All patients that arrive must first wait to register before registering. Then, they wait to see the doctor. She determines in T₁ = 5 minutes if the patient only needs a prescription (which is the case for 90% of patients). If the patient needs admission, they will spend T₂ = 30 minutes with the doctor. Assume that 10% of patients who come to the emergency require admission (and, therefore, that 100% of patients that arrive at the emergency department leave only after seeing the doctor).

On average, 55 patients per hour arrive at the emergency and 7 are waiting to register. Registration takes 2 minutes on average. There are also an average of 34 patients that are waiting to see the doctor. The consultation with the doctor takes 5 minutes on average for 90% of patients (those who only require a prescription) and 30 minutes on average for 10% of patients (those who require admission).

Compute

[1 point] the average time a patient spends in the emergency department, and

[1 point] the average number of patients in the emergency department.

[2 points] The hospital is now considering adding a nurse practitioner at Registration. The nurse would take care of patients who only require a prescription. After seeing the nurse, those patients would leave the hospital with their prescription (without seeing the doctor). However, the activity time at Registration would increase from 2 to 10 minutes for all patients. Moreover, the average number of patients that wait to see the doctor would drop from 34 to 3.4. Assume the rest of the process remains the same and recompute the average time a patient spends in the emergency department and the average number of patients.

Hint: Apply Little's Law at each of the four stages of the process to determine the missing values of I or T.

A (simplified) emergency department of a hospital is described below