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We study the resource-allocation problem in a multiuser orthogonal frequency-division multiplexing (OFDM)-based cognitive radio (CR) system using a cross-layer approach. The goal is to provide satisfactory quality of service (QoS) to both real-time and non-real-time applications, despite the rapid variations in available resources caused by the activities of the primary users. The dynamic nature of the available spectrum gives rise to two resource allocation issues: 1) problem feasibility and 2) false urgency. To solve the problem-feasibility issue, which arises when resources are insufficient to meet all user QoS requirements, we adopt a goal-programming approach. The false-urgency issue that was caused by variations in available system resources is effectively avoided by a proposed rate-requirement calculation mechanism based on the status of the packets in queue and system resource availability. Simulation results show that the proposed cross-layer resource-allocation algorithm for CR systems performs better than existing algorithms that were designed for multiuser OFDM systems.