Hub location problems (HLP) have multiple applications in logistic systems, the airways industry, supply chain network design, and telecommunication. In the HLP, the selected nodes as hubs perform the principal role in processing the inflow arising from other nodes. So, congestion would be inevitable at hub nodes. This paper considers a p-hub median problem with multiple hub node servers delivering service at variable rates. Since the service rates are limited and variable, a queue is formed at each hub server. To tackle this problem, we developed a mixed-integer linear programming model that optimizes the selected hub nodes to reduce congestion under an allowable defined queue length at each server and minimize the total costs of the model, including transportation and hub establishment costs. We utilized the Civil Aeronautics Board (CAB) dataset containing 25 USA cities, which is a valuable source for designing numerical examples in the HLP, to prove the model's efficiency. The results obtained from the designed sample problems show that strategic decisions on defining the number of hubs and maximum acceptable queue length at each hub server will significantly impact the hub location network design.