3. Consider a queue where, per time step, it new customers arrive with probability a, for n E No, and, in case of a non-empty queue, the customer at the head of the queue is served and leaves. a) Choose an appropriate measurable space to model the lengths of the queue at all times 1' E No. Dene the corresponding transition kernel. b) Suppose that initially the queue is empty. Derive a recursive formula for the probability of length I; E No of the queue at time i E No. Derive a formula for the probability of lengths (k1,. . . ,k) of the queue at times 1,. . . ,1