CWB Clinic is a local clinic located at Clear Water Bay. The inter$-$arrival time of patients is exponentialwith rate $\backslash$lambda $=30$ patients per hour. Upon arrival, each patient completes the registration with a receptionist,which takes an exponential service time with rate $$R $=40$ patients per hour. After this, $1/333\%$ of patients

are assigned to a physician, and $2/367\%$ of patients are assigned to a nurse clinician. The service time withthe nurse clinician is exponential with rate $$N $=30$ patients per hour, and the service time with the physicianis exponential with rate $$P $=15$ patients per hour. All patients are discharged after their treatment with the

physician, but $10\%$ of the patients from the nurse clinician are directed to thephysician. The patient $$ows aredescribed in the following $$gure: $($a$)$ What is the average waiting time in each of the queues?

Hint: Burke$$s theorem says that when the inter$-$arrival time of M$/$M$/1$ isexponential with rate $\backslash$lambda $,$ theinter$-$departure time is also exponential with rate $\backslash$lambda $.$ This means that the interarrival times for the nurse clinician and the physician are also exponentially distributed.

$($b$)$ What is the overall average time in the system?