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In this paper, we analyze the delay performance of a secondary user (SU) under dynamic spectrum access. We design simple time-threshold policies for the SU to minimize the average delay while satisfying the collision probability constraint of the primary user (PU). Such policies perform closely to an optimized policy found by a Markov Decision Process (MDP) formulation, while facilitating analytical analysis of the delay and collision probability. For general PU busy and idle period distributions, we analyze the performance of threshold policies through a one-dimensional Markov chain, and develop analytical expressions to approximate the delay and collision probability. The accuracy of the Markov chain analysis and the analytical approximations is examined under various busy and idle distributions. We investigate the impact of busy and idle distributions on system performance. We find that while the idle distribution determines the time capacity of SU access, the busy distribution significantly affects the delay performance of the threshold policies. The effect of imperfect sensing is also studied.