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 four 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 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:

ARRIVAL RATE

TIME (PATIENTS/HOUR)

9 A. M. - 3 P. M .......6

3 P. M. - 8 P. M. .......4

8 P.M. – midnight ....... 12

These arrivals follow a Poisson distribution, and treatment times, 12 minutes on the average, follow the exponential pattern.

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