Question

Chicago Hospital manages their emergency room (ER) registration through a check-in process. When it’s a patient’s turn, a doctor will see the patient first, then assign the patient either with a prescription (end of the process) or with admission to the hospital. Currently, there are 13 patients (per hour) in average arrive at the ER. Among them 4 patients will be admitted to the hospital. The hospital finds out that, in average, there are 1.2 people waiting to be registered, and 3.2 registered patients are waiting to see a doctor. The registration process takes 4 minutes per patient in average. Those patients who get prescription take an average of 8 minutes seeing a doctor. Those who are admitted to the hospital spend 30 minutes in average with a doctor. Assume the process is stable and the average inflow rate equals the average outflow rate. (1) On average, how long does a patient spend in the ER (from the time entering to the time leaving)? (2) On average, how many patients are being examined by doctors? (3) On average, how many patients are there in the ER?

Please write legibly!

Answer 1

A flowchart has been added for better understanding. Process has been mapped first using icons for operations (rectangles), buffers or waiting lines (triangles).

Assume a stable system. This implies that average inflow equals average outflow at every stage. In this case you are given inventory numbers I and flow rate R= 13 patients/hr. There are two flow rate:
  
- Potential admits: flow rate = 4/hr.

- Simple prescription: flow rate = 9/hr.

To find the average flow times, we use Little's law at each activity for which the flow time is not known:

- Buffer 1: R = 13/hr (both flow units move through there), I = 1.2, so that waiting time in buffer 1 = T = I/R = 1.2/13 hr = 0.092 hrs = 5.52 min.

- Registration: flow time T = 4 min = 4/60 hr. All flow units move through this stage. Thus flow rate through this stage is R = 13 / hr. Average inventory at registration is given by I = RT = 13*2/60 = 0.43 patients.

- Buffer 2: R = 13/hr (both flow units move through there), I = 3.2, so that waiting time in buffer 2 = T = I/R = 3.2/13 hr = 0.246 hrs = 14.76 min.

- Doctor time: depends on the flow unit:

- potential admits: T = 30 min

- prescription ones: T = 8 min

1.
Now we have everything to find the total average flow times: find the critical path for each flow unit. In this case, each flow unit only has one path,
so that is the critical path. We find its flow time by adding the activity times on the path:

- For a potential admit, average flow time (buffer 1 + registration + buffer 2 + doctor) = 5.52 + 4 + 14.76 + 30 = 54.28 min

- For a person with a prescription, average flow time (buffer 1 + registration + buffer 2 + doctor) = 5.52 + 4 + 14.76 + 8 = 32.28 min.

On average, how long does a patient spend in the emergency room :

We know the flow time of each flow unit. The average flow time over all flow units is the weighted average: 4 patients spend 54.28 min while 13 spend 32.28 min. Thus, the grand average is:

T = ((4 * 54.28) + (13 * 32.28))/13 = 48.98 min.

2.
On average, how many patients are being examined by a doctor :

The average inventory at the doctor's activity is the result. First calculate inventory of each type of flow unit:

(a) Potential admits: R = 4 patients/hr, T = 30 min = 0.5 hr, thus, I = RT = 4/hr*0.5 hr = 2 patients

(b) Simple prescription: R = 9 patients/hr, T = 8 min = (8/60) hr, thus I = RT = 9*(8/60) = 1.19 patients

Thus, total inventory at the doctor is 2 + 1.19 = 3.19 patients.

3.
On average, how many patients are in the ER :

Total inventory in ER = inventory in buffer 1 + inventory in registration + inventory in buffer 2 + inventory with doctors = 1.2 + 0.43 + 3.2 + 3.19 = 8.02 patients.