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In this letter, we consider the uplink random access problem in a wireless multimedia network (WMN) with audio, video, and best effort applications. Since these multimedia applications have different quality-of-service (QoS) requirements, we model their utilities with concave, step, and quasi-concave functions. We assume that the access point performs admission control and assigns transmission probabilities to the users for random access, based on solving a non-convex network utility maximization problem. We propose a novel enumeration algorithm to obtain the global optimal solution by solving a number of computationally tractable convex optimization problems. We characterize the total number of iterations of the algorithm analytically. Simulation results show that our proposed algorithm achieves a higher average network aggregate utility than a carrier sense multiple access (CSMA) scheme implemented in a slotted time system.