Problem 9 (10 pts) In many of the queueing models, we assume that the size of the queue is in_nite. In reality however, this is seldom the case. We can imagine that we can place a limit on the number of customers in the system. As such, we can think of the M/M/1/N queue, where N represents the maximum total number of customers in the system.

a) De_ne the states as the number of customers in the system. Draw a transition diagram clearly identifying the transition rates between states.

b) It turns out that in this situation, the probability of being in states P0 and Pn is given below: