Anoffice supply store keeps a small inventory of printer cartridges. Theink in them can dry out,so they have an expiration date, which in turn means that the store manager does not want to keep too many on hand. On the other hand, the manager does not want to be out of stock when customer wants to buy a cartridge. Assume that customer demand occurs one at a time (that is, customer never want to buy more than one at a time) according to poisson process with a mean rate of two per week. Assume that store manager can order replacement stock one carton at a time, where a carton caontian six cattridges. After palcing an order, the store has to wait for for a period of time(order processing plus delievery) that is negatively exponentially distributed with a mena of a week. Such an order willbe placed by themanageras soon as thestockfall to onecartridge.

a> Set up a ergodic Markov process model to track the level of inevtory. The state should corresponds to the number of cartridges in stock. Each customer demands decreases the stock by one unit. The repalcement event will increase it by six.

b>What is the probability that a customer wanting one of these cartridges finds the store out of stock ?