Customers arrive at a counter and form a single line to be served. During each time interval (Lt + 1), the number of new arrivals are independent and follow a Poisson distribution with parameter A. At the beginning of each time interval, one customer is served, unless the queue is empty. Let (Xht Z 1) represent the length of the queue. Then (Xt)t21 is a queuing chain; by construction, X: + t+1 1, X: > 0, t+1a X: = 0, where (63,75 2 1) are independent and identically distributed as g N Poisson()\\) for some A > 0. Xt+1 = (a) [2] Write down the transition matrix P for the queuing chain (Xn). (b) [2] Is the chain (Xt)t21 irreducible? Justify your answer. (c) [5] Suppose there exists 71' = (7T('i),'i Z 0) that satises 'JTP = 71'. Show that the probability generating function of 77, denoted by 1377(3), can be written as p.43) : 7r(0)Pg(s)(s 1) s P(5) ' (0.1) where 135(5) is the probability generating function of g N Poisson(/\\). (d) [4] Hence show that, when A