(Inventory and Littles Law)
A hospital emergency room is organized so that patients first register with a receptionist before seeing a doctor. Patients wait in line prior to registration, and after registration patients are directed to a waiting room, where they again wait to see a doctor. Once the patient sees a doctor, he/she must be diagnosed, with 10% of all patients receiving a diagnosis that requires hospital admission. The other 90% are discharged with a simple prescription. Diagnosing a patient who is subsequently admitted requires 30 minutes on average, whereas patients who are discharged with a prescription require just 5 minutes for diagnosis on average. The registration process takes an average of 2 minutes per patient, regardless of the eventual diagnosis. The average arrival rate of patients equals 55 patients per hour, which also equals the average flow rate of diagnosed patients (i.e., the process is stable).
Answer the following questions:
(a) Suppose that on average 7 patients are in line waiting to be registered and on average 34 registered patients are in the waiting room waiting to see a doctor. Approximately how long (on average) will a patient who is subsequently admitted to the hospital spend in the emergency room (i.e., beginning with waiting for registration and ending with a completed diagnosis)?
(b) On average, how many patients are being examined by a doctor at any given time?