Continuation: Unit-Supply Inventory System. A dual of the model in the preceding exercise is an inventory system, where in each period, the inventory is replenished by one unit and it decreases by Un...

Continuation: Unit-Supply Inventory System. A dual of the model in the preceding exercise is an inventory system, where in each period, the inventory is replenished by one unit and it decreases by Un (if possible). Then the inventory at the beginning of period n+1 is
_Xn+1 = (_Xn−U_n+1)+. Assume Un are i.i.d. with distribution p_i
= P{U₁
= i}, i ≥ 0. Justify that
_Xn
is a Markov chain and specify its matrix of transition probabilities.