This article concerns operations of businesses that own inventories of rental items, and can hire additional items from secondary sources whenever they face a temporary exhaustion of their inventories. This set-up is relevant to many operations: the items may be tools, trucks, containers, communication channels, or individuals who provide services such as repairmen. A fundamental problem that emerges in the design of these operations is to determine the optimal size of the inventory of items the business should own. To solve this