2. (40 points) Consider the following inventory model for an expensive item for which demand occurs according to a Poisson process with rate 10/month. The store capacity is equal to 3 so that at most 3 items can be held in stock. Every time inventory is depleted, an order of size 3 is placed to the supplier. Each order is received after a random delivery time which has an arbitrary distribution with mean 1/30 months. There is no backordering and customers that arrive during the lead time leave never to return. Let N, be the number of replenishments made until time t and 2 be the number of items in stock at time t. So, starting with Zo = 3 items in store, the inventory level decreases by making sales to arriving customers one at a time until inventory level becomes Zo = 0. At this time, an order of size 3 is placed to the supplier which is then received in the store after the random delivery time. You may assume that a replenishment has just been made at time 0. (a) (10 points) Find N lim (b) (15 points) Find P{Z = 3) (You do not have to find the renewal function R explicitly and you may suppose that it is given. Give your answer in convolution form.) (c) (15 points) Compute lim++ P{Z1 =3) 2. (40 points) Consider the following inventory model for an expensive item for which demand occurs according to a Poisson process with rate 10/month. The store capacity is equal to 3 so that at most 3 items can be held in stock. Every time inventory is depleted, an order of size 3 is placed to the supplier. Each order is received after a random delivery time which has an arbitrary distribution with mean 1/30 months. There is no backordering and customers that arrive during the lead time leave never to return. Let N, be the number of replenishments made until time t and 2 be the number of items in stock at time t. So, starting with Zo = 3 items in store, the inventory level decreases by making sales to arriving customers one at a time until inventory level becomes Zo = 0. At this time, an order of size 3 is placed to the supplier which is then received in the store after the random delivery time. You may assume that a replenishment has just been made at time 0. (a) (10 points) Find N lim (b) (15 points) Find P{Z = 3) (You do not have to find the renewal function R explicitly and you may suppose that it is given. Give your answer in convolution form.) (c) (15 points) Compute lim++ P{Z1 =3)