$(20$ points$)$ Consider a discrete time periodic inventory model and let ${}_n$ be the total demand in period $n$ and let $x_n$ be the inventory quantity on hand at the end of period $n.$ Assume that $(s,S)$ inventory policy is used. That is$,$ if the end of period stock is not greater than $s,$ then a quantity is immediately procured to bring the level up to $S.$ If the end of period stock exceeds $s,$ then no replenishment takes place. Suppose that $s=2$ and $S=4$ and ${}_1,{}_2,$dots,${}_n$ are independent random variables where

$P({}_i=0)=0\mathrm{.}1,P({}_i=1)=0\mathrm{.}4$ and $P({}_i=2)=0\mathrm{.}5$ for $i=1,2,3,$dots,$n$

Then $x_0,x_1,$dots,$x_n$ is a Markov Chain.

First define states of the Markov chain then determine the transition probability matrix of the chain.