Consider a single-server discrete-time queueing system that operates in the following manner. Let Xn denote the number of customers in the system at time n ∈ {0, 1, 2,...}. If a customer is receiving service in time n, then the probability that he finishes receiving service before time n + 1 is q, where 0 ≤ q ≤ 1. Let the random variable Yn denote the number of customers that arrive between time n and n + 1, where the PMF of Yn is given by PYn (k) = P[Yn = k] = e−λ λk k!

k = 0, 1,...

a. Give an expression for the relationship between Xn+1, Xn, and Yn.

b. Find the expression for the transition probabilities P[Xn+1 = j|Xn = i].