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Sensing/monitoring of spectrum-availability has been identified as a key requirement for dynamic spectrum allocation in cognitive radio networks (CRNs). An important issue associated with MAC-layer sensing in CRNs is how often to sense the availability of licensed channels and in which order to sense those channels. To resolve this issue, we address (1) how to maximize the discovery of spectrum opportunities by sensing-period adaptation and (2) how to minimize the delay in finding an available channel. Specifically, we develop a sensing-period optimization mechanism and an optimal channel-sequencing algorithm, as well as an environment- adaptive channel-usage pattern estimation method. Our simulation results demonstrate the efficacy of the proposed schemes and its significant performance improvement over nonoptimal schemes. The sensing-period optimization discovers more than 98 percent of the analytical maximum of discoverable spectrum-opportunities, regardless of the number of channels sensed. For the scenarios tested, the proposed scheme is shown to discover up to 22 percent more opportunities than nonoptimal schemes, which may become even greater with a proper choice of initial sensing periods. The idle-channel discovery delay with the optimal channel-sequencing technique ranges from 0.08 to 0.35 seconds under the tested scenarios, which is much faster than nonoptimal schemes. Moreover, our estimation method is shown to track time-varying channel-parameters accurately.