$.$ Patients arrive at a medical facility which caters to people with motor disabilities by wheelchair friendly vans. The arrivals occur according to a Poisson process at the rate of $10$ per hour. These vans wait at the medical facility till the patient sees the doctor and is ready to leave. Each van has room for only one patient. The patients are seen by doctors in the order they arrive. The time spent by a doctor with a patient has an exponential distribution with a mean of $10$ minutes. The waiting cost of the vans is $$100$ per hour. The medical facility pays doctors at the rate of $$20$ per $60$ minutes.

a$.$ What is the least number of doctors required to have a stable system?

b$.$ The facility is trying to decide whether to employ $2$ or $3$ doctors. If cost minimization is their objective, how many doctors should they employ?

c$.$ If the medical facility starts using their own vans and there is no waiting cost due to vans waiting for patients and the only waiting cost is the cost of patients waiting which is $$80$ per hour, should $2$ or $3$ doctors be employed if cost minimization is their objective?