Optimal Control in a Netflix-like Closed Rental System

We analyze the problem of product allocation to customers following the introduction of a new product in a closed rental system, such as Netflix. We consider two types of customers who di?er in their rental time distributions. All customers desire the newly introduced product, and if a customers request for this product is denied, she receives a substitute product and requests for the new product upon return. We study the control problem of minimizing the mean delay encountered by customers before obtaining the new product in a large market setting. We show that asymptotically this problem is equivalent to solving a linear program that depends on the entire rental duration distribution as opposed to mean alone. The optimal policy turns out to be a mixed priority rule that prioritizes the slower customer class while maintaining a base allocation to the faster customer class to ensure quick return.