## 3 Analysis of Success Probability
### Success Probability in Hybrid Cognitive Networks
#### 3.1.1 Analysis in the Primary Network  
Considering a typical (i.e., any arbitrarily selected) primary transmitter-receiver pair in the hybrid CRN, the success probability can be obtained as  
\[\mu_{p}^{h}=(1-p_{m})[\varepsilon(1-P_{o,p}^{U})+(1-\varepsilon)(1-P_{o,p}^{O} )]+p_{m}(1-P_{o,p}^{OI}), \tag{4}\]  
where \(p_{m}\) is the probability of misdetection of PT signal by ST and \(\varepsilon\) is the ST switching rate between underlay and overlay modes. The switching rate \(\varepsilon=0\) indicates a fully or conventional overlay mode, \(\varepsilon=1\) indicates a fully underlay mode, while \(0<\varepsilon<1\) indicates the hybrid model. Similarly, \(P_{o,p}^{U}\) is the PU outage probability as in the conventional underlay mode, \(P_{o,p}^{O}\) is the PU outage probability as in the conventional overlay mode and \(P_{o,p}^{O}\) is the PU outage probability when there is a misdetection of PU signal. Generally, the success probability at a receiving node is equivalent to \(\mu=1-P_{outage}\), where \(P_{outage}\) is the outage probability at such a receiving node. The outage probability at any typical PR \(y_{k}^{p}\) is obtained from (2) as [2]:  
\[P_{outage}=P(SINR_{y_{k}^{p}}\leq\theta_{p})=1-\exp(-s\sigma^{2})\mathcal{L}_{ I_{tp}}(s)\mathcal{L}_{I_{tp}}(s), \tag{5}\]  
where \(\mathcal{L}_{I}\) is the Laplace transform of aggregate interference \(I\) taken at \(s=\frac{\theta_{p}r_{k}^{p}}{P_{p}}\). The parameter \(\theta_{p}=2^{\tau-1}\), where \(\tau\) is the data rate for primary transmission. From (4), it is clear that we must first obtain the analysis for the conventional underlay and overlay transmissions before the analysis for the hybrid transmission can be obtained. 1. Underlay model: In the presence of at least one PT, STs coexist with active PTs as in the underlay mode, provided that its location is not within the exclusion region \(D\) of any active PT. Hence the PU outage probability at any typical PR \(y_{k}^{p}\) is obtained from (5) as \[P_{o,p}^{U}(y_{k}^{p})=1-\exp(-s\sigma^{2})\mathcal{L}_{I_{tp}}(s)\mathcal{L}_ {I_{tp}}(s).\]In a simplified CRN, only a single PT \(x_{k}^{p}\) transmits on any channel, thus \(\mathcal{L}_{I_{pp}}=1\). However, in a massive or large scale CRN considered in this paper, multiple PTs can co-exist within the same channel, hence \(I_{pp}=\sum_{x_{i}^{p}\in\Phi_{p}\setminus x_{k}^{I}}P_{p}h_{x_{i}^{p}y_{k}^{p} \mid x_{i}^{p}y_{k}^{p}\mid\mid x_{i}^{p}y_{k}^{p}\mid\mid-\alpha}\)[2]. \(\mathcal{L}_{I_{pp}}\) is expressed as  
\[\mathcal{L}_{I_{pp}}(s)=E\Big{\{}\exp\Big{(}-s\sum_{x_{i}^{p}\in\Phi_{p} \setminus x_{k}^{I}}P_{p}h_{x_{i}^{p}y_{k}^{p}}\mid\mid x_{i}^{p}y_{k}^{p} \mid\mid-\alpha}\Big{)}\Big{\}}, \tag{6}\]  
where \(||ab||\) is the large-scale path loss attenuation between the transmitter \(a\) and receiver \(b\). Under Rayleigh fading assumption, \(h_{ab}\) is assumed to be independent and identically distributed exponential random variables with unit mean, i.e., \(h_{ab}\sim\exp(1)\). By applying the probability generating function of PPP to (6),  
\[\mathcal{L}_{I_{pp}}(s)=\mathcal{L}(\lambda_{p}^{I},P_{p},s)=\exp\Big{\{}- \lambda_{p}^{I}v_{p}\frac{\pi^{2}\delta}{sin(\pi\delta)}P_{p}^{\delta}s^{ \delta}\Big{\}}, \tag{7}\]  
where \(\delta=\frac{2}{\alpha}\) and \(\lambda_{p}^{I}=\lambda_{p}P_{a}\). The expression of \(P_{a}\) can be obtained through the void probability technique, \(P_{a}=\exp(-\lambda_{p}\pi D^{2})\), which captures the repulsion between active PTs. The resulting point process can as well be approximated as a PPP \(\Phi_{p}^{I}\) of intensity \(\lambda_{p}^{I}\). The derivation of (7) is straightforward from [29]. Similarly,  
\[\mathcal{L}_{I_{pp}}(s)=\mathcal{L}(\lambda_{s}^{I},P_{s}^{UI},s),\]  
where \(\lambda_{s}^{I}=\lambda_{s}P_{b}\), knowing that an ST will only transmit when its location is outside the region of radius \(D\) of any active PTs. Hence, \(P_{b}=\exp(-\lambda_{p}^{I}\pi D^{2})\). The resulting point process in this case is better represented as hole processes.