## 2 Network Model  
We considered a massive CRN as a network with multiple primary and secondary transmitter-receiver pairs within the Euclidean space as shown in Figure 1, where primary transmitters (PTs) and secondary transmitters (STs) are distributed following two independent homogeneous PPPs (HPPPs) and \(\Phi_{s}\) of intensities \(\lambda_{p}\) and \(\lambda_{s}\) respectively. The term massive or large-scale CRN is often used when multiple PUs and SUs are transmitting within the same channel. In such a network, intra-network and inter-network interference can influence the performance of the network and must be captured in the network modeling. A typical primary connection is made up of a PT \(x_{k}^{p}\in\Phi_{p}\) and a primary receiver (PR) \(y_{k}^{p}\), while any typical secondary connection is made up of an ST \(x_{k}^{s}\in\Phi_{s}\) and a secondary receiver (SR) \(y_{k}^{s}\). In order to meet the channel access demands in secondary networks, STs are permitted to transmit to their respective receivers at any period of time following the hybrid spectrum access technique. Since PUs' deployments are rather planned in real-life deployments, we assumed that each PT is at least a distance \(D\) from its nearest PT to ensure minimum repulsion between PU transmitters that are normally observed in real-life deployments. Although  
Figure 1: Users’ distribution in massive CRN. PTs' distribution may not actually follow PPP in practical systems, existing efforts [2; 24] showed that the PPP can provide an accurate approximation for the distribution of PTs. Each PT has a dedicated PR, whose location is at a distance \(r_{p}\) in a random orientation. Similarly, each ST has a dedicated SR, whose location is at a distance \(r_{s}\) in a random orientation. Based on the displacement theorem [25], the locations of PRs and SRs are also known to follow independent HPPPs of intensities \(\lambda_{p}\) and \(\lambda_{s}\) respectively. It was further assumed that each transmitter is connected to its nearest receiver [26]. We considered a discrete, time-slotted system, in which STs are required to perform channel sensing at the beginning of each time slot in order to detect the state of PTs within such channels, before generating packets for their respective receivers. We assumed that packet generation occurs right before transmission and that the transmission of one packet occupies a one-time slot. Any ST transmits in the underlay mode with an appropriate power control technique with transmit power \(P_{s}=P_{s}^{LI}\) if the sensing outcome reveals the presence of at least one active PT on the channel and transmits in the overlay mode with a higher rate (i.e., ST full transmit power, \(P_{s}=P_{s}^{O}\)) if otherwise. At the detection of the presence or absence of primary signal, ST switches between underlay and overlay modes in order to maximize its throughput, while ensuring that PTs' QoS constraint is satisfied. Since SUs are expected to be energy-efficient devices, continuous channel sensing can degrade the efficiency of each SU. Hence, channel sensing may be carried out by some randomly dedicated SUs at each time slot. The outcome of the channel sensing can then be made available to all SUs via a dedicated error-free link. STs' transmissions are allowed in the underlay mode at any time slot as long as the interference received at any PR within such location is below the pre-defined interference temperature. Hence, the transmit power of any typical ST in the underlay mode (i.e., \(P_{s}^{LI}\)) is subject to power control technique and can be obtained as [1]  
\[P_{s}^{U}= \begin{cases}P_{s}^{\max},&\text{if }h_{sp}||sp||^{-\alpha}\leq\frac{I _{s}}{\frac{I_{s}}{\rho_{s}^{\max}}}\\ \frac{I_{s}}{h_{sp}||sp||^{-\alpha}},&\text{if }h_{sp}||sp||^{-\alpha}>\frac{I _{s}}{P_{s}^{\max}},\end{cases}