Ted$$ Glickman, the administrator at D$.$C$.$ General Hospital emergency room faces the problem of providing treatment for

patients who arrive at different rates during the day. There are $4$ 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 and responsive$$ treatment, and Ted feels$$ that, on the$$ average, patients should not have to sit in the waiting area for more than $6$ minutes before being seen by a doctor. Patients are treated on a$$ 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