HI I need help with this one question for my homework! I hope you have better luck than I did

Question 3 (15 marks) Customers arrive at a counter and form a single line to be served. During each time interval (t,t + 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 (Xt,t Z 1) represent the length of the queue. Then (Xt)t21 is a queuing chain; by construction, Xt + t+1 1, X1, > 0, Et+la Xt : 07 where (gm 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. (0) [5] Suppose there exists 7r 2 (7r(i),i 2 0) that satises 7rP = 7r. Show that the probability generating function of 71', denoted by PAS), can be written as \"(OlPASXS - 1) 5 - 135(5) ' where P(s) is the probability generating function of 6 ~ Poisson()\\). 13,,(5) = (0.1) (d) [4] Hence show that, when A