Being a beneficial service in emergency situations, UAV-aided relay communications have received tremendous attentions in the recent years. In this paper, we consider a three-node UAV-based relay network, in which a mobile UAV establishes communication between two remaining nodes in the system. Despite numerous works available for such a system on the optimization of trajectory and other communication resources, none of these have addressed the delay performance of the system at the granular packet level. Furthermore, it is already established that the equipment of buffer at the relay node provides more flexibility in the delivery of packets through the exploitation of better channel quality. On the other hand, with the continued popularity of multimedia and similar other applications, it is very likely that the source node in the system receives bursty traffic from different external networks. Given that the source and UAV nodes have finite packet-level buffers and the trajectory of the UAV is known, under a bursty traffic model, we aim to study the average end-to-end packet delay and buffer overflow performance of the system. While capturing the predictable channel variation due to the movement of the UAV, we establish a queuing model for the source and UAV nodes based on the stochastic process. Then, from the dynamics of the established queuing model, we derive the average end-to-end packet delay and queue overflow probability of such a system. Through extensive numerical simulation, we justify the accuracy and effectiveness of the proposed analytical model while comparing with three static deployment scenarios of the UAV. Through providing sufficient analytical evidence as well as the numerical results, we exhibit that a mobile relay system outperforms the static relay one in terms of both the delay and buffer overflow metrics.