Suppose that customers arrive to a system according to a Poisson process with rate . There are

Suppose that customers arrive to a system according to a Poisson process with rate λ. There are an infinite 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 first departure.

(a) Find P(N = 1).

(b) Find P(N = 2).

(c) Find P(N = j).

(d) Find the probability that the first to arrive is the first to depart.

(e) Find the expected time of the first departure.