Question

The administrator at the City Hospital’s emergency room faces a problem of providing treatment for patients that 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, etc.) or else rescheduled to work at other hours.

It is important to provide quick and responsive treatment, and the administrator feels that, on the average, patients should not have to sit in the waiting area for more than five 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

9 a.m. – 3 p.m.               6 patients/hour

3 p.m. – 8 p.m.           4 patients/hour

8 p.m. – Midnight           12 patients/hour

__________________________________________

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 in order to maintain the level of patient care expected? (Show all your work.)

Answer 1

9 a.m. - 3 p.m.

L = arrival rate = 6 / hour
M = service rate = 1 in 12 min = 5/ hour

P₀ = Probability of empty system and W_q = waiting time in queue are given by -

Trying these formulae for the number of servers (S) = 2,3,4... we get,

S	Wq (min)	P0
0
1	NA	NA
2	6.7500	0.2500
3	0.9412	0.2941
4	0.1588	0.3002

So, the no. of doctors should be 3 in order to keep Wq below 5 min

3 p.m. - 8 p.m.

L = arrival rate = 4 / hour
M = service rate = 1 in 12 min = 5/ hour

S	Wq (min)	P0
0
1	48.000	0.2000
2	2.2857	0.4286
3	0.2838	0.4472

So, the no. of doctors should be 2 in order to keep Wq below 5 min.

8 p.m. – Midnight

L = arrival rate = 12 / hour
M = service rate = 1 in 12 min = 5/ hour

S	Wq	P0
0
1	NA	NA
2	NA	NA
3	12.9438	0.0562
4	2.1528	0.0831
5	0.5239	0.0889

So, the no. of doctors should be 4 in order to keep Wq below 5 min.