# Body-to-Body Cooperation in Internet of Medical Things: Toward Energy Efficiency Improvement
## 3 Link and Outage Probability Analysis  
In this section, the propagation model and the outage probability between two nodes are described. The average signal-to-noise ratio (\(SNR_{i,j}^{an}\)) from node \(i\) to node \(j\) is given as [24]:  
\[SNR_{i,j}^{an}=\ SNR_{i,j}\ X_{i,j}=\ \frac{P_{i,j}\ k_{i,j}}{P_{N}+P_{I}}\ X_{i,j} \tag{1}\]  
where:  
\[k_{i,j}=\frac{G\ \lambda}{\left(4\pi\right)^{2}M_{l}\ N_{f}} \tag{2}\]  
\(P_{i,j}:\) is the transmission power;  
\(P_{N}:\) is the noise power;  
\(X_{i,j}:\) is a complex Gaussian random variable with unit variance;  
\(|X_{sd}|^{2}:\) is given as \(E\left[|\ X_{ij}\ |^{2}\right]=\ d_{ij}^{-\alpha}\);  
\(d_{ij}:\) is the distance between two nodes;  
\(G:\) is the total gain of the transmit and receive antennae;  
\(\lambda:\) is the wavelength;  
\(N_{f}:\) is the noise figure at the receiver; and  
\(M_{l}:\) is the link margin. The outage probability is defined as the probability that the transmission rate is less than or equal to the required transmission rate \(\beta\). The outage probability can be calculated as [22]:  
\[P_{i,j}^{out}=P\Big{(}\beta_{i,j}\ \leq\beta\Big{)}=1-\exp\Big{(}-U_{i,j}\ d_{ij}^{+ \alpha_{1}}\Big{)} \tag{3}\]  
where \(\beta_{i,j}=\log_{2}\Big{(}1+\ SNR_{i,j}\ X_{i,j}\Big{)}\) and \(U_{i,j}\) is expressed as:  
\[U_{i,j}=\ \frac{\ \Big{(}2^{\beta}-1\Big{)}}{SNR_{i,j}} \tag{4}\]  
_Outage Probability Analysis of IWC-IoMT_  
The outage probability of the IWC-IoMT protocol is evaluated in this section. This analysis is based on the rate region of two-tandem transmission, as described in Figure 3, where the end-to-end outage probability of the sensor 1 to the \(T2\) link is evaluated. The rate region of the two sensors and the two CNs is shown in Figure 3. Note that the \((\beta_{1},\beta_{1})\) in Figure 3 is separated into four regions. We can describe the error and correct reception regions as follows: let us consider the error region of sensor 1, where in Figure 3, region 3 (R3) and region 4 (R4) corresponding to data received from sensor 1 is erroneous; while the data received from sensor 2 can be error-free in R3. In the same way, region 2 (R2) and region 4 (R4) corresponding to data received from sensor 2 is erroneous; while the data received from sensor 1 can be error-free in R2. On the other hand, region 1 (R1) corresponding to data received from sensor 1 and sensor 2 is correct. The communication model of the IWC-IoMT, shown in Figure 2, is a two-hop communication, where the first phase is the cooperative mode and the second phase is two non-tandems (transmissionover two different time slots) direct transmission mode. Consequently, the outage probability of the IWC-IoMT can be expressed as:  
\[ \begin{split} P_{*}^{out}=\left[\underbrace{\left(P_{s1,cn1}^{out} \cap P_{s1,cn2}^{out}\right)}_{\text{1 st phase}}\cap\underbrace{\left(1-P_{SH}^{ out}\right)}_{\text{2 nd phase}}\right]\cup\ \left[\underbrace{\left(\left(1-P_{s1,cn1}^{out}\right)\cap \left(1-P_{s1,cn2}^{out}\right)\right)}_{\text{1 st phase}}\cap\underbrace{ \left(P_{SH}^{out}\right)}_{\text{2 nd phase}}\right],\\ P_{*}^{out}=\left[\underbrace{\left(P_{s1,cn1}^{out}P_{s1,cn2}^{ out}\right)}_{\text{1 st phase}}\underbrace{\left(1-P_{SH}^{out}\right)}_{\text{2 nd phase}}\right]+\left[\underbrace{\left(\left(1-P_{s1,cn1}^{ out}\right)\left(1-P_{s1,cn2}^{out}\right)\right)}_{\text{1 st phase}}\underbrace{\left(P_{SH}^{out}\right)}_{\text{2 nd phase}}\right].\end{split}  \tag{5}\]  
In Equation (5), the first term corresponds with the event when the first phase is in the outage, while the second phase is not in the outage and the second term corresponds to the event, which is contrast to the first event. \(P_{s1,cn1}^{out}\) is the outage probability of the \(S1-CN1\) link ( \(X_{1,1}\)link) and can be expressed as:  
\[P_{s1,cn1}^{out}=P\left(R_{3}^{cn1}\right)+P\left(R_{4}^{cn1}\right) \tag{6}\]  
where \(P\left(R_{3}^{cn1}\right)\) and \(P\left(R_{4}^{cn1}\right)\) are the rate regions of sensor 1 to \(CN1\) (first phase).