# Power Efficient Random Access for Massive NB-IoT Connectivity
## 4 Average Power Consumption in Existing NB-IoT RAP
### If the Device Starts RAP in CE0  
In the existing NB-IoT RAP, the sequence of RAP reattempts as well as the number of designated reattempts in each of the CE levels are fixed. We consider that \(L_{0}\) and \(L_{1}\) are, respectively, the maximum number of attempts that the device performs in CE0 and CE1 before it can move to the next higher CE level for RAP reattempts. \(\Delta P_{0}\), \(\Delta P_{1}\) and \(\Delta P_{2}\) delineate the power ramping at each attempt in CE0, CE1 and CE2, respectively. The power consumption of the device would not only depend upon the total number of attempts \(n\) but also on its relation with \(L_{0}\) and \(L_{1}\). 1. For \(n\leq L_{0}\): For the existing NB-IoT RAP, if the device starts RAP in CE0 and succeeds in CE0 itself (i.e., \(n\leq L_{0}\)), then the power consumed (\(PE_{0}[n\leq L_{0}]\)) by this device can be expressed as \[P_{E_{0}}[n\leq L_{0}]=\sum_{u=1}^{n}R_{0}(P_{0}+(u-1)\Delta P_{0})\] (12) In the existing NB-IoT RAP, the device first reattempts RAP in the same channel conditions (i.e., \(q_{0}\) for CE0) for \(L_{0}\) attempts before moving to the next higher CE levels if the failure persists. The probability that the device that starts RAP in CE0 and succeeds in CE0 itself in \(n\leq L_{0}\) reattempts, \(\mathbb{P}_{r}(S_{00})(n)\), can be expressed as \[\mathbb{P}_{r}(S_{00})\{n=1\}=1-\mathbb{P}_{r}(C))\] (13) \[\mathbb{P}_{r}(S_{00})\{1<n\leq L_{0}\}=\mathbb{P}_{r}(C)\bigg{(}1-(1 -\mathbb{P}_{r}(C))q_{0}\bigg{)}^{(n-2)}\bigg{(}(1-\mathbb{P}_{r}(C))q_{0}\bigg{)}\] (14) It is noteworthy that unlike PE-RAP, in \(\mathbb{P}_{r}(S_{00})\{1<n\leq L_{0}\}\) evaluations, both probabilities due to collision \(\mathbb{P}_{r}(C)\) and channel condition \(q_{0}\) are considered. At any RAP reattempt the failure could have occurred due to collision or due to poor channel conditions. However, for the selection of its first attempt, the device measured RSRP and hence for \(\mathbb{P}_{r}(S_{00})\{n=1\}\) the failure is considered only due to collision. The probability that a device that starts RAP in CE0 but is not able to succeed for the designated \(L_{0}\) reattempts in CE0 can be expressed as \[\mathbb{P}_{r}(F_{00})=\mathbb{P}_{r}(C)\bigg{(}1-(1-\mathbb{P}_{r}(C))q_{0} \bigg{)}^{(L_{0}-1)}\] (15) Subsequently, if the RAP failure persists then the device moves on to CE1 for RAP reattempts. 2. For \(L_{0}<n\leq L_{0}+L_{1}\): If the device starts RAP in CE0 and succeeds in \(L_{0}<n\leq L_{0}+L_{1}\) attempts, then power consumed by this device can be expressed as \[P_{E_{0}}[L_{0}<n\leq L_{0}+L_{1}]=R_{0}P_{0}L_{0}+R_{0}\Delta P_{0}\frac{(L_{ 0}-1)(L_{0})}{2}+\sum_{v=1}^{n-L_{0}}R_{1}(P_{1}+(v-1)\Delta P_{1})\] (16) The probability that the device succeeds in CE1, having started in CE0, in \(n=L_{0}+1,L_{0}+2\ldots L_{0}+L_{1}\) attempts can be expressed as in Equation (17). \[\mathbb{P}_{r}(S_{01})\{L_{0}<n\leq L_{1}\}=\mathbb{P}_{F_{00}}\bigg{(}1-((1- \mathbb{P}_{r}(C))(q_{0}+q_{1}))\bigg{)}^{n-L_{0}-1}\bigg{(}(1-\mathbb{P}_{r} (C))(q_{0}+q_{1})\bigg{)}\] (17) We can express \(\mathbb{P}_{r}(F_{01})\) as the probability that a device that starts RAP in CE0 but is not able to succeed for the designated \(L_{0}\) reattempts in CE0 as well as the configured \(L_{1}\) reattempts in CE1. It can be obtained as \[\mathbb{P}_{r}(F_{01})=\mathbb{P}_{F_{00}}\bigg{(}1-((1-\mathbb{P}_{r}(C))(q_{ 0}+q_{1}))\bigg{)}^{L_{1}}\] (18) The RAP failure in Equation (18) considers collision probability (\(\mathbb{P}_{r}(C)\)) as well as channel probabilities \(q_{0}\) & \(q_{1}\). Finally the device tries reattempts in CE2. 3.