# Exploiting the Capture Effect to Enhance RACH Performance in Cellular-Based M2M Communications
## 4 Analysis Model
### System Model  
We consider a single cell with radius \(X^{cell}\). Let \(\mathcal{N}\) and \(N\) denote the set of devices sending Msg1 to participate in RA at the same RA slot, and the size of \(\mathcal{N}\), respectively. Devices in \(\mathcal{N}\) are uniformly distributed in a cell, hence, \(X_{u}=x\) which denotes the distance between the eNB and \(u\), follows a uniform distribution. A device \(u\in\mathcal{N}\), which failed the RA procedure \((t-1)\) times and participates in the \(t^{th}\) RA at this RA slot has a trial number \(T_{u}=t\). Similarly, \(R_{u}=r\) denotes the ramping number of a device defined by Equation (5) at the current RA slot. As seen in Equation (5), the Msg3 transmission power of \(u\) is upper-bounded by \(P_{Cmax}\) as \(P_{target}+(r-1)PRS+PL\leq P_{Cmax}\). Note that \(PL\) is related to \(x\). For simplicity, \(PL\) can be defined using the log-distance path loss model without fading as \begin{table}
\begin{tabular}{c l} \hline \hline
**Notation** & **Description** \\ \hline \(\mathcal{N}/N\) & a set of devices participate RA/the size of \(\mathcal{N}\) \\ \(M_{v}\) & the number of devices that select preamble \(v\) among terminals excluding device \(u\) \\ \(V_{u}\) & a preamble selected by device \(u\) \\ \(T_{u}\) & the RA trial number of device \(u\) \\ \(R_{u}\) & the ramping number of device \(u\) \\ \(S_{u}\) & the event that device \(u\) succeeds in the RA procedure \\ \(P_{u}(P_{u}{}^{rx})\) & transmission (received) power of (from) device \(u\) \\ \(K_{max}(x)\) & the maximum ramping number of a device located at a distance \(x\) from eNB \\ \(PRS\) & Power ramping step \\ \(X_{u}\) & distance from eNB to device \(u\) \\ \(Z_{u}\) & the zone where device \(u\) belongs \\ \(N_{T=t}^{k}(N_{R=t}^{k})\) & a set of devices located within _zone_\(k\) with trial number \(t\) (with ramping number \(r\)) \\ \(\delta_{F,k}(t)\) (\(\delta_{S,k}(t)\)) & the average energy consumption of device located within _zone_\(k\) with the trial number \(t\), \\  & having single failed (successful) RA events, \\ \(w_{S}(t,k)\) (\(w_{F}(k)\)) & the energy consumption of the device that succeeded (failed) in the connection \\ \(W\) & the energy consumption of device which is completed the RA procedure \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Notation summary. \(PL(x)=PL_{0}+10\alpha\log(\frac{x}{x_{0}})\), where \(PL_{0}\) and \(x_{0}\) are the reference-path loss and distance, respectively. From the upper-bound condition and the path-loss model, the upper-bound of the ramping number of the device located at a distance \(x\) from the eNB is as follows:  
\[r\leq\max\left(1,\left\lfloor\frac{P_{Cmax}-PL(x)-P_{target}}{PRS}+1\right\rfloor \right)=K_{max}(x). \tag{6}\]  
**Definition 1**. : \(K_{max}(x)\) _is the maximum allowable ramping number of a device located at a distance \(x\) from the eNB. Then, the ramping number of \(u\), \(R_{u}\), can be defined as follows:_  
\[R_{u}=\min\left(K_{max}(x_{u}),T_{u}\right). \tag{7}\]  
However, Equation (7) is hard to deal with. Hence, we introduce the concept of _zones_ for tractability as shown in Figure 1.  
**Definition 2**. : _Zone \(k\) is the area in which devices with \(K_{max}=k\) are located. In addition, \(Z_{u}\) is the zone in which device \(u\) belongs. Thus, \(Z_{u}=k\) means that a device \(u\) is in zone \(k\) (also, \(u\) can only increase its ramping number up to \(k\)). Therefore, zone \(N_{max}\) is the closet to the eNB, whereas zone 1 is the farthest from the eNB._  
If the distribution of \(X_{u}\) is known (i.e., the distribution of devices in a cell), the probability that device \(u\) belongs to _zone \(k\)_, \(\mathbb{P}(Z_{u}=k)\), is (see Figure 1)  
\[\mathbb{P}(Z_{u}=k)=\int_{b_{k+1}}^{b_{k}}\mathbb{P}(X_{u}=x)dx,\quad k\in[1, N_{max}]. \tag{8}\]  
Note that \(K_{max}(x)\) is a decreasing function for \(x\). Therefore, there exists a boundary distance \(b_{k}=K_{max}^{-1}(k),\ k=1,..,N_{max}\), where \(b_{N_{max}+1}=0\). **Definition 3**.