Unit-Demand Inventory System. Consider an inventory or input-output system in discrete time, where Xn denotes the quantity of items in the system at the beginning of the nth period. At the beginning...

Unit-Demand Inventory System. Consider an inventory or input-output system in discrete time, where
_Xn
denotes the quantity of items in the system at the beginning of the nth period. At the beginning of each period, the inventory decreases by one unit provided the inventory level is positive, and otherwise the inventory remains at 0 until the end of the period. At the end of the nth period, the inventory is replenished by an amount Vn, where Vn are i.i.d. with distribution pi = P{V₁
= i}, i ≥ 0. Under these assumptions
_Xn+1 = (_Xn
−1+V_n) if
_Xn
> 0 and
_Xn
= Vn if
_Xn
= 0. Justify that
_Xn
is a Markov chain and specify its matrix of transition probabilities. Is this Markov chain a special case of another one in this chapter?