5.3. A Batch Processing Model. Customers arrive at a facility and wait there until K customers have accumulated. Upon the arrival of the Kth customer, all are instantaneously served, and the process repeats. Let eo, e . . . denote the arrivals in successive periods, assumed to be independent random variables whose distribution is given by Pr{6,=0}=a, 1} = 1 -a, where 0 < a < 1. Let X,, denote the number of customers in the system at time n. Then {X_n} is a Markov chain on the states 0, 1, . . . , K - 1. With K = 3, give the transition probability matrix for {X_n} Be explicit about any assumptions you make.