# Power Efficient Random Access for Massive NB-IoT Connectivity
## 3 Collision--CE Level Based RAP (PE-RAP)
### Average Power Consumption in PE-RAP  
As there are three CE levels, \(q_{0}\), \(q_{1}\) and \(q_{2}\), respectively, represent the probabilities of channel conditions for CE0, CE1 and CE2 such that \(q_{0}+q_{1}+q_{2}=1\). The device that fails RAP can re-select any of the CE levels for the RAP reattempt based on the probabilities \(q_{0},q_{1}\) and \(q_{2}\). Let the device succeed in its RAP after \(n\) number of reattempts where, unlike existing system, each RAP trial is independent of the previous attempt. \(\mathbb{P}_{r}(C)\) represents the collision probability such that a collision results in a failed RAP attempt. Since the choice of preamble by any active NB-IoT device is random and independent of the other devices, more than one active device may end up selecting the same preamble which would result in the collision. Then \(\mathbb{P}_{r}\{n=1\}=(1-\mathbb{P}_{r}(C))\) gives the probability that the RAP is successful in one attempt. Thus, the probability of an RAP being a success in \(n\) attempts can be expressed as  
\[\mathbb{P}_{r}\{n\}=\mathbb{P}_{r}(C)^{n-1}(1-\mathbb{P}_{r}(C)) \tag{1}\]  
Of total \(n\) RAP reattempts to achieve success, the device could have performed any of its RAP attempts in any of the CE levels. Unlike the existing NB-IoT RAP the order and count of reattempts in different CE levels is not crucial in the proposed system. Let \(i\) be the number of RAP reattempts that the device performs in CE0 and \(j\) be the RAP reattempts in CE1. The remaining out of \(n\) are performed in CE2. In each CE level the number of repetitions are different. The required preamble repetitions are small in level CE0 and are greater for level CE2. If \(R_{0}\), \(R_{1}\) and \(R_{2}\), respectively, delineate the number of repetitions in CE0, CE1 and CE2, then \(R_{0}<R_{1}<R_{2}\). A high number of repetitions translates into more power consumption. The power consumption is different in different CE levels. If \(P_{0}\), \(P_{1}\) and \(P_{2}\), respectively, delineate the power levels for each repetition in CE0, CE1 and CE2, then the average power consumed by the device over \(n\) reattempts can be expressed as  
\[P_{avg}[n]=\sum_{i=0}^{n}\sum_{j=0}^{n-i}\binom{n}{i}\binom{n-i}{j}\left((iR_ {0}P_{0}+jR_{1}P_{1}+(n-i-j)R_{2}P_{2})\right)q_{0}^{i}(1-q_{0}-q_{2})^{i}(q _{2})^{n-i-j} \tag{2}\] \begin{table}
\begin{tabular}{p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Parameter** & **Symbol** & **Description** \\ \hline Number of Repetitions & \(R_{0},R_{1},R_{2}\) & \(R_{0},R_{1},R_{2}\), respectively to delineate the number of repetitions in CE0, CE1 and CE2 \\ \hline Probability of Device belonging to a CE level & \(q_{0}\), \(q_{1}\), \(q_{2}\) & \(q_{0}\), \(q_{1}\) and \(q_{2}\) respectively gives the probability of channel condition for CE0, CE1 and CE2 such that \(q_{0}\)+ \(q_{1}\)+\(q_{2}=1\) \\ \hline Number of Devices & \(N\) & Number of NB-IoT devices \\ \hline Preambles/Subcarriers & \(M_{s}\) & Number of subcarriers (preambles) \\ \hline Collision Probability & \(\mathbb{P}_{r}(C)\) & Probability of collision of a device \\ \hline ACB Factor & \(\rho\) & Access Class Barring (ACB) factor transmitted by network \\ \hline Number of Devices Passes ACB check & \(m_{k}\) & Devices Passes ACB Check and transmit during this time slot \\ \hline Arrival & B(a, b) & Beta function with parameters a \& b \\ \hline Power Level & \(P_{0},P_{1}\), \(P_{2}\) & Power levels of CE0, CE1 and CE2 \\ \hline Number of attempts & \(L_{0}\), \(L_{1}\) & Maximum number of attempts that a device performs in CE0 and CE1 \\ \hline Power Consumed & \(\mathbb{P}_{r}\{n\}\) & Power consumed by device after \(n\) attempts \\ \hline Probability of RAP & \(P_{avg}[n]\) & Probability of an RAP being a success in n attempts \\ \hline Average Power Spent & \(P_{E_{avg}}\), \(P_{E_{avg}}\), \(P_{E_{avg}}\) & Average power spent in CE0, CE1 and CE2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters used for power efficient random access procedure PE-RAP analysis.