1. Write a computer program to simulate the queue in Exercise 20. Have your program keep track of the proportion of the time that the queue length is j for j = 0, 1, . . . , n and the average queue length. Show that the behavior of the queue length is very different depending upon whether the traffic intensity s has the property s <>1, s = 1, or s > 1.
2. In the queueing problem of Exercise 20, let S be the total service time required by a customer and T the time between arrivals of the customers.
(a) Show that P (S = j) = (1 - r)j-1r and P (T = j) = (1 - p)j-1p, for j > 0.
(b) Show that E(S) = 1/r and E(T ) = 1/p.
(c) Interpret the conditions s 1, s = 1 and s > 1 in terms of these expected values.