# Toward an Advanced Human Monitoring System Based on a Smart Body Area Network for Industry Use
## 3 System Model
### Mathematical Model  
The energy efficiency in the PHY layer was derived from [32] as follows:  
\[\eta\equiv\frac{P_{succ}L_{info}}{E_{link}} \tag{2}\]  
In this equation, \(L_{info}\) is the information bit length in the PSDU, \(E_{link}\) is the energy consumption of the communication link and \(P_{succ}\) is the transmission success ratio. \(P_{succ}\) can be expressed as follows [32]:  
\[P_{succ}=1-PER=\Big{(}1-P_{fail,preamble}\Big{)}(1-P_{e,PLCpheader})(1-P_{e, PSDU}) \tag{3}\]  
In this equation, \(PER\) is the packet error ratio, \(P_{e,preamble}\) is the PLCP header detection failure ratio, \(P_{e,PLCpheader}\) is the PLCP header error ratio and \(P_{e,PSDU}\) is the PSDU error ratio. Additionally, \(E_{link}\) can be simply described as follows [32]:  
\[E_{link}=L_{PPDU}(P_{tx}+P_{rx})/R_{s}+(\epsilon_{enc}+\epsilon_{dec}) \tag{4}\]  
\[L_{PPDU}=L_{preamble}+L_{SFD}+L_{PLCpheader}+L_{PSDU} \tag{5}\]  
In this equation, \(L_{PPDU}\), \(L_{preamble}\), \(L_{SFD}\), \(L_{PLCPheader}\) and \(L_{PSDU}\) are the lengths of the PPDU, the preamble, the SFD, the PLCP header and the PSDU, respectively; \(P_{tx}\) and \(P_{rx}\) are the transmitter and receiver power consumption, respectively; and \(\epsilon_{enc}\) and \(\epsilon_{dec}\) are the encoding and decoding energies, respectively. As a channel model, this research assumed the additive white Gaussian noise (AWGN) channel and IEEE model CM3, which is a channel model of a wearable WBAN [32]. Similarly, the IEEE model CM3 was applied to the path loss model. The path loss model of IEEE model CM3 when using the 2.4 GHz band is as follows [34]:  
\[PL(d)=a\times log_{10}d+b+N \tag{6}\]  
In this equation, \(a\) and \(b\) are the coefficients of linear fitting, \(d\) is the Tx-Rx distance in mm and \(N\) is a normally distributed variable with a standard deviation \(\sigma_{N}\). The \(K\) factor in decibels (\(K_{dB}\)) and each parameter when assuming frequency-flat fading are as shown in Equation (7) [34]:  
\[K_{dB}=K_{0}-m_{k}PL(d)+\sigma_{k}n_{k} \tag{7}\]  
The meaning of each parameter was defined in [34]. \(K_{0}\) is the fit with the measurement data for the \(K\) factor for low path loss, \(m_{k}\) is the slope of the linear correlation between the path loss and the \(K\) factor, \(\sigma_{k}\) is the log-normal variance of the measured data between the path loss and the \(K\) factor and \(n_{k}\) is the zero mean and unit variance Gaussian random variable. Next, the interbeacon interval \(T_{D}\) is described as follows [14; 15; 16; 27]:  
\[T_{D}=T_{SAP}+T_{CM}+T_{IP} \tag{8}\]  
\[T_{SAP}=N_{s}T_{s} \tag{9}\]  
\[N_{s}=\sum_{i=1}^{N_{node}}N_{s,i} \tag{10}\]  
\[T_{CM}=N_{CM}T_{s} \tag{11}\]  
In this equation, \(T_{sAP}\), \(T_{CM}\) and \(T_{IP}\) are the durations of the SAP, the C/M period and the inactive period, respectively. Then, \(N_{s,i}\) represents the number of available time slotsfor each node in the SAP. \(N_{node}\) is also the number of nodes in the network. In addition, the number of available time slots, except the inactive period \(N_{D}\), is calculated as follows:  
\[P_{succ}=1-PER=\Big{(}1-P_{fail,preamble}\Big{)}(1-P_{e,PLCpheader})(1-P_{e, PSDU}) \tag{3}\]0  
\[P_{succ}=1-PER=\Big{(}1-P_{fail,preamble}\Big{)}(1-P_{e,PLCpheader})(1-P_{e, PSDU}) \tag{3}\]1  
In this equation, \(L_{PPDU}\) and \(L_{PPDU,ACK}\) are the PPDU lengths of the data and ACK frames, respectively. \(T_{IFS}\) is the duration of the interframe spaces. \(R_{b}\) is also the bit rate. Let \(r_{sAP}\):\(r_{CM}\) be the preset ratio of \(N_{s}\) and \(N_{CM}\).