## 4 Critical Data-Based Incremental Cooperative Communication (CD-ICC)
### Formulation of the CD-ICC  
As described earlier, the CD-ICC comprised from two events and is mathematically expressed as:  
\[P_{CD-ICC}=\underbrace{P(X)}_{Critical\ data\ event}+\underbrace{P(Y)}_{Normal\ data\ event} \tag{6}\]  
where, \(P(X)\) is the probability of the gathered data were critical. The critical data delivered to destination utilizing ICC and is mathematically expressed as:  
\[P(X)=P(P(\xi\geq\xi_{thd}),P^{s}_{ICC}) \tag{7}\]  
where, the \(P(\xi\geq\xi_{thd})\) is the probability of critical data index \(\xi\) greater than threshold value \(\xi_{thd}\), and \(P^{s}_{ICC}\) the successful transmission probability of the ICC. In the (7), events are independent, then the \(P(X)\) can be written as:  
\[P(X)=P_{\xi}(\xi_{thd})P^{s}_{ICC} \tag{8}\]  
The \(P^{s}_{ICC}\) can be expressed as:  
\[P^{s}_{ICC}=P^{s}_{sd}+(1-P^{s}_{sd})P^{s}_{sr}P^{s}_{rd} \tag{9}\]  
where, the term \(P^{s}_{sd}\) represent \(S-D\) link isn't in the outage, while \(\left(1-P^{s}_{sd}\right)\) represent the \(S-D\) link in the outage, and the terms \(P^{s}_{sr}\) and \(P^{s}_{rd}\) represent \(S-R\) and \(R-D\) links are not in the outage. Afterwards, and with help of (5), we obtain \(P^{s}_{ICC}\) as:  
\[P^{s}_{ICC}=0.5 erfc\left(\frac{\omega}{U_{sd}}\right)+0.25 erfc\left(\frac{ \omega}{U_{sr}}\right)erfc\left(\frac{\omega}{U_{rd}}\right)\left(1-0.5 erfc \left(\frac{\omega}{U_{sd}}\right)\right) \tag{10}\]  
where, \(\xi\) is normal random variable with zero mean and unity variance, hence the probability that the critical data index \(\xi\) greater than threshold value \(\xi_{thd}\) is given as:  
\[P_{\xi}(\xi_{thd})=P(\xi\geq\xi_{thd})=0.5\int\limits_{\xi_{thd}}^{\infty}g(y )dy \tag{11}\]  
where, \(g(y)\) is given as:  
\[g(y)=\frac{1}{\sqrt{2\pi}}\exp\left(-\frac{y^{2}}{2}\right) \tag{12}\]  
insert (12) in (11), we obtain (11) as:  
\[P_{\xi}(\xi_{thd})=0.5\int\limits_{\xi_{thd}}^{\infty}\frac{1}{\sqrt{2\pi}} \exp\left(-\frac{y^{2}}{2}\right)dy \tag{13}\]  
solving the integral of (13), we obtain (13) as:  
\[P_{\xi}(\xi_{thd})=erfc(\xi_{thd}) \tag{14}\]  
where, \(\xi_{thd}\) is threshold that gathered data is critical, and it expressed as:  
\[\xi_{thd}=\left|\frac{\xi_{min}-\xi_{max}}{\xi_{max}}\right| \tag{15}\]  
A parameter called the critical data index threshold, \(\xi_{thd}\), determines the degree of criticality. \(\xi_{max}\) is the maximum critical data index and is equal to 7, while \(\xi_{min}\) is the minimum critical data index andcan vary between 0 and 7. \(\xi_{min}\) depends on the gathered data from the human body, and if the data is critical, then \(\xi_{min}\) takes a high value, and vice versa. Table 2 show the probability of the critical data index with different values of \(\xi_{min}\).  
It is clear that, as the \(\xi_{thd}\) is high, the probability of the critical data index is low and vice versa. Inserting (10) and (14) in (8), we obtain \(P(X)\) as:  
\[\begin{array}{l}P(X)=erfc(\xi_{thd})\Big{(}0.5 erfc\Big{(}\frac{\omega}{U_{sd }}\Big{)}\\ +0.25 erfc\Big{(}\frac{\omega}{U_{rd}}\Big{)}erfc\Big{(}\frac{\omega}{U_{rd}} \Big{)}\Big{(}1-0.5 erfc\Big{(}\frac{\omega}{U_{rd}}\Big{)}\Big{)}\Big{)}\end{array} \tag{16}\]  
The second term of the (6) represent the event that the gathered data by the sensor were not critical and with help of (5) and (14), the \(P(Y)\) can be expressed as:  
\[P(Y)=\frac{1}{2}(1-erfc(\xi_{thd}))erfc\bigg{(}\frac{\omega}{U_{sd}}\bigg{)} \tag{17}\]  
finally, summing up the \(P(X)\) and \(P(Y)\) together, we obtain \(P_{CD-ICC}\) as:  
\[\begin{array}{l}P_{CD-ICC}\\ =\underbrace{erfc(\xi_{thd})\Big{(}0.5 erfc\bigg{(}\frac{\omega}{U_{sd}} \bigg{)}+0.25 erfc\bigg{(}\frac{\omega}{U_{rd}}\bigg{)}erfc\bigg{(}\frac{\omega }{U_{rd}}\bigg{)}\Big{(}1-\ 0.5\ erfc\bigg{(}\frac{\omega}{U_{sd}}\bigg{)}\bigg{)} \Big{)}}_{\text{\tiny{critical data event}}}\\ +\underbrace{0.5(\ 1-\ erfc(\xi_{thd}))\ erfc\bigg{(}\frac{\omega}{U_{ sd}}\bigg{)}}_{\text{\tiny{normal data event}}}\end{array} \tag{18}\]