Research and development of unmanned aerial vehicles (UAVs) have accommodated the challenges in providing satisfactory coverage and rate performance to data-hungry devices and applications, especially in urgent situations and remote environments. Furthermore, a multi-layer UAV network divides the total distance between a user equipment and macrocell base station into multiple hops, which enables data to transmit at a higher bit rate over each hop with a lower total power consumption. However, multi-hopping through a multi-layer UAV network causes delay penalty to packets. In this correspondence paper, we classify data packets in UAVs at each layer of a multi-layer UAV network into incumbent packets and relayed packets, and propose a traffic service scheme for variable classes of packets. Additionally, packet delay for each class of packets at each layer of the multi-layer UAV network is investigated and minimum total packet delay is achieved by optimally allocating spectrum and power resources among layers of the UAV network. Numerical results demonstrate the effectiveness of the proposed algorithm.