In recent years, realizing safe and efficient air traffic management (ATM) has become one of the most important social issues because of the rapid increase in air traffic volume. To realize safe and efficient ATM, it is necessary for related stakeholders to share aeronautical big data by connecting to aircrafts and ground systems efficiently and globally. However, the current aeronautical communication systems are not adapted to be used in support of global collaborative decision making (CDM) through information exchange. In order to accelerate the implementation of information collaborative environment and improve operational awareness in the aeronautical domain, this study focuses on an information platform for sharing aeronautical big data, and a communication medium for achieving Internet of aeronautical things. This paper proposes a new packet-scheduling algorithm for a next-generation aeronautical mobile communication system that includes various types of services and devices. The algorithm balances service assurance and service fairness using strict priorities for real-time services and loose priorities for non-real-time services. Performance evaluations were conducted using a network simulator to confirm the effectiveness of the proposed algorithm. First, a simple single-device scenario was utilized to confirm the effectiveness of the quality of service function for an aeronautical mobile communication system. Second, a multiple-device scenario was employed to evaluate the performance of the proposed scheduler. Finally, an airport scenario was used to demonstrate the effectiveness of our algorithm in a more practical situation. Furthermore, we show that the airport scenario is useful for evaluating the overall performance of a new aeronautical mobile communication system in the system introduction phase or cell design phase.