If the shop has no snakes in stock at the end of a day, the owner contacts his snake supplier to order four more snakes. The snakes are delivered the following morning before the shop opens. The snake supplier makes a charge of C for the delivery. (i) Write down the transition matrix for the number of snakes in stock when the shop opens in a morning, given the number in stock when the shop opened the previous day. [2] (ii) Calculate the stationary distribution for the number of snakes in stock when the shop opens, using your transition matrix in part (i). [4] (iii) Calculate the expected long term average number of restocking orders placed by the shop owner per trading day. [2] If a customer arrives intending to purchase a snake, and there is none in stock, the sale is lost to a rival pet shop. (iv) Calculate the expected long term number of sales lost per trading day. [2] The owner is unhappy about losing these sales as there is a profit on each sale of P. He therefore considers changing his restocking approach to place an order before he has run out of snakes. The charge for the delivery remains at C irrespective of how many snakes are delivered. (v) Evaluate the expected number of restocking orders, and number of lost sales per trading day, if the owner decides to restock if there are fewer than two snakes remaining in stock at the end of the day. [5] (vi) Explain why restocking when two or more snakes remain in stock cannot optimise the shop's profits. The pet shop owner wishes to maximise the profit he makes on snakes. [2] (vii) Derive a condition in terms of C and P under which the owner should change from only restocking where there are no snakes in stock, to restocking when there are fewer than two snakes in stock. [2] [Total 19]