24. Recall that an M/M/1 queueing system is a GI/G/c system in which customers arrive according to a Poisson process with rate λ, and service times are exponential with mean 1/μ. For an M/M/1 queueing system, each time that a customer arrives to the system or a customer departs from the system, we say that a transition occurs. Let Xn be the number of customers in the system immediately after the nth transition. Show that {Xn : n = 1, 2, . . .} is a Markov chain, and find its probability transition matrix.

Find the period of each state of the Markov chain.