Joint pricing and inventory control for a production–inventory queueing system

We consider a production–inventory queueing system consisting of a queue and an inventory, where there is a single type of product and a single firm. Customers can observe the number of products in the inventory and the number of waiting customers in the queue upon arrival. Customers decide whether to wait for the product or leave without a purchase according to their utility, which depends on the product’s price and estimated waiting time. If the number of products in the inventory is lower than a certain threshold, the firm produces the products. The product’s production time and the customers’ reward from purchasing the product have general distributions. We investigate the customers’ equilibrium strategies, profit maximization and social welfare maximization. Specifically, we show that a customer’s equilibrium strategy exists for a given joint pricing and inventory control. In general, there can exist multiple equilibria. However, if the production time distribution is decreasing mean residual life, the equilibrium is unique. We also present a method for computing equilibrium strategies. In addition, we compute the maximum profit rate and the profit-maximizing solution. We also compute the maximum social benefit rate and the welfare-maximizing solution. Finally, we present various numerical experiments that include comparisons of the maximum profit rate and the maximum social benefit rate, as well as of the profit-maximizing solution and the welfare-maximizing solution.