# Power-Efficient Random Access Design for Machine Type Communication
## 4 Link Budget Analysis and Simulation Results
### Simulation Results  
Figure 9 shows the decoding performance of the LTE-\(n\)MTC random access request signal for various coverage classes. The transmission time intervals (TTIs) for the LTE-\(n\)MTC access request \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & **Coverage Class (\(\alpha\))** & & **4 (140 dB)** & **3 (145 dB)** & **2 (150 dB)** & **1 (155 dB)** \\ \hline \multirow{4}{*}{\(\Gamma\) **(dB)**} & **LTE-nMTC** & Tx1 & 0 & 0 & 0 & 0 \\ \cline{2-7}  & & Tx 1 & \(-11\) & \(-16\) & \(-21\) & \(-26\) \\  & **LTE-eMTC** & Tx (2, 3) & \(-4\) & \(-9\) & \(-14\) & \(-19\) \\  & & Rx (1, 2) & 1 & \(-4\) & \(-9\) & \(-14\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Random access channel operating SNRs under the LTE-nMTC design and LTE-eMTC. signal to close the various coverage class links are summarized in Table 5 for reference. The TTI is the minimum transmission length determined such that the SNR at 1% packet error rate (PER) is less than \(\Gamma\). From Table 5, we can see that the preamble transmission time in LTE-eMTC (Tx1) is about one third of our proposed LTE-nMTC random access message design. This is a matter of course that LTE-eMTC preamble only contains partial of the random access message, i.e., the round trip delay and the absence of a device without detail information. In order to deal with the potential collision, the device has to transmit the contention resolution key (Tx2) again after it detects the RAR. Therefore, the transmission time increases and exceeds our proposed design. We can see that the device total transmission time in the LTE-eMTC random access procedure is at least 1.5 times the transmission time of our proposed random access message. Moreover, the random access procedure in LTE-eMTC is divided into four steps, including uplink and downlink, which will increase the access delay comparing to our one-step random access message design. Let's consider coverage class 1 as an example. After considering the collision probability  
\[p_{c}=1-\Big{(}1-\frac{w}{LW}\Big{)}^{N-1}=9.6\% \tag{25}\]  
and decoding error (\(p_{\rm e}=1\%\)), the average transmission time for LTE-\(n\)MTC can be expressed as  
\[t_{\rm Tx}=\frac{T_{\rm Tx}}{(1-p_{\rm e})(1-p_{\rm c})}=\frac{0.2578\;{\rm s} }{(1-1\%)(1-9.6\%)}\approx 0.288\;{\rm s}, \tag{26}\]  
where \(T_{\rm Tx}\) is the TTI for LTE-\(n\)MTC coverage class 1, and the 9% collision probability is obtained from (15). Hence, total power consumption per random access according to (1) is  
\[C_{\rm LTE\text{-nMTC}}=t_{\rm Tx}I_{\rm Tx}=0.288\;{\rm s}\;\times\;260\;{\rm mA }\approx 0.021\;{\rm mAh} \tag{27}\]  
or 0.00042% two AA battery capacity (\(C_{\rm AA}=5000\;{\rm mAh}\)). \begin{table}
\begin{tabular}{c c c c c c c} \hline \multicolumn{2}{c}{**Coverage Class (a)**} & \multicolumn{2}{c}{**4 (140 dB)**} & \multicolumn{2}{c}{**3 (145 dB)**} & \multicolumn{2}{c}{**2 (150 dB)**} & \multicolumn{2}{c}{**1 (155 dB)**} \\ \hline \multirow{6}{*}{**TTI (seq)**} & **LTE=nMTC** & Tx1 & 0.0074 & 0.0221 & 0.09945 & 0.2578 \\  & **LTE=nMTC** & (5 symbol groups) & (12 symbol groups) & (54 symbol groups) & (140 symbol groups) \\ \cline{1-1}  & **LTE=eMTC** & Tx1 & 0.002 & 0.007 & 0.03 & 0.08 \\ \cline{1-1}  & **RX1** & 0.116 & 0.274 & 0.638 & 1.564 \\ \cline{1-1}  & **LTE=eMTC** & Tx2 & 0.008 & 0.045 & 0.2 & 0.85 \\ \cline{1-1}  & **RX2** & 0.011 & 0.034 & 0.138 & 0.574 \\ \cline{1-1}  & **TX3** & 0.001 & 0.008 & 0.06 & 0.13 \\ \hline \end{tabular}
\end{table} 
Table 5: Transmission time intervals of LTE-nMTC and LTE-eMTC random access message. Figure 9: PER (packet error rate) of LTE-nMTC random access under different coverage classes. For LTE-eMTC, it is approximately (ignoring the collision, miss of preamble detection, and random access response and contention resolution decoding error)  
\[