutes (.6 hours) since the last emergency. For example, if there is an emergency at 1:00 and a second one at 1:17 then she will not be able to get to sleep until at least 1:53, and it will be even later if there is another emergency before that time. (a) Compute the long-run fraction of time she spends sleeping, by formulating a renewal reward process in which the reward in the ith interval is the amount of time she gets to sleep in that interval. (b) The doctor alternates between sleeping for an amount of time si and being awake for an amount of time ui. Use the result from (a) to compute Eui. (c) Solve problem (b) by noting that the doctor trying to sleep is the same as chicken crossing the road in Exercise 2.23.

Exercise 2.23

When did the chicken cross the road? Suppose that traffic on a road follows a Poisson process with rate cars per minute. A chicken needs a gap of length at least c minutes in the traffic to cross the road. To compute the time the chicken will have to wait to cross the road, let t1, t2, t3…….. be the interarrival times for the cars