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Traffic congestion in communication networks can dramatically deteriorate user-perceived Quality-of-Service (QoS). The integration of the Random Early Detection (RED) and priority scheduling mechanisms is a promising scheme for congestion control and provisioning of differentiated QoS required by multimedia applications. Although analytical modelling of RED congestion control has received significant research efforts, the performance models reported in the current literature were primarily restricted to the RED algorithm only without consideration of traffic scheduling scheme for QoS differentiation. Moreover, for analytical tractability, these models were developed under the simplified assumption that the traffic follows Short-Range-Dependent (SRD) arrival processes (e.g., Poisson or Markov processes), which are unable to capture the self-similar nature (i.e., scale-invariant burstiness) of multimedia traffic in modern communication networks. To fill these gaps, this paper presents a new analytical model of RED congestion control for prioritized multi-class self-similar traffic. The closed-form expressions for the loss probability of individual traffic classes are derived. The effectiveness and accuracy of the model are validated through extensive comparison between analytical and simulation results. To illustrate its application, the model is adopted as a cost-effective tool to investigate the optimal threshold configuration and minimize the required buffer space with congestion control.