# Body-to-Body Cooperation in Internet of Medical Things: Toward Energy Efficiency Improvement
## 3 Link and Outage Probability Analysis
The terms \(P\left(R_{3}^{cn1}\right)\) and \(P\left(R_{4}^{cn1}\right)\) in Equation (6) can be further expressed as:  
\[ \begin{split} P\left(R_{3}^{cn1}\right)=P\left[\log_{2}(1+SNR\ X _{1,1})\textless\beta_{1}\,\log_{2}\left(1+\frac{SNR\ X_{2,1}}{1+\ SNR\ X_{1,1}}\right)\textgreater \beta_{2}\ \right],\\ P\left(R_{3}^{cn1}\right)=\int\limits_{X_{1,1-0}}^{U_{1,1}}\exp(-X _{1,1})\ d\ X_{1,1}\int\limits_{X_{2,1}=U_{2,1}+X_{1,1}\left(2^{\beta_{2}-1} \right)}^{\infty}\exp(-X_{2,1})\ dX_{2,1},\\ P\left(R_{3}^{cn1}\right)=\frac{1}{2^{\beta_{2}}}\left[\exp(-U_{2,1 })-\ \exp(-\frac{2^{\beta_{1}+\ \beta_{2}-1}}{SNR})\right].\end{split}  \tag{7}\]  
and:  
\[ \begin{split} P\left(R_{4}^{cn1}\right)=P\left[\log_{2}\left(1+ \frac{SNR\ X_{1,1}}{1+\ SNR\ X_{1,1}}\right)\textless\beta_{1}\,\log_{2}\left(1+\frac{SNR\ X_{2,1}}{1+\ SNR\ X_{1,1}}\right)\right.\\ \left.\beta_{2}\,\log_{2}\left(1+\ SNR\ X_{1,1}+\ SNR\ X_{2,1} \right)\textless\beta_{1}+\beta_{2}\ \right],\\ P\left(R_{4}^{cn1}\right)=\int\limits_{0}^{U_{1,1}}\int\limits_{0}^{U _{2,1}}\exp(-X_{1,1}-X_{2,1})\ d\ X_{1,1}\ dX_{2,1}\\ +\int\limits_{U_{1,1}}^{U_{1,1}}\exp(-X_{1,1})\ d\ X_{1,1}\int \limits_{\frac{X_{1,1}^{\prime}}{2^{\beta_{1}+\ \beta_{2}-1}}}^{\textless X_{1,1}}\exp(-X_{2,1})\ d\ X_{2,1},\\ P\left(R_{4}^{cn1}\right)=1-\left(\frac{1}{2^{\beta 1}}\exp(-U_{1,1})+\frac{1}{2^{\beta 2}}\exp(-U_{ 2,1})\right.\\ \left.+\exp\left(-\frac{2^{\beta_{1}+\ \beta_{2}-1}}{SNR}\right) \left(\left[1-\frac{1}{2^{\beta 1}}-\frac{1}{2^{\beta 2}}\right]+U_{1,1}U_{2,1} SNR\right)\right)\end{split}  \tag{8}\]  
Inserting Equations (8) and (7) into Equation (6), we obtain \(P_{s1,cn1}^{out}\). In what follows, \(P_{s1,cn2}^{out}\) is the outage probability of the \(S1-CN2\) link ( \(X_{1,2}\) link), and can be derived by similar calculations shown in Equations (6)-(8). \(P_{s1,cn2}^{out}\) is expressed as:  
\[P_{s1,cn2}^{out}=P\left(R_{2}^{cn2}\right)+P\left(R_{4}^{cn2}\right) \tag{9}\]  
The term \(P\left(R_{2}^{cn2}\right)\) and \(P\left(R_{4}^{cn2}\right)\) in Equation (6) can be further expressed as:  
\[P\left(R_{2}^{cn2}\right)=\frac{1}{2^{\beta_{2}}}\left[\exp(-U_{2,1})-\ \exp(-\frac{2^{\beta_{1}+\ \beta_{2}-1}}{SNR})\right] \tag{10}\]and:  
\[ \begin{split} P\Big{(}R_{4}^{cn2}\Big{)}=1-\Big{(}\frac{1}{2^{\beta 1}}\exp(-U_{1,1})+\frac{1}{2^{\beta 2}}\exp(-U_{2,1})\\ +\exp\Big{(}-\frac{2^{\beta_{1}+\beta_{2}-1}}{SNR}\Big{)}\Big{(} \big{[}1-\frac{1}{2^{\beta 1}}-\frac{1}{2^{\beta 2}}\Big{]}+U_{1,1}U_{2,1} SNR\Big{)}\Big{)}\end{split}  \tag{11}\]  
Inserting Equations (10) and (11) into Equation (9), we obtain \(P_{s1,cn2}^{out}\). \(P_{SH}^{out}\) is the outage probability of the second phase from \(CN1\) and \(CN2\) to \(T2\). The second phase represents the transmission from the \(CN\) nodes to \(T2\) at different time slots, and \(T2\) combines the received data via MRC. Thus, the outage occurs whenever:  
\[\beta(SNR_{1}X_{1},SNR_{2}\ X_{2})<\beta_{SH} \tag{12}\]  
We assume that each \(CN\) node transmits what is received from sensor 1 over a period of time denoted as \(\omega\). Thus, \(CN1\) transmits the data to \(T2\) via \(\omega\), and \(CN2\) transmits the data to \(T2\) via \(1-\omega=\ \overline{\omega}\), where the ratio \(\omega\) (\(0<\omega<1\)). Then, Equation (12) can be further expressed as:  
\[\beta(SNR_{1}X_{1},SNR_{2}\ X_{2})=\log_{2}\Big{(}(1+SNR_{1}\ X_{1})^{\omega} \ (1+SNR_{2}\ X_{2}\ )^{\overline{\omega}}\Big{)} \tag{13}\]  
where the outage probability \(P_{SH}^{out}\) can then be evaluated over the outage region in Equation (13) as:  
\[ \begin{split} P_{SH}^{out}=P\left[(1+SNR_{1}\ X_{1})^{\omega}\ (1+SNR_{2}\ X_{2}\ )^{\overline{\omega}}<\beta_{SH}\ \right],\\ P_{SH}^{out}=\int\limits_{0}^{U_{SH}}\exp(-X_{1})dX_{1}\int \limits_{0}^{U_{SH2}}\exp(-X_{2})dX_{2},\\ P_{SH}^{out}=(1-\exp(-U_{SH1}))\ (1-\exp(-U_{SH2})).\end{split}  \tag{14}\]  
where \(U_{SH1}\) and \(U_{SH2}\) can then be expressed as:  
\[