## 4 Energy Consumption
### Average Energy Consumed by RN1 during Awake Cycles
#### 4.4.3 Energy Consumption in Type 1C Cycles  
The energy consumed by RN1 in type 1C _awake_ cycles can be determined as  
\[E_{aw}[1C] =\sum_{i=1}^{Q_{2}}\sum_{k=1}^{M_{2}}\pi_{2}(i,k)(k+1)P_{\text{S},k,2}\Big{[}E_{3}^{aw,1}-\widehat{f}_{k,2}\;t_{DATA}(P_{rx,1}-P_{sl})\Big{]}\] \[+\sum_{k=1}^{M_{2}}\pi_{2}(0,k)R_{\text{S},k-1,2}\Big{[}E_{3}^{aw, 1}-\widehat{f}_{k-1,2}\;t_{DATA}(P_{rx,1}-P_{sl})\Big{]}\] \[+\sum_{i=1}^{Q_{2}}\sum_{k=2}^{M_{2}}\pi_{2}(i,k)\widehat{P}_{f,k,2}(T-T_{\text{sc},1}c)P_{rx,1} \tag{13}\] \[+\sum_{k=2}^{M_{2}}\pi_{2}(0,k)\widehat{P}_{f,k,2}^{\prime}(T-T_{ \text{sc},1})P_{rx,1}\,,\] \[\widehat{f}_{k,2} =\frac{1}{G_{k,2}}\sum_{i=1}^{Q_{2}}\alpha_{1}\cdot\pi_{2}(i,k)\,, \;\;\;G_{k,2}=\sum_{i=1}^{Q_{2}}\pi_{2}(i,k)\,.\]  
where \(\widehat{P}_{f,k,2}\) is the probability that PFs transmitted by RN2 collide while contending with other additional \(k\) C2 SNs, \(\widehat{P}_{f,k,2}^{\prime}\) is the probability that two or more of the \(k\) active SNs, other than RN2, transmit with collision while RN2 is inactive, \(\widehat{f}_{k,2}\) is the average PF size transmitted by RN2, conditioned on RN2 being contending with other \(k\) C2 SNs in the same cycle, \(\alpha_{2}=min(i,F_{2})\) is the number of _DATA_ packets aggregated in the current PF, \(F_{1}\) is the maximum number of _DATA_ packets that C2 SNs can aggregate, \(\widehat{P}_{f,k,2}\), \(\widehat{P}_{f,k,2}\), \(\widehat{f}_{k,2}\) and \(\widehat{f}_{k-1,2}\) can be determined in the same way as their C1 counterparts.  
As before, note that \(\widehat{f}_{k,2}\;\widehat{f}_{k-1,2}\) are obtained to determine the average time RN1 will remain in the _sleep_ mode while waiting for the end of the PF transmitted by a C2 SNs. Note also that \(\widehat{f}_{k,2}=1\) when the SPT scheme is deployed.  
Each of the terms in expression (13) defines the average energy consumed by RN1 due to the occurrence of different events. The first term accounts for the average energy consumed when one of the \(k+1\) active C2 SNs (the RN2 and other \(k\) C2 SNs) transmits a PF successfully. The second term accounts for the average energy consumed when one of the \(k\) active C2 SNs other than RN2, transmits a PF successfully in a cycle where RN2 is inactive. The third and fourth account for the average energy consumed when two or more of \(k\) C2 SNs, different from the RN2, transmit with collision when the RN2 is active or inactive, respectively.  
4.4.4. Energy Consumption in in Type 1D Cycles  
The average energy consumed by RN1 in type 1D _awake_ cycles can be determined as  
\[E_{aw}[1D]=\pi_{1}(0,0)\pi_{2}(0,0)(T-T_{sc,1})P_{rx,1}\,. \tag{14}\]  
4.4.5. Average Energy Consumed by RN1 in _awake_ Cycles  
The total average energy consumed by RN1 in _awake_ cycles can be determined as  
\[\begin{split}& E_{aw,1}=E_{aw}[1A]+E_{aw}[1B]+R_{1,0}E_{aw}[1C]+E_{aw}[1D] \,,\\ & E_{aw,1}=E_{tx,s}^{aw}[1A]+E_{tx,f}^{aw}[1A]+E_{oh}^{aw}[1A]+E_{aw }[1B]+R_{1,0}E_{aw}[1C]+E_{aw}[1D]\,.\end{split} \tag{15}\]  
Please note that \(E_{aw}[1C]\) is weighted by the fraction of cycles where C1 SNs remain inactive, \(R_{1,0}\). In these cycles, active C2 SNs, if any, will contend for medium access.