## 5 Numerical Results and Simulation  
We now provide some details and outcomes of numerical simulations carried out in order to demonstrate the performance of both PUs and SUs under the proposed approach. The delivery rate--the success probability for any test transmitter--is determined as the probability that its intending paired receiver does not experience outage as a result of other users' activities at any observation period. We compared the performance of the hybrid spectrum access scheme with conventional overlay and underlay spectrum access schemes. This is necessary to demonstrate the advantages of the hybrid spectrum access model over the widely studied conventional overlay and underlay spectrum access models. We further carried out the Monte Carlo simulation averaged over 50,000 channel realizations in order to validate the presented analytical approach. Except otherwise stated, the parameters used for simulation are presented in Table 1.  
In order to ensure that the PUs' QoS constraints are satisfied, it is important to guarantee a high packet delivery rate at PRs. PUs service delivery rate can however be degraded by the activities of SUs when channel sensing is not well coordinated. As shown in Figure 3, an increase in switching rate reduces the PU service completion rate (i.e., the delivery rate) owing to more activities of SUs in the underlay mode, while an increase in the probability of misdetection significantly affects the services of active PUs since STs transmit with full transmit power in the presence of PUs. It is hence important to ensure that channel sensing is well-coordinated among SUs to improve PUs detections. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Definition** & **Value** \\ \hline \(\lambda_{s}\) & Distribution of secondary transmitter-receiver pairs & 0.3 \\ \(\lambda_{p}\) & Distribution of primary transmitter-receiver pairs & 0.03 \\ \(r_{p}\) & Distance between any tagged primary transmitter-receiver pair & 0.5 m \\ \(r_{s}\) & Distance between any tagged secondary transmitter-receiver & 0.1 m \\ \(P_{p}\) & PT’s transmission power signal & 0 dB \\ \(p_{s}^{O}\) & ST’s transmission power signal in the overlay mode & \(-\)32 dB \\ \(p_{s}^{U}\) & ST’s transmission power signal in the underlay mode & \(-\)36 dB \\ \(\alpha\) & Path-loss exponent & 4 \\ \(D\) & Coverage region of any PT & 1.2 m \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Simulation parameters. Intuitively, the outage probability (PUs and SUs) should increase with an increase in the SINR threshold \(\theta_{T\!r}\), a parameter that is proportional to the data transmission rate. As expected, an increase in the required data rate of the primary link increases the outage probability in the primary network. This shows that the PU service delivery rate will reduce with an increase in data rate irrespective of the spectrum access mode. As presented in Figure 4, when \(\varepsilon=0.25\) and \(p_{m}=0.2\), the transmission under the overlay spectrum access scheme produces a lower chance of outage than the transmission under the hybrid spectrum access scheme. This is expected since interference in the conventional large-scale overlay scheme is dominated by interference from active PTs only. Transmissions in the conventional overlay scheme, however, means low throughput for secondary transmissions, making the hybrid scheme a more preferred spectrum access model from SUs' perspective. The effect of the required data rate on the success probability in the secondary network is presented in Figure 5. While an increase in the required data rate can reduce the success probability, accessing a channel under the hybrid scheme is shown to improve the overall success probability in secondary networks owing to the fact that, SUs are able to access channel for transmissions at any time. Also, SUs can transmit with full transmit power in the absent of PUs and can switch to the underlay spectrum access scheme when at least one PU is present.