On average, 5 patients per hour (Poisson) come to the walk-in health clinic on campus. The average time taken to serve a patient is 27 minutes (exponential). The clinic has 2 doctors available to see patients. When its waiting room is full (15 places), the clinic refers new patients to the emergency department of the nearby hospital.

a) Calculate :
Doctor utilization, the average number of patients in the queue, the average time a patient spends in the queue, the probability that a patient who presents will be diverted to emergency, and the probability that there are more than 5 patients in the queue when a patient presents.

b) We want to reduce the average time spent in the waiting room to under 40 minutes. We have 3 options:
1- reduce the average service time to 20 minutes per patient by hiring a nurse practitioner ($100/hour)
2- reduce the size of the waiting room. 10 per seat taken.
3- add a doctor ($200/hour)
In addition, the clinic incurs an opportunity loss of $200 for each patient diverted to emergency.

Repeat the calculations in a) and calculate the total cost for each option (scenarios 1, 2, 3). Based on the costs, which option do you recommend?