A car company sells 50 different basic models (additional options are added at the dealership after purchases are made). Customers are of two basic types: (1) those who are willing to order the configuration they desire from the factory and wait several weeks for delivery and (2) those who want the car quickly and therefore buy off the lot. The traditional mode of handling customers of the second type is for the dealerships to hold stock of models they think will sell. A newer strategy is to hold stock in regional distribution centers, which can ship cars to dealerships within 24 hours. Under this strategy, dealerships hold only show inventory and a sufficient variety of stock to facilitate test drives. Consider a region in which total demand for each of the 50 models is Poisson with a rate of 1,000 cars per month. Replenishment lead time from the factory (to either a dealership or the regional distribution center) is 1 month. (a) First consider the case in which inventory is held at the dealerships. Assume that there are 200 dealerships in the region, each of which experiences demand of 1,000/200 = 5 cars of each of the 50 model types per month (and demand is still Poisson). The dealerships monitor their inventory levels in continuous time and order replenishments in lots of one (i.e., they make use of a base stock model). How many vehicles must each dealership stock to guarantee a fill rate of 99 percent? (b) Now suppose that all inventory is held at the regional distribution center, which also uses a base stock model to set inventory levels. How much inventory is required to guarantee a 99 percent fill rate?