# A Sigfox Energy Consumption Model
## 4 Modeling Sigfox Device Current Consumption
### Bidirectional Transactions
Finally, let \(C\) denote the event where none of the three uplink frame transmissions is correctly received, thus the base station does not send a downlink message, and \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**State Number**} & \multirow{2}{*}{**Description**} & \multicolumn{2}{c}{**Duration**} & \multicolumn{2}{c}{**Current Consumption**} \\ \cline{3-6}  & & **Variable** & **Value (ms)** & **Variable** & **Value (mA)** \\ \hline
1 & Wake up & \(T_{wu}\) & 305 & \(I_{wu}\) & 10.7 \\
2 & Transmission & \(T_{tx}\) & [1200,2080] (4) & \(I_{tx}\) & 27.6 \\
3 & Wait next transmission & \(T_{wntx}\) & 493 & \(I_{wntx}\) & 1.2 \\
4 & Wait next reception & \(T_{wntx}\) & 16493 & \(I_{wntx}\) & 1.3 \\
5 & Reception & \(T_{rx}\) & 12690 (11) & \(I_{rx}\) & 18.5 \\
6 & Wait confirm. & & & & \\
7 & Transmission & \(T_{wctrl\_tx}\) & 1430 & \(I_{wctrl\_tx}\) & 1.2 \\
8 & Cool down & \(T_{cd}\) & 495 & \(I_{cd}\) & 1.2 \\
9 & Sleep & \(T_{sleep\_bi}\) & [0,\(T_{Period}\)) (9) & \(I_{slap}\) & \(16\times 10^{-3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: States, variables and their values for Sigfox bidirectional transaction. For a bit rate of 100 bit/s, \(T_{tx}\) can take values between 1200 ms (1-byte payload) and 2080 ms (12-byte payload). The reception duration ranges from 387 ms up to \(T_{DL\_WIN\_MAX}\) = 25 s (in US and EU regions), with an average of \(T_{rx}\) = 12.69 s. \(T_{slope\_bi}\) ranges in this study from 0 up to \(\sim\)\(T_{Period}\) (\(T_{slope\_bi}\) tends to \(T_{Period}\) for very high \(T_{Period}\) values). therefore the device remains listening during the maximum downlink window duration. Based on the three events described, \(I_{avg\_bi}\) can be computed as shown next:  
\[I_{avg\_bi}=I_{A}\cdot p_{A}+I_{B}\cdot p_{B}+I_{C}\cdot p_{C} \tag{12}\]  
where \(I_{A}\), \(I_{B}\) and \(I_{C}\) correspond to the average current consumption over a transaction period where events \(A\), \(B\) and \(C\) occur, respectively, and \(p_{A}\), \(p_{B}\), and \(p_{C}\) denote the corresponding respective probabilities. These current consumption and probability variables can be calculated as described next. \(I_{A}\) corresponds to the calculation of \(I_{avg\_bi}\) as determined by using Equations (8)-(11), since both the uplink and the downlink frame transmissions are successful in event \(A\). \(I_{B}\) can be obtained similarly to (8), although the states related with sending the uplink confirmation, i.e., states 6 and 7 in Table 3, need to be excluded from the calculation as shown next:  
\[I_{B}=\frac{1}{T_{Period}}\left(\sum_{j=1}^{N_{targets\_bi}}T_{j}\cdot I_{j}- \sum_{j=6}^{7}T_{j}\cdot I_{j}\right) \tag{13}\]  
where state 9 in event \(B\), i.e., the sleep interval in event \(B\), denoted \(T_{sleep\_bi\_B}\), is calculated by using (14) and (15):  
\[T_{sleep\_bi\_B}=T_{Period}-T_{act\_bi\_B} \tag{14}\]  
\[T_{act\_bi\_B}=T_{wu}+3\cdot T_{tx}+2\cdot T_{wmx}+T_{rx}+T_{cd} \tag{15}\]  
Finally, \(I_{C}\) can be determined on the basis of (13)-(15), but setting \(T_{rx}\) to the maximum duration of the downlink receive window, denoted \(T_{DL\_WIN\_RX\_MAX}\), since in event \(C\) the device stays in receive mode for that time. The probabilities \(p_{A}\), \(p_{B}\), and \(p_{C}\), can be calculated as follows:  
\[p_{A}=\left(1-FLR_{UL}{}^{3}\right)\cdot(1-FLR_{DL}) \tag{16}\]  
\[p_{B}=\left(1-FLR_{UL}{}^{3}\right)\cdot FLR_{DL} \tag{17}\]  
\[p_{C}=FLR_{UL}{}^{3} \tag{18}\]  
where \(FLR_{DL}\) denotes the FLR for downlink transmission. With regard to the rest of performance parameters considered in this paper, let \(T_{lifetime\_bi}\) and \(EC_{delivery\_bi}\) denote the device lifetime and the energy cost of data delivery, respectively, for a device that triggers bidirectional transactions.