A store stocks an item, for which there is a random demand \(D\) each day. We suppose that demands on successive days are i.i.d. random variables with the discrete uniform distribution on \(\{0,1,2\}\). When the demand exceeds the stock, excess demand is lost. If there are no items left in stock at the end of the previous day, then an order for \(S\) items is placed. The order is filled by morning. Let \(X_{n}\) be the number of items in stock at the beginning of day \(n\).

(a) Write the transition matrix of the chain \(\left(X_{n}\right)\).

(b) Write the stationary equations and verify that the following is a solution:

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xk=xs (1-(-1/2)S-k+1), k = 1, 2, ..., S, where x is the limit as no of P[X = k]. (c) Find xs. (The information in this problem would be needed, for instance, to minimize, with respect to the reorder quantity S, the sum of long-run expected storage cost and cost due to lost sales.)