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We herein analyze the capacity of voice over IP (VoIP) and propose a new method for finding the minimum detection and false-alarm probabilities to ensure the quality-of-service (QoS) requirement of VoIP users in a cognitive radio system. We propose a model for the system supporting the VoIP service as a 2-D discrete-time Markov chain (DTMC). The VoIP traffic and wireless channels in the cognitive radio system are described as a Markov-modulated Poisson process (MMPP) model and a Markov channel model, respectively. In addition, we introduce a simple spectrum-sensing model based on energy detection. By means of the DTMC approach, we demonstrate various analytical and simulation results under the constraint of imperfect spectrum sensing, such as the packet dropping probability, average throughput, and VoIP capacity.