# Body-to-Body Cooperation in Internet of Medical Things: Toward Energy Efficiency Improvement
## 4 Energy Efficiency of the IWC-IoMT  
Energy efficiency is denoted as the achievable information transmission per joule of energy consumption. In this section, the energy efficiencies of DT-IoMT, TH-IoMT and IWC-IoMT are derived based on the previous calculations. The energy efficiency between two nodes or the energy efficiency of the DT-IoMT is defined as [25]:  
\[E_{i,j}=\Big{(}1-\ P_{i,j}^{out}\Big{)}\frac{\beta_{i,j}}{\bigcup\limits_{i,j }^{out}}\ \ \ \ [\text{bit/joule}] \tag{16}\]  
Figure 3: Rate region of IWC-IoMT protocol, where \(\beta_{1}=\ \beta_{s1,cn1}\), and \(\beta_{2}=\ \beta_{s1,cn2}\).  
where \(P_{i,j}^{tot}\) is the total power consumption and is comprised of \(P_{amp}\), \(P_{tx}\) and \(P_{rx}\), the \(P_{amp}\) is the power amplifier consumption for the transmission, and \(P_{tx}\) and \(P_{rx}\) are the power consumption of the internal circuitry for transmitting and receiving, respectively. \(P_{i,j}^{tot}\) is expressed as:  
\[P_{i,j}^{tot}=P_{amp}+P_{tx}+P_{rx} \tag{17}\]  
In what follows, the energy efficiency of the TH-IoMT, \(E_{TH}\) is expressed as:  
\[E_{TH}=\frac{\left(1-P_{s1,cm1}^{out}\right)\left(1-P_{cm1,T2}^{out}\right) \beta_{TH}}{P_{TH}^{tot}} \tag{18}\]  
where in the two-hop cooperation, two nodes are involved in the data transmission and the total power consumption, \(P_{TH}^{tot}\), and the achievable link capacity is equal to the minimum of the two hops, \(\beta_{TH}\), which are expressed as:  
\[P_{TH}^{tot}=2\left(P_{amp}+P_{tx}+P_{rx}\right) \tag{19}\]  
\[\beta_{TH}\approx\beta_{s1,cm1}\approx\beta_{cm1,T2} \tag{20}\]  
When we insert Equations (19) and (20) into Equation (18), we obtain \(E_{TH}\). Next, the energy efficiency of the IWC-IoMT protocol for sensor 1 is expressed as:  
\[E_{\star}=\frac{\left(1-P_{s1,cm1}^{out}\right)\left(1-P_{s1,cm2}^{out}\right) \beta_{FH}}{P_{amp}+P_{tx}+2P_{rx}}+\frac{\left(1-P_{cm1,T2}^{out}\right)\left( 1-P_{cm2,T2}^{out}\right)\beta_{SH}}{2P_{amp}+2P_{tx}+P_{rx}} \tag{21}\]  
where \(\left(1-P_{s1,cm1}^{out}\right)\left(1-P_{cm1,T2}^{out}\right)\) represent the successful transmission probability from \(S1\) to \(CN1\), and from \(S1\) to \(CN2\), respectively, and \(\left(1-P_{cm1,T2}^{out}\right)\left(1-P_{cm2,T2}^{out}\right)\) represent the successful transmission probability from \(CN1\) to \(T2\), and from \(CN2\) to \(T2\), respectively. The first term total power, \(P_{amp}+P_{tx}+2P_{rx}\), two nodes are involved in the data reception and a single node is involved in data transmission. Then, the second term total power, \(2P_{amp}+2P_{tx}+P_{rx}\), two nodes are involved in the data transmission and a single node is involved in data reception. Furthermore, the achievable link capacity of the first and second terms is expressed as:  
\[\beta_{FF}=\beta_{s1,cm1}=\beta_{s1,cm2} \tag{22}\]  
\[\beta_{SF}=\beta_{cm1,T2}=\beta_{cm2,T2} \tag{23}\]  
where \(\beta_{FF}\) is the transmission rate from the sensors to the coordinators (first phase), and \(\beta_{SF}\) is the transmission rate from the coordinators to \(T2\) (second hop). Finally, if we insert Equations (21) and (22) into Equation (20), we obtain \(E_{\star}\).