## 2 System Model
For a given detection threshold, an inherent tradeoff [19] exists in the frame structure between the sensing time and transmission time over the considered channel. It is assumed that the packet transmissions of SU and PU are synchronized at each slot. We assume that the PU is authorized to use a fixed spectrum, which is divided into a set of narrowband channels \(j\in[1,N]\). We assume that PU traffic across all the primary channels is independent and identically distributed. Therefore, the utility performance in a case that the handoff channel is randomly selected is the same as that in a case that the handoff channel is sequentially selected. Hence, activity of a PU over the channels is sensed in a sequential order, starting from the first channel. If the current channel is idle, a packet is transmitted to the corresponding receiver. If the channel is busy, the SU switches to the next channel with the aid of spectrum handoff, and senses the PU activity in the next channel. The activity of PU on a channel is modeled by discrete-time semi-Markov (two-state) process [36; 37; 38; 39; 40], as shown in Figure 3. The states alternate between the _OFF_ and _ON_ states. The _ON_ state represents that channel is occupied by the PU, and the _OFF_ state represents that channel is not occupied by the PU. Without the loss of generality, we assume that the length of _OFF_ and _ON_ durations is represented by the random variables \(A\) and \(B\), respectively, and follows exponential distribution with means \(R\) and \(S\), respectively. At any time, the probabilities that the PU is in _ON_, and _OFF_ states are given by \(P_{ON}=S/(S+R)\), and \(P_{OFF}=R/(S+R)\), respectively. Furthermore, it is assumed that the states of the considered channel are independent of those of the other channels. Figure 2: Pattern of \(N\)-channel usage by SU. To model the imperfections of the channel sensing by incorporating sensing errors [33; 34; 35; 36], the states of the considered channel are represented by missed-detections and false-alarm probabilities, and are calculated by the activity of PU and estimated sensing outcomes. A detection event occurs when a SU successfully senses an occupied channel, and a false alarm indicates a busy status for an idle channel (i.e., the transmission opportunity is wasted). As shown in our previous work [19], a higher detection probability results in better protection for the PU, and a higher false alarm probability means less chance for the SU to utilize the channel. The utilization of the spectrum can be improved with the selection of the optimal sensing operating point depending on the sensing parameters (e.g., SNR) and sensing duration [33]. In the considered ED scheme, the detected signal is squared and integrated over the observation interval, and then the output is compared to a fixed threshold to make a local decision. In this study, the sum utilization of spectrum, i.e., joint utilization of spectrum by the SU and PU, is considered. Thus, the sensing performance under the QoS constraints for PU or SU can be formulated [37]. We consider the sensing performance subject to a QoS constraint for the PU, i.e., the probability of detection is set at a desired value (to give a desired level of protection to the PU), and the probability of false alarm is obtained at a given SNR or sensing duration. The receiver operating characteristic with given assumptions is expressed as  
\[{P_{FA}}^{j}=Q\bigg{(}\sqrt{2\gamma^{j}+1}Q^{-1}(P_{D}^{j})+\gamma^{j}\sqrt{N_ {s}^{j}}\bigg{)} \tag{1}\]  
where \(j\in[1,N]\) is the channel index, \(P_{D}^{j}\) (detection probability) and \(P_{FA}^{j}\) (false alarm probability) are the sensing pair on the \(j\)th channel, \(\gamma^{j}\) is the SNR of the PU signal at the SU on the \(j\)th channel, \(N_{s}^{j}\) is the number of sensing samples of the SU on the \(j\)th channel, and \(Q(\cdot)\) and \(Q^{-1}(\cdot)\) denote the complementary distribution function of the standard Gaussian and its inverse, respectively.