Consider a medical clinic that can be modeled as a G/G/1 queue, where the waiting room acts as a buffer and the doctor acts as the server. It takes an average of 8 minutes for the doctor to process a patient (e.g., measure vitals, perform tests, etc), with a standard deviation of 4 minutes. Due to Covid restrictions, the clinic is trying to limit the number of patients in the waiting room. They are determining if they should schedule patients or hire a second doctor.

Question

Assume that, on average, 7 patients arrive every hour, and the standard deviation of inter-arrival times is 10 minutes. Assume that a second doctor arrives to form a resource pool with the first doctor.

On average, how many patients are in the waiting room?