We investigate the effect of bursty traffic in an LTE and Wi-Fi aggregation (LWA)-enabled network, where part of the LTE traffic is offloaded to Wi-Fi access points (APs) to boost the performance of LTE networks. A Wi-Fi AP maintains two queues containing data intended for the LWA-mode user and the native Wi-Fi user, and it is allowed to serve them simultaneously by using superposition coding (SC). With respect to the existing works on LWA, the novelty of our study consists of a random access protocol allowing the Wi-Fi AP to serve the native Wi-Fi user with probabilities that depend on the queue size of the LWA-mode data. We analyze the throughput of the native Wi-Fi network, accounting for different transmitting probabilities of the queues, the traffic flow splitting between LTE and Wi-Fi, and the operating mode of the LWA user with both LTE and Wi-Fi interfaces. Our results provide fundamental insights in the throughput behavior of such aggregated systems, which are essential for further investigation in larger topologies.