Question

A hospital emergency room (ER) is currently organized so that all patients register through an initial check-in process. At his or her turn, each patient is seen by a doctor and then exits the process, either with a prescription or with admission to the hospital. Currently, 50 people per hour arrive at the ER, 10% of whom are admitted to the hospital. On average, 30 people are waiting to be registered and 40 are registered and waiting to see a doctor. The registration process takes, on average, 2 minutes per patient. Among patients who receive prescriptions, average time spent with a doctor is 10 minutes. Among those admitted to the hospital, average time is 33 minutes.

On average, how long does a patient stay in the ER? On average, how many patients are being examined by doctors? On average, how many patients are in the ER?

. A triage system has been proposed for the ER described in the above. As mentioned, 50 patients per hour arrive at the ER. Under the proposed triage plan, patients who are entering will be registered as before. They will then be quickly examined by a nurse practitioner who will classify them as Simple Prescriptions or Potential Admits. While Simple Prescriptions will move on to an area staffed for regular care, Potential Admits will be taken to the emergency area. Planners anticipate that initial examination by the triage nurse will take 3 minutes. They expect that, on average, 20 patients will be waiting to register and 5 will be waiting to be seen by the triage nurse. Recall that registration takes an average of 2 minutes per patient. Planner expect the Simple Prescriptions area to have, on average, 15 patients waiting to be seen. As before, once a patient’s turn comes, each will take 10 minutes of a doctor’s time. The hospital anticipates that, on average, the emergency area will have only 2 patient waiting to be seen. As before, once a patient’s turn comes, he or she will take 33 minutes of a doctor’s time. Assume that, as before, 90% of all patients are Simple Prescriptions. Assume, too, that the triage nurse is 100% accurate in her classifications.

1. Under the proposed plan, how long, on average, will a patient stay in the ER?
2. On average, how long will a Potential Admit stay in the ER? On average, how many patients will be in the ER?

. Refer again to Problem above. Once the triage system is put in place, it performs quite close to expectations. All data conform to planners’ expectations except for one set – the classifications made by the nurse practitioner. Assume that the triage nurse has been sending 93% of all patients to the Simple Prescriptions area when in fact only 90% should have been so classified. The remaining 3% are discovered when transferred to the emergency area by a doctor. Assume all other information from Problem 4 is valid. On average, how long does a patient stay in the ER? On average, how long does a Potential Admit stay in the ER? On average, how many patients are in the ER?

Answer 1

Answer – Process flow diagram of the current scenario will be: -

Buffer 3 Potential (waiting) Admits R 50/hour T 2 mins 0.825/min - I 30 I 40 Buffer 1 Triage Nurse Registration Buffer 2 (waiting) (waiting) Simple Prescriptions Buffer 4 (waiting)

Using the formula, I = R * T

	R	I	T
Buffer 1	50	20	24
Registration	50	1.666	2
Buffer 2	50	5	6
Triage Nurse	50	2.499	3
Buffer 3	5	1	12
Potential Admission	5	2.499	30
Buffer 4	45	15	20
Simple Prescription	45	3.75	5
		51.414

Answer a – on average a patient will stay in the ER =

                                T_PA = 24+2+6+3+12+30 = 77 minutes

                                T_SP = 24+2+6+3+20+5 = 60 minutes

                                T_ER = 77(0.9) + 60(0.1) = 75.3 minutes

Answer b – On average, a potential admit to stay in the ER is equal to

                                T_PA = 26+2+6+3+12+30 = 77 minutes

                   The number of patients in the ER = I_ER

                                                I_ER = 52