## 4 Performance Analysis for Different ERSR Radius Sets
### User Relay Selection Ratio Analysis
#### 4.1.3 Total Number of UR in \(A_{n}\)  
When USU is at \(\gamma_{u}\), the number of URs is given by  
\[M_{UR}(\gamma_{u})=M_{u}(\gamma_{u})\lambda_{s}^{A_{n}}(\gamma_{u}). \tag{16}\]  
The amount of UR in \(A_{n}\) totals to  
\[M_{UR}^{A_{n}}{}^{\prime}=\int_{\frac{DI_{m}^{A_{n}}}{2}}^{\frac{DI_{m}^{A_{n} }}{2}}M_{UR}(\gamma_{u})d\gamma_{u}. \tag{17}\]  
Substituting (15) and (16) into (17), the expression of the amount of UR is (18). \[M_{UR}^{A_{n}}{}^{\prime}= \int_{\frac{DI_{m}^{A_{n}}}{2}}^{\frac{DI_{m}^{A_{n}}}{2}}M_{UR}( \gamma_{u})d\gamma_{u} \tag{18}\] \[= \int_{\frac{DI_{m}^{A_{n}}}{2}}^{\frac{DI_{m}^{A_{n}}}{2}}\left(M_ {1}+M_{2}\right)\left[1-\exp\left(-\frac{\gamma_{u}{}^{\beta}}{\rho_{0}} \delta_{0}\right)\right]\frac{2\gamma_{u}}{R^{2}}\frac{\left(M_{1}+M_{2} \right)r_{n}}{R^{2}}\exp\left(-\frac{\gamma_{u}{}^{\beta}}{\rho_{0}}\delta_{0} \right)d\gamma_{u}\] \[= \frac{2r_{n}{}^{2}(M_{1}+M_{2})^{2}}{\beta R^{4}}\Big{[}\frac{ \rho_{0}}{\delta_{0}}\Big{]}^{\frac{2}{\beta}}\left\{\left[\Gamma\left(\frac{2 }{\beta},\frac{\left(DI_{ex}^{A_{n}}\right)^{\beta}\delta_{0}}{2^{\beta}\rho_ {0}}\right)-\Gamma\left(\frac{2}{\beta},\frac{\left(DI_{ex}^{A_{n}}\right)^{ \beta}\delta_{0}}{2^{\beta-1}\rho_{0}}\right)\right]\right.\] \[-\left.\left.\left[\Gamma\left(\frac{2}{\beta},\frac{\left(DI_{ in}^{A_{n}}\right)^{\beta}\delta_{0}}{2^{\beta}\rho_{0}}\right)-\Gamma\left(\frac{2}{ \beta},\frac{\left(DI_{in}^{A_{n}}\right)^{\beta}\delta_{0}}{2^{\beta-1}\rho_{ 0}}\right)\right]\right\}.\]  
However, in the annular area, when the density of USU is higher than that of SU, the phenomenon that SU is repeatedly selected as UR exists. The number of repeatedly selected UR should be expurgated from \(M_{UR}^{A_{n}}{}^{\prime}\). We define average repetition rate (ARR) to indicate the average times that SU is repeatedly selected, which is the ratio of SUs \(M_{s}^{A_{n}}\) to USUs \(M_{u}^{A_{n}}\). It is denoted by  
\[R^{A_{n}}=\frac{M_{s}^{A_{n}}}{M_{u}^{A_{n}}}, \tag{19}\]where \(M_{u}^{A_{n}}\) is  
\[ \begin{split} M_{u}^{A_{n}}&=(M_{1}+M_{2})\underbrace{ \int_{\frac{DI_{uc}^{A_{n}}}{2}}^{\frac{DI_{uc}^{A_{n}}}{2}}f(\gamma_{u})d\gamma _{u}}_{P^{A_{n}}}-M_{s}^{A_{n}}\\ &=(M_{1}+M_{2})\ \frac{\left(DI_{cv}^{A_{n}}\right)^{2}-\left(DI_{ in}^{A_{n}}\right)^{2}}{4R^{2}}-M_{s}^{A_{n}}.\end{split}  \tag{20}\]  
\(P^{A_{n}}\) is the probability that an MG user locates in \(A_{n}\). \(M_{s}^{A_{n}}\) is given by  
\[ \begin{split} M_{s}^{A_{n}}&=(M_{1}+M_{2}) \underbrace{\int_{\frac{DI_{uc}^{A_{n}}}{2}}^{\frac{DI_{uc}^{A_{n}}}{2}}P_{s}( \gamma_{u})f(\gamma_{u})d\gamma_{u}}_{P_{s}^{A_{n}}}\\ &=(M_{1}+M_{2})\int_{\frac{DI_{uc}^{A_{n}}}{2}}^{\frac{DI_{uc}^{A _{n}}}{2}}\exp\left(-\frac{\gamma_{u}^{\beta}}{\rho_{0}}\delta_{0}\right)\frac {2\gamma_{u}}{R^{2}}d\gamma_{u}\\ &=\frac{2\left(M_{1}+M_{2}\right)}{\beta R^{2}}\left[\frac{\rho_{ 0}}{\delta_{0}}\right]^{\frac{2}{\beta}}\left[\Gamma\left(\frac{2}{\beta}, \frac{\left(DI_{uc}^{A_{n}}\right)^{\beta}\delta_{0}}{2^{\beta}\rho_{0}}\right) -\Gamma\left(\frac{2}{\beta},\frac{\left(DI_{in}^{A_{n}}\right)^{\beta}\delta _{0}}{2^{\beta-1}\rho_{0}}\right)\right].\end{split}  \tag{21}\]  
After simple algebraic operations, \(R^{A_{n}}\) is given by (22). \[R^{A_{n}}=\frac{\left[\frac{\rho_{0}}{\delta_{0}}\right]^{\frac{2}{\beta}} \left[\Gamma\left(\frac{2}{\beta},\frac{\left(DI_{cv}^{A_{n}}\right)^{\beta} \delta_{0}}{2^{\beta}\rho_{0}}\right)-\Gamma\left(\frac{2}{\beta},\frac{\left( DI_{in}^{A_{n}}\right)^{\beta}\delta_{0}}{2^{\beta-1}\rho_{0}}\right)\right]}{ \frac{\beta}{8}\left[\left(DI_{cv}^{A_{n}}\right)^{2}-\left(DI_{in}^{A_{n}} \right)^{2}\right]-\left[\frac{\rho_{0}}{\delta_{0}}\right]^{\frac{2}{\beta}} \left[\Gamma\left(\frac{2}{\beta},\frac{\left(DI_{cs}^{A_{n}}\right)^{\beta} \delta_{0}}{2^{\beta}\rho_{0}}\right)-\Gamma\left(\frac{2}{\beta},\frac{\left( DI_{in}^{A_{n}}\right)^{\beta}\delta_{0}}{2^{\beta-1}\rho_{0}}\right)\right]}. \tag{22}\]  
Based on above analysis, after expurgating repeated calculations, the number of URs in \(A_{n}\) is  
\[M_{UR}^{A_{n}}=\left\{