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We present a physical cognitive radio system implementation under the GNU Radio platform with the aim of evaluating a reinforcement learning spectrum management scheme. In our experiments we examine the packet transmission success rate of the cognitive user for a variety of channel utilisation parameters. We derive analytical expressions using Markov chain analysis for the learning convergence time and secondary user packet transmission success rate in the general case of large-scale networks. Our results show that the reinforcement learning scheme significantly improves system performance.