An office supply store keeps a small inventory of printer cartridges. the ink 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 a customer wants to buy a cartridge. assume that customer demands occur one at a time (that is, customers never want to buy more than one at a time) according to a poisson process with a mean rate of two per week. assume that the store manager can order replacement stock one carton at a time, where a carton contains six cartridges. after placing an order for replacement, the store has to wait for a period of time (order processing plus delivery) that is negative exponentially distributed with a mean of one week. such an order will be placed by the manager as soon as the stock fall to one cartridge.

a. set up an ergodic markov process model to track the level of inventory. the states should correspond to the number of cartridges in stock. each customer demands decreases the stock by one unit. the replacement event will incrase the stock by six.

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