## 2 System Model  
### Primary Network and Cognitive Radio ad hoc Network Model  
We consider an underlay SN-CRN coexisting with a cellular primary network involving \(N_{P}\) PUs and \(N_{S}\) SUs in a torus area \(\Omega_{S}=\left[0,\sqrt{N_{S}/\rho}\right]^{2}\) (\(\rho\) is the spatial density of SUs) sharing the spectrum within the same frequency band simultaneously, as depicted in Figure 1. Particularly, PUs have the full privilege of accessing their allocated frequency band whereas SUs can opportunistically utilize idle channels unoccupied by the PUs. In the cellular primary network, PUs send their data traffic to the primary base station (PBS) via the licensed uplink channels constituting a channel set \(\mathcal{C}=\{1,2,\cdots,N_{c}\}\). We employ the independent and identically distributed alternating ON-OFF process to model the occupation time length of PUs in uplink channels. Specifically, the OFF state indicates the idle state where the unoccupied uplink channels or the SOPs can be freely occupied by SUs. In CRANET, SUs denoted by a set \(\mathcal{N}_{S}=\{1,2,\cdots,N_{S}\}\) can only leverage the OFF state to access the SOPs over the idle authorized uplink channels. Due to the randomness of data traffic and the dynamic behavior of PUs, we suppose that the SOPs are available for usage by SU \(i\) with a probability of \(\delta_{i}\), for \(i\in\mathcal{N}_{S}\). With the aid of the ON-OFF process to characterize the status of the uplink channels, the occupancy probability of the \(k\)-th uplink channel by PUs can be given by \(\alpha_{k}/(\alpha_{k}+\beta_{k})\), where \(\alpha_{k}\) is probability that the \(k\)-th uplink channel transits from OFF to ON state, and \(\beta_{k}\) is probability that the \(k\)-th uplink channel transits from ON to OFF state, for \(k\in\mathcal{C}\). We also assume that the occupancy probability of uplink channels by PUs can be acquired by SUs through a priori knowledge of the local spectrum sensing. By bearing in mind the mutually independent occupancy probability of the \(k\)-th uplink channel, the SOP usage probability \(\delta_{i}\) of SU \(i\) during time interval \([t_{0},T]\) is formulated as follows:  
\[\delta_{i}=\prod_{k=1}^{N_{c}}\biggl{(}1-\frac{\alpha_{k}}{\alpha_{k}+\beta_{ k}}\biggr{)},\quad i=1,\cdots,N_{s} \tag{1}\]  
Figure 1: Illustration of the rectangular grid based SN-CRN. SU: secondary user; PU: primary user; PBS: primary base station. Let \(\theta_{i}(t)\) and \(B_{i}\) denote the amount of data traffic in the buffer of SU \(i\) at time \(t\in[t_{0},T]\) and the buffer size of SU \(i\), respectively. The buffer of SU \(i\) generally consists of two buffer segments that can hold the offered data load including data traffic injected from upstream SUs and the data sequence of SSR transmitted from SSNs, respectively. Given a time interval \(\Delta t\), the first buffer segment of SU \(i\) called as the forward buffer holds the amount of data traffic injected from upstream SUs denoted by \(\theta_{i}^{F}(\Delta t)\). Meanwhile, the second buffer segment of SU \(i\) known as the internal buffer is used to store the amount of the data sequence of SSR transmitted from SSNs denoted by \(\theta_{i}^{I}(\Delta t)\). Thus, the amount of aggregate data traffic from upstream SUs and SSNs in the buffer of SU \(i\) with a time interval \(\Delta t\) can be formulated as \(\varphi_{i}(\Delta t)=\theta_{i}^{F}(\Delta t)+\theta_{i}^{I}(\Delta t)\). Then, for a given time interval \(\Delta t\), the amount of data traffic \(\theta_{i}(t)\) in buffer of SU \(i\) at time \(t\) evolves as follows:  
\[\theta_{i}(t+\Delta t)=\min\{\theta_{i}(t)+\varphi_{i}(\Delta t),B_{i}\}-( \Lambda_{i}(\Delta t)+\chi_{i}(\Delta t)) \tag{2}\]  
where \(\Lambda_{i}(\Delta t)\) and \(\chi_{i}(\Delta t)\) stand for the amount of data traffic successfully delivered by SU \(i\) and the amount of the data sequence of SSR removed by SU \(i\) within a time interval \(\Delta t\), respectively.