# Exploiting the Capture Effect to Enhance RACH Performance in Cellular-Based M2M Communications
## 4 Analysis Model
### Average Energy Consumption  
#### 4.4.1 Energy Consumption Model  
An average energy consumption model for the devices needs to be defined to analyze the energy consumption of the proposed scheme. For the energy-consumption parameters, we have partially referenced the model used in [10]. In [10], the authors assumed that an RA collision is detected by the eNB during the Msg1 reception, whereas we assumed that a collision is detected when the eNB cannot decode Msg3. Therefore, we refer to and modify the energy consumption model defined in [10]. In this study, the description of how parameters are defined is omitted. See [10] for a detailed description of the parameters. Instead, we provide simplified description in Figure 4. Let \(P_{1},P_{2}\), and \(P_{3}\) denote the power consumption for waiting RA slot and performing back-off, receiving messages from the eNB, and transmission, respectively. Here, for analytic convenience, we assume that all transmission powers are equal, i.e., \(P_{3}=P_{msg1}=P_{msg3}\) as in Equation (5). Then, the average energy consumption for failed RA and successful RA events can be modeled as follows.  
1. The energy consumed during a failed RA event is equal to that consumed in the following steps: waiting for the first RA slots(\((\tau_{RA\_REP}/2)P_{1}\)), transmitting Msg1 (\(P_{3}\)), waiting for the RAR window (\(\tau_{RAR}P_{1}\)), waiting for the Msg2 (\((W_{RAR}/2)P_{2}\)), receiving the Msg2 (\(\tau_{msg2}P_{2}\)), processing time for Msg3 (\(T_{proc}P_{2}\)), transmitting Msg3 (\(P_{3}\)), and expiring the contention-resolution timer (\(W_{cont\_sol}P_{2}\)).
2. The energy consumed during a successful RA event is equal to that consumed during the following steps. From transmitting Msg1 to receiving Msg2, the power consumed by the successful RA follows that consumed by the failed RA procedure and equals (\((\tau_{RA\_REP}/2)P_{1}+P_{3}+\tau_{RAR}P_{1}+(W_{RAR}/2)P_{2}+\tau_{msg2}P_{ 2}+T_{proc}P_{2}+P_{3}\)). In addition, the following steps are: for waiting Msg4 (\((W_{cont\_sol}/2)P_{2}\)), the power consumed when receiving Msg4 (\(\tau_{msg4}P_{2}\)), processing time for acknowledgement (ACK) (\(T_{proc2}P_{2}\)), and the power consumed by transmitting the ACK of Msg4 (\(P_{3}\)).  
Note that 1/2 terms denote the average time values. Parameter values in the above model are described in Table 2.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Notation** & **Definition** & **Values** \\ \hline \multicolumn{3}{c}{**Parameters Relative to RA Procedure**} \\ \hline \(V\) & the number of preambles & 20, 40, 60 \\ \(N_{max}\) & maximum number of RA trial & 4, 8, 12 \\ \(X^{cell}\) & radius of single cell & 1500 m \\ \(X_{u}\) & The distribution of devices & uniform \\ \hline \multicolumn{3}{c}{**Parameters Relative to Power Model**} \\ \hline \(PIRTP\) & Preamble initial receive target power & \(-104\) dBm \\ \(\mathcal{P}_{Cmax}\) & Maximum transmit power & 23 dBm \\ \(PRS\) & Power ramping step & 1, 2, 3, 4 dB \\ \(delta\) & offset value & 0 dBm \\ \(PL_{0}\) & reference path-ols & 127 dBm \\ \(x_{0}\) & reference point distance & 1500 m \\ \(\alpha\) & path-loss exponent & 2 \\ \hline \multicolumn{3}{c}{**Parameters Relative to Energy Consumption Model [10]**} \\ \hline \(\tau_{RA,REP}\) & periodicity of RACH & 5 ms \\ \(\tau_{RAR}\) & processing delay & 2 ms \\ \(W_{PAR}\) & RAR window size & 5 ms \\ \(\tau_{msg2},\tau_{msg4}\) & requiring time for receiving \(\text{Msg2}/\text{Msg4}\) & 1 ms \\ \(\mathcal{W}_{com,sol}\) & size of Contention-resolution timer & 48 ms \\ \(\mathcal{P}_{1}\) & power consumption when the device is waiting & \(-37\) dBm \\ \(\mathcal{P}_{2}\) & power consumption when the device is receiving & \(-22\) dBm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation parameters.  
Figure 4: The power consumption model for **(a)** single failed RA event (**b**) single successful RA event.