Suppose that customers arrive to a system according to a Poisson process with rate λ. There are an in?nite number of servers in this system so a customer begins service upon arrival. The service times of the arrivals are independent exponential random variables with rate μ, and are independent of the arrival process. Customers depart the system when their service ends. Let N be the number of arrivals before the ?rst departure.
(a) Find P(N = 1).
(b) Find P(N = 2).
(c) Find P(N = j).
(d) Find the probability that the ?rst to arrive is the ?rst to depart.
(e) Find the expected time of the ?rst departure.