In this paper, cognitive transmission under quality of service (QoS) constraints is studied. In the cognitive radio channel model, it is assumed that the secondary transmitter sends the data with two different transmission and power policies, depending on the activity of the primary users and the probability of detection, which is determined by channel sensing performed by the secondary users. A state transition model with three states is constructed to model this cognitive transmission channel. Statistical limitations on the buffer lengths are imposed to take into account the QoS constraints. The transmission and power policies are determined through the constraint to provide a minimum-rate transmission to the primary users for a certain percentage of the time. The maximum throughput under these statistical QoS constraints is identified by finding the effective capacity of the cognitive radio channel. The impact upon the effective capacity of several system parameters, including the channel sensing duration, detection threshold, detection and false alarm probabilities, QoS parameters, and data transmission and power policies are investigated.