Consider the M/M/1 queue with impatient customers model as presented in Example 8.9. Give your answers in terms of the limiting probabilities Pn,n ≥ 0.

(a) What is the average amount of time that a customer spends in the system.

(b) If en denotes the probability that a customer who finds n others in the system upon arrival will be served, find en,n ≥ 0.

(c) Find the conditional probability that a served customer found n in the system upon arrival. That is, find P(arrival finds n| arrival is served).

(d) Find the average amount of time spent in queue by a customer that is served.

(e) Find the average amount of time spent in queue by a customer that departs before entering service.