A novel converged optical network unit (ONU)-base station (BS) architecture has been contemplated for next-generation optical-wireless networks. It has been demonstrated through high quality studies that data traffic carried by both wired and wireless networks exhibit self-similar and long range dependent characteristics; attributes that classical teletraffic theory based on simplistic Poisson models fail to capture. Therefore, in order to apprehend the proposed converged architecture and to reinforce the provisioning of tightly bound quality of service (QoS) parameters to end-users, we substantiate the analysis of the QoS behavior of the ONU-BS under self-similar and long range dependent traffic conditions using custom queuing which is a common queuing discipline. This paper extends our previous work on priority queuing and brings novelty in terms of presenting performance analysis of the converged ONU-BS under realistic traffic load conditions. Further, the presented analysis can be used as a network planning and optimization tool to select the most robust and appropriate queuing discipline for the ONU-BS relevant to the QoS requirements of different applications.