## 5 Performance Evaluation
### Energy Efficiency of Nodes and PC  
In this subsection, we described the results of energy efficiency of the proposed MAC by analyzing the energy consumption from both PC and nodes. Depending on the parameters described in Table 1, the overall energy consumption of PC denoted as \(E^{C}\) could be illustrated as follows:  
\[ \begin{array}{ll}E^{C}&=E^{C}_{active}+E^{C}_{sleep}+\frac{E^{C}_{known}}{\eta} \\ &=E^{C}_{sync}+E^{C}_{d-TP}+E^{C}_{u-TP}+E^{C}_{CP}+E^{C}_{sleep}+\eta^{-1}E^{C}_ {bacon}\end{array}  \tag{11}\]  
The energy consumption of Sync and Beacon periods within one superframe could be simply described as below equations where \(P\) parameter would be classified into \(P_{Tx}\), \(P_{Rx}\), and \(P_{idle}\) on each radio transceiver condition (\(Tx\), \(Rx\), and idle listening status). Note that the duration of each period is not equal to the interval value. \[ \begin{array}{ll}E^{C}_{sync}&=P_{Tx}D_{sync}\\ E^{C}_{bacon}&=P_{Tx}D_{bacon}\end{array}  \tag{12}\]  
In TP, the data transmission from a PC can be incurred or not. Therefore, the analysis of energy consumption in u-TP and d-TP has to consider the occurrence probability of data transmission event and dependently assigned uplink and downlink time slot numbers denoted as \(k_{u}\) and \(k_{d}\), as below Equations (13) and (14), respectively. \[ \begin{array}{ll}E^{C}_{u-TP}&=(P_{idle}T_{IFS}+P_{Rx}T_{Data}+P_{Tx}T_{ Ack})\lambda_{u}k_{u}\\ &+P_{idle}(T_{Data}+T_{Ack})(1-\lambda_{u})k_{u}\\ &+(P_{idle}T_{IFS}+P_{Rx}T_{Data}+P_{Tx}T_{Ack})\lambda_{e}k_{u}\\ &=k_{u}(\lambda_{u}+\lambda_{e})\\ &\times\Big{(}P_{idle}\Big{(}T_{IFS}+\Big{(}(\lambda_{u}+\lambda_{e})^{-1}-1 \Big{)}\big{(}T_{Data}+T_{Ack}\big{)}\Big{)}+P_{Rx}T_{Data}+P_{Tx}T_{Ack}\Big{)} \end{array}  \tag{13}\]  
\[ \begin{array}{ll}E^{C}_{d-TP}&=k_{d}(P_{idle}(T_{IFS}+(1-\lambda_{d})(T_{ Data}+T_{Ack}))\\ &+P_{Tx}(\lambda_{d}T_{Data}+\lambda_{e}T_{Ack})+P_{Rx}(\lambda_{d}T_{Ack}+ \lambda_{e}T_{Data}))\end{array}  \tag{14}\]  
In addition, the overall energy consumption of the PC in CP could be illustrated as below equation where both the energy consumptions for emergency data and normal data transmission were denoted as \(E_{Emergency}\) and \(E_{Normal}\), respectively. \[ \begin{array}{ll}E^{C}_{CP}=\lambda_{e}E_{Emergency}+(1-\lambda_{e})\big{(} \lambda_{reg}+\lambda_{slot}+\lambda_{remain}\big{)}E_{Normal}\end{array}  \tag{15}\]  
For more simplification of the analysis, we solve the above equation where the length values of all packets including the slot request, registration request, and request for remaining data transmission have the same size. \[ \begin{array}{ll}E^{C}_{CP}&=\lambda_{e}(P_{Rx}T_{Data}+P_{Tx}T_{Ack})\\ &+(1-\lambda_{e})\big{(}(\lambda_{reg}+\lambda_{slot}+\lambda_{remain}\big{)}( P_{Rx}T_{Data}-P_{idle}(T_{Data}+T_{Ack}))+\lambda_{remain}P_{Tx}T_{Ack}+P_{idle}(T_{ Data}+T_{Ack}))\end{array}  \tag{16}\]  
In sleep period, a PC should perform LPL mechanism for allowing data reception of emergency data from any node. Hence, we could describe the energy consumption of a PC with the values of duty cycle interval, wake-up duration, and long preamble length depending on LPL respectively denoted as \(T_{datty}\), \(T_{wakeup}\), and \(\omega_{p}\), as below equation. \[ \begin{array}{ll}E^{C}_{Sleep}=P_{Rx}\lambda_{e}\Big{(}\frac{\omega_{p}}{2R }+T_{Data}\Big{)}+(1-\lambda_{e})P_{idle}D_{Sleep}\frac{T_{wakeup}}{T_{datty}} \end{array}  \tag{17}\]  
Meanwhile, the average energy consumption value of nodes could be also described as below. \[ \begin{array}{ll}E^{N}=E^{N}_{sync}+E^{N}_{d-TP}+E^{N}_{u-TP}+E^{N}_{CP}+E^{ N}_{sleep}+\eta^{-1}E^{N}_{bacon}\end{array}  \tag{18}\]The energy consumption of a node could be induced by the same approach from Equation (12) with just cross match between the transmitter and receiver, as below equation. \[ \begin{array}{ll}E_{sync}^{N}=P_{Rx}D_{sync}\\ E_{beacon}^{N}=P_{Rx}D_{beacon}\end{array}