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Dynamic spectrum access (DSA) strategy selection and the associated resource allocation are critically important issues for cognitive networks, because they need to not only satisfy the interference constraint caused to the primary users (PU), but also meet the delay quality-of-service (QoS) requirements for the secondary users (SU). In this paper, we develop the optimal resource allocation schemes for the underlay and overlay DSA strategies, respectively, in delay-QoS constrained cognitive networks. Specifically, for the underlay strategy, we find that 1) when the maximum average interference power is less than the maximum average transmit power, the cognitive network will gradually converge to an interference-power constrained system as the QoS constraint becomes more stringent; 2) when the maximum average interference power is larger than the maximum average transmit power, the cognitive network reduces to a transmit-power constrained system not varying with the QoS requirement. For the overlay strategy, we observe that 1) a unique optimal sensing time exists under the given QoS constraint; 2) the optimal sensing time increases as the QoS constraint gets more stringent. Following these results, we further propose a selection criterion across underlay and overlay DSA strategies. By applying this criterion, the SU can determine whether to use underlay or overlay for DSA under the given QoS constraint and the PUs' spectrum-occupancy probability.