How many doctors should be on duty during each period to maintain the level of patient care expected?

Richard Mange, the administrator at an Onienta Hospital emergency room, faces the problem of providing treatment for patients who arrive at different rates during the day. There are four doctors available to treat patients when needed. If not needed, they can be assigned other responsibilities (such as doing lab tests. reports, X-ray diagnoses) or else rescheduled to work at other hours.
It is important to provide quick mid responsive treatment, and Richard feels that, on the average, patients should not have to sit in the waiting area for more than 5 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 as follows:

Time

Arrival Rate

9 AM-3 P.M

6 patients/hour

3 P.M-8 P.M

4 patients/hour

8 P.M.-midnight

12 patients /hour

Arrivals follow a Poisson distribution, and treatment ‘lines, 12 minutes on the average, follow the exponential pattern.
a) How many doctors should be on duty during each period to maintain the level of patient care expected?
b) What condition would exist if only one doctor were on duty between 9 A.M. and 3 P.M.

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