This paper introduces a statistical model that allows the users of a random access-based cognitive radio network sharing a single wireless channel to evaluate the quality-of-service (QoS) capability of the network. The model captures the variation of the channel service process taking into account the statistics of the channel availability in addition to the randomness of the data traffic of the cognitive radio network users. The proposed model allows the cognitive radio network users to employ the effective bandwidth theory and its dual, the effective capacity concept, in order to evaluate the maximum packet delay bound that can be satisfied with a certain violation probability. As a result, the users can adapt the performance of their real-time applications to fit the cognitive radio network status. Simulation results validate the model and demonstrate its accuracy.