Answer the question below showing formulas used and calculations:

At the Royal London Hospitals emergency room, on average 20 patients show up per hour and are processed at the same rate. Arriving patients wait in queue till being called in for registration. Afterwards, they are assigned a bed where they wait till an intern is ready to see them. After consultation, patients are either identified as routine case, prescribed medication and released, or admitted to the hospital for further tests. On average, there are 30 patients waiting for registration and another 20 waiting to be seen by an intern. A recent study determined that 75% of the patients simply require routine treatment.

i. Given the current system, how long do patients spend in the emergency room before being released or admitted?

After some complaints, Royal London Hospital implements a new system that separates the emergency room into two treatment areas, one for routine care and another for emergency cases. A nurse assigns the patient to one of the two tracks after registration. Routine patients are afterwards seen by an intern who either prescribes medication and releases them or moves the patient to the emergency track in case the nurse has made a mistake. Emergency patients are seen by an intern who either admits them for further treatment or determines that they are routine cases, prescribes medication and releases them. It is determined that there are, on average, 20 people waiting to be registered, 5 waiting to be seen by the nurse, 10 waiting to be seen by an intern in the emergency area, and 10 waiting to be seen by an intern in the routine care area.

ii. Assuming that the nurse makes no mistakes, how long do patients that are finally admitted spend in the emergency room?

iii. How long do patients spend on average in the emergency room?

iv. How could the average flow time of the process be further improved?