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This paper presents an architecture and algorithms for optimizing the performance of Web services. For a given service, session-based admission control is combined with stage-wise request queuing, where the stages represent sub-tasks within sessions. The scheduling of requests is governed by generalized processor sharing. We present a performance model, relying on online estimation of parameters describing client-server interaction. A reward function corresponding to the service provider's objective is maximized using techniques for nonlinear optimization. In a case study, we model and optimize the resource sharing at a Web server hosting an electronic store. The performance advantages of our approach are quantified numerically, and the robustness to parameter estimation errors is assessed by sensitivity analysis.