3.5.3 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 0; 1; : : : denote the arrivals in successive periods, assumed to be independent random variables whose distribution is given by

where 0 Then, fXng is a Markov chain on the states 0;1; : : : ;K ????1. With K D 3, give the transition probability matrix for fXng. Be explicit about any assumptions you make.

Pr{0} =, Pr{k = 1}=1-,