A converged ONU-BS architecture has been proposed for next-generation optical-wireless networks. Further, it has been convincingly demonstrated through high quality studies that both wired and wireless data traffic are self-similar and long range dependent. Therefore, to help the understanding of the converged architecture and to facilitate the provisioning of tightly bound QoS parameters to end-users, we present an analysis of the QoS behavior of the converged architecture under self-similar and long range dependent traffic conditions using low latency queuing (LLQ) which is a common queuing discipline that is likely to be used in next-generation networks. This work extends our previous work on priority queuing (PQ) and brings novelty in terms of presenting performance analysis of the converged ONU-BS under realistic traffic load conditions and with a more robust queuing discipline.