A 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.

(i) What is the long-run fraction of time the doctor Spends sleeping? For that purpose, formulate a renewal reward process in which the reward in the ith interval is the amount of time the doctor gets to sleep in that interval. (ii) The doctor alternates between sleeping for an amOunt of time 55 and being awake fer an amount of time wi. Use (i) to compute Ewi