## 5 Delay and Duty Cycle Analysis of CD-ICC  
### Delay Analysis of CD-ICC  
The average e2e delay of the IEEE 802.15.6 of CD-ICC is evaluated in this subsection. Where, the average e2e delay is defined as the total time required of the medium access delay to transmit data. The average e2e delay includes average contention time due to collision (\(T_{C}\)), the average successful transmission time with no collision and no fading (\(T_{suc}\)), and average failure time due to fading but no collision (\(T_{fail}\)) [36]:  
\[T_{e2e}=T_{C}+T_{suc}+T_{fail} \tag{19}\]  
the average contention time due to collision can be expressed as:  
\[T_{C}=T_{data}+T_{I-ACK}+T_{CW}+2T_{pSIFS}+2T_{a} \tag{20}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\xi_{min}\) & \(\xi_{max}\) & \(\xi_{thd}\) & \(P\) (\(\xi\)\(\geq\)\(\xi_{thd}\)) \\ \hline
0 & 7 & 1 & 0.15 \\
1 & 7 & 0.857 & 0.2255 \\
2 & 7 & 0.714 & 0.3126 \\
3 & 7 & 0.571 & 0.4194 \\
4 & 7 & 0.428 & 0.540 \\
5 & 7 & 0.285 & 0.6869 \\
6 & 7 & 0.142 & 0.8408 \\
7 & 7 & 0.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Threshold and probability of the Critical Data Index. the time required to transmit data packet (see Figure 2a) can be expressed as:  
\[T_{data}=T_{P}+T_{PHY}+T_{MAC}+T_{ BODY}+T_{FCS}. \tag{21}\]  
the transmissi0n time required for \(I-ACK\) (see Figure 2b) can be expressed as:  
\[T_{I-ACK}=T_{P}+T_{PHY}+T_{MAC}+T_{FCS}. \tag{22}\]  
where \(T_{CW}\) average back0 ff time and is expressed as:  
\[T_{CW}=T_{s}\ CW \tag{23}\]  
Average successful transmission time with no collision and no fading can be expressed as:  
\[T_{suc}=P_{\xi}\ (\xi_{thd})\ \Big{(}T_{act}^{sd}\ P_{sd}^{s}+\ \Big{(}T_{act}^{sd}+T_{act}^{rd}\Big{)}\ (1-\ P_{sd}^{s}\ )\ P_{sr}^{s}\ P_{rd}^{s} \Big{)}+\big{(}1-\ P_{\xi}\ (\xi_{thd})\big{)}\ P_{sd}^{s}\ T_{act}^{sd} \tag{24}\]  
where \(T_{act}^{sd}\) is the RF activity time of \(S-D\) link, \(T_{act}^{sd}=\ T_{C}+T_{on}\). Re-write (24) as:  
\[T_{suc}=P_{\xi}\ (\xi_{thd})\ \Big{(}T_{act}^{sd}+T_{act}^{rd}\Big{)}\ (1-\ P_{sd}^{s}\ )\ P_{sr}^{s}\ P_{rd}^{s}+\ P_{sd}^{s}\ T_{act}^{sd} \tag{25}\]  
Equation (24) comprises two terms. The first term is e2e required time of the transmission when the gathered data by the sensor are critical, delivered through ICC. The second term is e2e required time of the transmission when the gathered data by the sensor are not critical, delivered over DTM. It is clear from (24), as the value of \(P_{\xi}(\xi_{thd})\) approaches one, either \(T_{act}^{sd}\) is the required time to transmit the data or \(T_{act}^{sd}+T_{act}^{rd}\) is the required time to transmit the data to the destination. On the other hand, as the value of \(P_{\xi}(\xi_{thd})\) approaches zero, then \(T_{act}^{sd}\) is the required time to transmit the data to the destination. \(T_{act}^{rd}\) is the RF activity time of the \(R-D\) link and can be expressed as:  
\[T_{act}^{rd}=T_{on}^{rd}+T_{CW}^{rd,*}+T_{data}^{rd}+T_{ACK}^{rd}+2T_{pSIFS}^ {rd}+2T_{R}^{rd}. \tag{26}\]  
Figure 2: (**a**) IEEE 802.15. Physical Protocol Data Unit frame structure; (**b**) IEEE 802.15. ACK packet. In this paper, we propose a new \(T_{CW}^{rd}\) expressed as follows:  
\[T_{CW}^{CD-ICC}=\left\lceil T_{s}\ \frac{CW_{max}}{CW_{min}}\right\rceil\left( \frac{d_{eq}}{2^{\rho-2}+d_{eq}^{-1}}\ \right)\ \
icefrac{{\rho-1}}{{\sqrt{d_{eq}}}}+(\rho-4)\ \Bigg{\rceil}\ \mu s \tag{27}\]  
where \(CW_{min}\) and \(CW_{max}\) are the minimum contention window and maximum contention window size, respectively. \(d_{eq}\) is expressed as \((d_{sr}+d_{rd})^{-\rho}\). The objective of the proposed back-off time is to make sure that the best relay sensors can participate in cooperation, and that the best relay sensor can access the channel first. However, when the relay sensor is willing to help the source, it may have a greater delay due to (27). Hence, each relay(s) will select the \(T_{CW}^{*}\) according to:  
\[T_{CW}^{rd,*}=\left\{ \begin{array}{l}CW\ T_{s}\ for\ T_{CW}^{CD-ICC}>T_{CW} \\ T_{CW}^{CD-ICC}\ for\ T_{CW}^{CD-ICC}\leq T_{CW}\end{array} \right.