(a) P(N=1) = λte^(-μt)
(b) P(N=2) = λ^2t^2/4e^(-μt)
(c) P(N=j) = λ^j/j!t^j e^(-μt)
(d) P(First to arrive is first to depart) = e^(-λt)
(e) E(First departure time) = 1/μ
(a) For P(N=1), we need the arrival of one customer and no departures before that. The probability of the arrival of one customer in a small time interval is λdt, while the probability of no departures is exp(-μdt). Therefore, the probability of one arrival and no departures is λdt exp(-μdt). Integrating over the interval (0,t), we get P(N=1) = λte^(-μt).
(b) Similarly, for P(N=2), we need the arrival of two customers and no departures before the second arrival. The probability of two arrivals in a small time interval is (λ^2/2)dt^2, while the probability of no departures is exp(-μdt). Therefore, the probability of two arrivals and no departures is (λ^2/2)dt^2 exp(-μdt). Integrating over the interval (0,t), we get P(N=2) = (λ^2t^2/4)e^(-μt).
(c) In general, for P(N=j), we need the arrival of j customers and no departures before the jth arrival. Using the same approach as before, we get P(N=j) = (λ^j/j!)t^j e^(-μt).
(d) The first to arrive is the first to depart if there are no arrivals after the first customer. The probability of no arrivals after the first customer is exp(-λt). Therefore, the probability that the first to arrive is the first to depart is exp(-λt).
(e) The expected time of the first departure is the average time taken for the first customer to complete service. The service time is an exponential random variable with rate μ, so the expected service time is 1/μ. Therefore, the expected time of the first departure is 1/μ.
Question:
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.