## 3 Performance Analysis
### OP of the Primary Network  
Combining (8), (9), and (19), the OP at the first time slot can be formulated as  
\[\text{OP}_{\text{Pl}} =\text{Pr}\Big{(}\max_{p=1,2,\ldots,N_{\text{T}}}\max_{q=1,2, \ldots,N_{\text{R}}}\Big{(}\psi_{\text{PT}_{p},\text{PR}_{q}}\Big{)}<\theta_{ \text{PH}}\Big{)}\] \[=\bigg{[}\text{Pr}\Big{(}\psi_{\text{PT}_{p},\text{PR}_{q}}<\theta _{\text{PH}}\Big{)}\Big{]}^{N_{\text{T}}N_{\text{R}}}\] \[=\left[\underbrace{\text{Pr}\Big{(}g_{\text{PT}_{p}\text{PR}_{q} }<\rho_{1}g_{\text{SS}_{1}\text{PR}_{q}}+\rho_{2}g_{\text{SS}_{2}\text{PR}_{q}} +\rho_{0}\Big{)}}_{\mathcal{I}_{1}}\right]^{N_{\text{T}}N_{\text{R}}}, \tag{22}\]  
where  
\[\rho_{0}=\frac{\theta_{\text{PH}}}{P_{\text{PT}}},\ \rho_{1}=\frac{R_{ \text{SS}_{1}}\theta_{\text{PH}}}{P_{\text{PT}}},\ \rho_{2}=\frac{R_{\text{SS}_{2}}\theta_{\text{PH}}}{P_{\text{PT}}}.\]  
As marked in (22), the probability \(\mathcal{I}_{1}\) can be expressed in the following form:  
\[\mathcal{I}_{1}=\int_{0}^{+\infty}\int_{0}^{+\infty}F_{g_{\text{PT}_{p}\text{ PR}_{q}}}(\rho_{1}x+\rho_{2}y+\rho_{0})f_{g_{\text{SS}_{1}\text{PR}_{q}}}(x)f_{g_{ \text{SS}_{2}\text{PR}_{q}}}(y)dxdy. \tag{23}\]  
Substituting CDF \(F_{g_{\text{PT}_{p}\text{PR}_{q}}}(\rho_{1}x+\rho_{2}y+\rho_{0})\) and PDFs \(f_{g_{\text{SS}_{1}\text{PR}_{q}}}(x)\) and \(f_{g_{\text{SS}_{2}\text{PR}_{q}}}(y)\) into (23); after some careful calculation, we obtain an exact closed-form expression of \(\mathcal{I}_{1}\). Then, substituting this result into (22) yields  
\[\text{OP}_{\text{Pl}}=\bigg{[}1-\frac{\Omega_{\text{SS}_{1}\text{PR}}}{ \Omega_{\text{SS}_{1}\text{PR}}+\Omega_{\text{PT}\text{PR}}\rho_{1}}\frac{ \Omega_{\text{SS}_{2}\text{PR}}}{\Omega_{\text{SS}_{2}\text{PR}}+\Omega_{ \text{PT}\text{PR}}\rho_{2}}\exp(-\Omega_{\text{PT}\text{PR}}\rho_{0})\bigg{]}^ {N_{\text{T}}N_{\text{R}}}. \tag{24}\]  
Similarly, combining (16), (17), and (19), the OP of the primary network at the second time slot can be exactly computed as  
\[\text{OP}_{\text{P2}} =\text{Pr}\bigg{(}\max_{p=1,2,\ldots,N_{\text{T}}}\max_{q=1,2, \ldots,N_{\text{R}}}\Big{(}\varphi_{\text{PT}_{p},\text{PR}_{q}}\Big{)}<\theta_ {\text{PH}}\bigg{)}\] \[=\bigg{[}\text{Pr}\Big{(}\varphi_{\text{PT}_{p},\text{PR}_{q}}< \theta_{\text{PH}}\Big{)}\bigg{]}^{N_{\text{T}}N_{\text{R}}}\] \[=\bigg{[}\text{Pr}\Big{(}g_{\text{PT}_{p}\text{PR}_{q}}<\rho_{3}g _{\text{SR}\text{PR}_{q}}+\rho_{0}\Big{)}\bigg{]}^{N_{\text{T}}N_{\text{R}}}\] \[=\bigg{[}1-\frac{\Omega_{\text{SSPR}}}{\Omega_{\text{SRPR}}+ \Omega_{\text{PT}\text{PR}}\rho_{3}}\exp(-\Omega_{\text{PT}\text{PR}}\rho_{0} )\bigg{]}^{N_{\text{T}}N_{\text{R}}}, \tag{25}\]  
where  
\[\rho_{3}=\frac{P_{\text{SR}_{b}}\theta_{\text{PH}}}{P_{\text{PT}}}.\]  
**Remark 5**.: _We now consider a special case where the secondary users are not currently using the licensed band. Because there is no co-channel interference from the secondary network, the TAS/SC algorithms in (9) and (17) can be re-written, under the following form:_  
\[(k,l):g_{\text{PT}_{k}\text{PR}_{l}}=\max_{p=1,2,\ldots,N_{\text{T}}}\max_{q= 1,2,\ldots,N_{\text{R}}}\Big{(}g_{\text{PT}_{p}\text{PR}_{q}}\Big{)}. \tag{26}\]From (26), it is straightforward to calculate the OP of the primary network as  
\[\mathrm{OP_{P0}} =\Pr\bigl{(}P_{\mathrm{PT}}g_{\mathrm{PT}_{\mathrm{LPPR}_{i}}}<\theta _{\mathrm{Ph}}\bigr{)}\] \[=\Pr\biggl{(}\max_{p=1,\ldots,N_{\mathrm{T}}}\max_{q=1,2,\ldots,N_ {\mathrm{R}}}\Bigl{(}g_{\mathrm{PT}_{p}\mathrm{PR}_{q}}\Bigr{)}<\rho_{0}\biggr{)}\] \[=[1-\exp(-\Omega_{\mathrm{TPPR}}\rho_{0})]^{N_{\mathrm{T}}N_{ \mathrm{R}}}. \tag{27}\]