The access network has remained the bottleneck in efforts to deliver bandwidth-intensive new-generation applications and services to subscribers. In the wired access network, the gigabit Ethernet passive optical network (GEPON) is a promising technology for relieving this bottleneck, while its counterpart in the wireless access network is worldwide interoperability for microwave access (WiMAX). A converged quadruplet-service-enabled (video, voice, data, and mobility) network, which takes full advantage of the strengths and weaknesses of each of these promising technologies, has been proposed. Besides, research and Internet measurements have revealed that actual Ethernet and wireless data traffic are self-similar and long-range dependent. Therefore, we review the quality of service (QoS) architecture for integrating WiMAX and GEPON access networks that we proposed in previous work. Then, we present an analysis of the queuing behavior of the QoS architecture under self-similar and long-range-dependent data traffic conditions and derive closed-form expressions of the expected waiting time in queue (queuing delay) and the packet loss rate per QoS traffic class. This work brings novelty in terms of presenting performance analysis of the proposed QoS-aware integrated architecture under realistic load conditions and facilitates the provisioning of tightly bound QoS parameters to end users of the converged access network.