M/M/1 queues Consider the infinite length M/M/1 queue discussed in Example 8.7.

(a) Given that the probability that there are n customers in the queue is P(n)¼(1
)n, where ¼l / , show that the average number of customers in the queue is N ¼ EðnÞ ¼

X1 n¼0 nPðnÞ ¼



1


:

(b) Plot N as a function of  when 0   <1. What happens when   1?

(c) Find the average delay that customers experience and the average waiting time that customers spend in the queue. (Hint: use Little’s theorem as in Problem 7.12.)