Model of an inventory system. For a certain type of product there is initially a stock of 4 units to satisfy customer demand directly from the shelf. Customers arrive according to a Poisson process at a rate of 2 customers per hour. Each customer demands one unit. Each time a customer demand occurs, a replenishment order for one unit is immediately placed at a supplier. The lead times of orders (i.e., the times that elapse from the moment an order is placed until the order arrives) are independent, exponentially distributed with a mean of 1 hour. If on arrival of a customer the shelf is empty, the customer waits patiently until the product becomes available. Customers are obtaining their product in order of arrival. a. Calculate the equilibrium probabilities pi that there are i (i = 0, 1, 2, . . .) outstanding orders Determine the mean number of products on stock. c. Determine the probability that an arriving customer finds an empty shelf and thus has to wait for a product. d. Determine the mean number of waiting customers and the mean waiting time.