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We survey advances in theories and models for Internet quality of service (QoS). We start with the theory of network calculus, which lays the foundation for support of deterministic performance guarantees in networks, and illustrate its applications to integrated services, differentiated services, and streaming media playback delays. We also present mechanisms and architecture for scalable support of guaranteed services in the Internet, based on the concept of a stateless core. Methods for scalable control operations are also discussed. We then turn our attention to statistical performance guarantees and describe several new probabilistic results that can be used for a statistical dimensioning of differentiated services. Lastly, we review proposals and results in supporting performance guarantees in a best effort context. These include models for elastic throughput guarantees based on TCP performance modeling …