Let Xn be the inventory quantity on hand at the end of period n$.$ Suppose

$($q$,$Q$)=(2,2)$ inventory rule is applied, and backordering is not allowed.

a$)$ Determine the transition probability matrix

b$)$ In the long run, during what fraction of periods are orders placed?

c$)$ If setup cost of placing an order is $$10,$ what is the expected setup

cost per period?

d$)$ Suppose for every item carried in the stocks from one period to the

next period costs $$1.$ What is the expected inventory holding cost per

period?