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In this paper, we study the performance of multi-cell OFDMA WiMAX systems, in both downlink and uplink. We calculate analytically the number of collisions when the number of users in each cell is known. We then calculate the QoS indicators (e.g., blocking rates, download time and bit error rates) taking into account the physical layer conditions (modulation, propagation and MIMO), the MAC layer techniques (HARQ and radio resource management algorithms) and the traffic characteristics, in a cross-layer approach. We finally evaluate the impact of using adaptive modulation and coding on the overall performance of the system. This analysis allows us to calculate the Erlang capacity of a WiMAX system. Our numerical applications then show how to choose the best admission control and modulation schemes that extend the Erlang capacity region.