Question: Customers arrive at a dentist's surgery according to a Poisson process with parameter λ. The dentist examines one customer at a time. Each examination takes a random time y with mean my = 5 min and exponential PDF, independently of the number of customers in the (huge) waiting room and of the arrival process.

a) Find the arrival rate λ for which the mean waiting time of a customer in the waiting room is less than or equal to 20 minutes.

b) For such a λ, determine the mean number of customers in the waiting room.

c) Find the probability that there are no more than two customers in the waiting room, in stationary conditions.

d) Now, suppose that at time t0 = 6:30 pm the ingress door is closed and no other customers are admitted into the dentist's surgery. Suppose that there are still x(t0) = 10 customers in the dentist's surgery. What's the mean time required to the dentist to finish the work?

e) What is the answer to the previous point of the number of customers at time t0 has asymptotic PMD?