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The IEEE 802.11e task group has reached a stable consensus on two basic contention-based priority mechanisms to promote for standardization: usage of different arbitration interframe spaces and usage of different minimum/maximum contention windows. The goal of this article is to provide a thorough understanding of the principles behind their operation. To this purpose, rather than limit our investigation to high-level (e.g. throughput and delay) performance figures, we take a closer look at their detailed operation, also in terms of low-level performance metrics (e.g., the probability of accessing specific channel slots). Our investigation on one hand confirms that AIFS differentiation provides superior and more robust operation than contention window differentiation. On the other hand, it highlights performance issues related to the coexistence between 802.11e contention-based stations with legacy 802.11 stations, and provides guidelines for the 802.11e parameter settings when such a coexistence is the goal.