## 4 Numerical Results and Discussion  
In this section, numerical results are provided to evaluate the performance of sum utilization of spectrum under different network topologies, PU activities, sensing parameters, and sets of channels. In our simulation setup, we consider a CRN co-existing with a PRN that is licensed to use a set of \(N=5\) frequency bands. The communication range of PU and SU is varied to allow different values of probabilities and sensing parameters. The licensed bands are occupied by a PU according to the PU model as shown in Figure 3. To obtain the numerical results examined in this section, the key parameters are chosen as follows: \(P_{s}^{1}=1000\) bits, \(R_{p}=100\) kbps, \(R=20\) ms to \(100\) ms, \(t_{s}=1\) ms to \(10\) ms, and \(\Omega=0.0001\) m\({}^{-2}\). The sensing SNR at the SU is considered to be \(-15\) dB. The SNR for the secondary transmissions over each channel is set to \(30\) dB, and hence, the \(BER\) value is set to \(0.00025\). Unless otherwise stated, we consider the fair model of the PU for the channel status, i.e., \(P_{\text{ON}}=P_{\text{OFF}}=0.5\). The targeted detection probability \(P_{\text{D}}\) is set to \(90\%\), to restrict the interference probability to the PU to \(10\%\) or less. The values of these parameters are considered accordingly to validate the characteristics of the considered set of channels and network behavior. In the simulation runs, the maximum of five channels, each occupied by a single PU, are considered. The observation period commences from the first narrowband accessible channel to the last one. Figures 5 and 6 illustrate the probability of sum utilization of spectrum with different sensing durations and sets of channels, under the topology-1 model and topology-2 model, respectively. \"Multi-channel\" refers to a scenario in which different numbers of narrowband channels are considered, and the sum utilization probability over different numbers of narrowband channels is summed. The number of transmitted bits in a packet in the multi-channel scenario is adjusted as per the available transmission duration over that channel, and the weighting factor is added to consider the utilization as per the transmitted number of bits. Figures 5 and 6 validate the inherent tradeoff in the utilization of spectrum of the SU for single-channel scenario over different sensing durations. With the targeted detection probability, the spectrum utilization performance for the PU is considered to be the same. The tradeoff exists for any value of PU states mean durations considering the fair model of the PU for its channel status, i.e., the same PU states probabilities. The tradeoff exists between the two probabilities, i.e., the probability of \"no activity during transmission\" and the probability of \"detecting the spectrum opportunity\". In detail, the increase in probability of \"detecting the  
spectrum opportunity\" over the shorter sensing durations is dominant over the decrease in probability of \"no activity during transmission.\" Similarly, the decrease in probability of \"no activity during transmission\" over the longer sensing durations is dominant over the increase in probability of \"detecting the spectrum opportunity\". Hence, our results validate that an optimal sensing duration exists for single-channel scenario. However, the optimal sensing duration depends on the handoff probability and considered sensing parameters. It can be seen that the performance of the sum utilization of spectrum improves with the number of channels. The reason is that the multiple-channel scenario considers the transmission of a packet in the next channel for each PU interruption, and hence, enhances the reliability of the transmitted packet. However, the performance improvement is not significant when a large number of channels is considered. The reason is that in such a case, the handoff probability is not significant.