The medical director of a large emergency clinic faces a problem of providing treatment for patients who arrive at different rates during the day. There are several doctors available to treat patients when needed. If not needed, they can be assigned to other responsibilities (for example, lab tests, reports, x-ray diagnoses) or else rescheduled to work at other hours. It is important to provide quick and responsive treatment, and the medical director feels that, on the average, patients should not have to sit in the waiting area for more than 20 minutes before being seen by a doctor. Patients are treated on a first-come, first-served basis and see the first available doctor after waiting in the queue. The arrival pattern for a typical day is:

Time Arrival Rate, Time patients/hour

8 a.m. 4 p.m. 10

4 p.m. 12 a.m. 18

12 a.m. 8 a.m. 9

These arrivals follow a Poisson distribution, and treatment times, 24 minutes on the average, follows an exponential distribution. Consider the service cost (by doctors) to be $130 per hour. For each patient, the waiting cost when waiting for service (only count the time waiting in line but not time in service) is estimated to be $65 per hour.

How many doctors should be on duty during each period to maintain the level of patient care expected? Please provide comprehensive analysis and explain your results clearly.