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

$66$

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

$9$

A$.$M$.$minus$3$

P$.$M$.$

$88$

$$patients$/$hour

$3$

P$.$M$.$minus$8$

P$.$M$.$

$44$

$$patients$/$hour

$8$

P$.$M$.$minus$$midnight

$1212$

$$patients$/$hour

Arrivals follow a Poisson$$ distribution, and treatment$$ times,

$1212$

minutes on the$$ average, follow the negative exponential pattern.

Part $2$

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

The least number of doctors required during the $9$

A$.$M$.$minus$3$

P$.$M$.$ shift to meet the set waiting time goal is

$$

$$

$2$

$1$

$3$

$4$

$.$

Part $3$

The least number of doctors required during the $3$

P$.$M$.$minus$8$

P$.$M$.$ shift to meet the set waiting time goal is

$$

$$

$4$

$3$

$2$

$1$

$.$

Part $4$

The least number of doctors required during the $8$

P$.$M$.$minus$$Midnight

shift to meet the set waiting time goal is

$$

$$

$2$

$3$

$4$

$1$

$.$

Part $5$

$$b$)$ What condition would exist if only one doctor were on duty between

$8$ P$.$M$\mathrm{.}8$P$.$M$.$

and

midnightmidnight$?$

A$.$

Since

lambdaless thanmu$,$

the line would back up indefinitely.

C$.$

Since

lambdagreater than$>$mu$,$

the average wait time would increase slightly.

D$.$

Since

lambdaless than