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We analyze the MAC access delay of the IEEE 802.11e EDCA mechanism under saturation. We develop a detailed analytical model to evaluate the influence of all EDCA differentiation parameters, namely AIFS, CWmin, CWmax and TXOP limit, as well as the backoff multiplier beta. Explicit expressions for the mean, standard deviation and generating function of the access delay distribution are derived. By applying numerical inversion on the generating function, we are able to efficiently compute values of the distribution. Comparison with simulation confirms the accuracy of our analytical model over a wide range of operating conditions. We derive simple asymptotics and approximations for the mean and standard deviation of the access delay, which reveal the salient model parameters for performance under different differentiation mechanisms. We also use the model to numerically study the differentiation performance and find that beta differentiation, though rejected during the standardization process, is an effective differentiation mechanism that has some advantages over the other mechanisms.