Flying ad hoc networks (FANETs), which are composed of unmanned aerial vehicles (UAVs), face severe quality-of-service (QoS) challenges because of their high mobility, rapidly changing topologies, and limited infrastructure support. Conventional MAC protocols, such as contention-based CSMA/CA, often fail to guarantee low latency and high throughput under these conditions. To address these challenges, this paper investigates the MAC-layer QoS problem in FANETs and proposes a polling-based MAC protocol integrated with the IEEE 802.11 point coordination function (PCF). The protocol adopts a centralized scheduling mechanism using a master node to coordinate data transmission on the basis of a first-in-first-out (FIFO) policy. While this centralized approach simplifies coordination and reduces contention, it introduces limitations regarding scalability and robustness in decentralized or dynamic environments, which are common in FANET scenarios. To analyse protocol behavior, a polling system model is developed with three service strategies: Gated Service, Exhaustive Service, and Limited-1 Service. Analytical models are derived for queue length and delay, and MATLAB simulations validate the feasibility of the proposed scheme. The results show that despite its structural constraints, the proposed protocol can achieve a balance between throughput, delay, and fairness in specific mission-driven UAV applications, such as coordinated surveillance and real-time task distribution. Further research will explore decentralized adaptations to improve protocol flexibility and fault tolerance in dynamic UAV networks.