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

Answer:

a.   

9 a.m.-3 p.m.; λ = 6 patients/hour; μ = 5 patients/hour/doctor

Want W_q to be ≤5 minutes = 0.0833 hour. W_q ≤ 0.0833 implies that

or L_q £ 0.0833l or L_q £ 0.50

Thus m = 3 channels or doctors are needed (with m = 2, L_q = 0.6748; with m = 3, L_q = 0.0904).

b.   

3 p.m.-8 p.m.; λ = 4 patients/hour; μ = 5 patients/hour/doctor

W_q ≤ 0.0833 hour implies that or ≤ 0.0833 or L_q ≤ 0.0833l or

L_q ≤ 0.03333. This means μ = 2 doctors.

c.      

8 p.m.-midnight; λ = 12 patients/hour; μ = 5 patients/hour/doctor

Want W_q ≤ 0.0833 hour or ≤ 0.0833 or L_q ≤ 0.0833l or L_q ≤

1.00. m= 4 doctors are needed.