3. (20 points) A clinic has four different stations (see the following figure), where patients might be directed for service before leaving. All patients first sign in with a single receptionist who works at a rate of 35 patients per hour. From there, they might go to the nurse station (probability 0.6) or to the exam room. There is a single nurse in the nurse station who works at the rate of 20 patients per hour, and there are also three doctors in the exam room whose average processing time (each doctor) is 12 minutes. All patients eventually go through the check-out station that has a single operator with an average service time of 1.5 minutes. If patients arrive according to a Poisson process at the clinic at the rate of 30 per hour: Nurse station Check out Sign-in Exam room a. Determine the average number of customers in the check-out station. (8 points) b. Assume that it takes exactly 1.5 minutes that a nurse (in the nurse station) handles a patient. What mean time between overall arrivals (arrivals at the clinic) can be accepted if the mean length of the queue in the nurse station is not to exceed 3 patients? (12 points) c. Extra Credit: Suppose that the overall arrival rate is expected to increase to 60 patients. If the service rates do not change, how many more doctors will be needed at the exam room just to keep up with the patient load? (5 points) Hint: Regardless of the service rate at each station, over the long run, the departure rate out of each station is equal to the rate into it.