# Exploiting the Capture Effect to Enhance RACH Performance in Cellular-Based M2M Communications
## 4 Analysis Model
### RA Throughput
#### 4.2.1 One-Shot RA Throughput Analysis  
Devices in \(\mathcal{N}\) send an Msg1 that is randomly selected from \(V\) preambles. Focusing on a certain device, \(u\in\mathcal{N}\) with \(T_{u}=t\) and \(Z_{u}=k\), and, therefore, \(R_{u}=r=min(t,k)\). Let \(V_{u}\) denote the preamble selected by device \(u\), and \(M_{v}\) denote the number of devices that select preamble \(v\) among \(\mathcal{N}\) excluding device \(u\). Then, the probability that device \(u\) and \(m\) devices in set \(\mathcal{N}\backslash\{u\}\) select preamble \(v\) can be expressed as follows:  
\[\mathbb{P}(M_{v}=m,V_{u}=v\mid N)=\frac{1}{V}{N-1\choose m}{\left(\frac{1}{V }\right)^{m}}{\left(1-\frac{1}{V}\right)^{N-m-1}},m\in[0,N-1],v\in[1,V]. \tag{11}\]  
Next, we derive the probability that device \(u\) succeeds in a RA (i.e., the Msg3 from \(u\) is successfully decoded by the eNB), which we denote as the event \(S_{u}\). Let the random variable \(P_{i}^{rx}\) denote the received signal power from device \(i\) at the eNB. Then, the event \((S_{u})\) occurs, even if \(u\) collides with \(m\) devices, when \(P_{u}^{rx}\) meets the capture condition defined in Equation (1). Hence, the conditional probability of \((S_{u})\) is  
\[\mathbb{P}(S_{u}\mid N,M_{v}=m,V_{u}=v,P_{u}^{rx}=p)=\mathbb{P}\left(\beta\sum _{i\in M_{v}}P_{i}^{rx}\leq p\mid N,m,v,P_{u}^{rx}=p\right). \tag{12}\]  
Since we assume that power control with full compensation of path loss is performed perfectly, \(P_{i}^{rx}\) is a function of the ramping number as in Equation (5). Therefore, the probability density function (PDF) of \(P_{i}^{rx}\) follows the distribution of the ramping number \((R_{i})\) of the RA-participating devices as Equation (10). In other words, the PDF of \(P_{i}^{rx}\) is  
\[\mathbb{P}\left(P_{i}^{rx}=P^{rx}(r)\mid N\right)=\mathbb{P}(R_{i}=r|N). \tag{13}\]Hence, the conditional probability in Equation (12) is redefined as \(R_{u}\):  
\[\begin{split}&\mathbb{P}\left(S_{u}|N,M_{v}=m,V_{u}=v,P_{u}^{rx}=P^{rx} (r)\right)\\ &=\mathbb{P}(S_{u}|N,M_{v}=m,V_{u}=v,R_{u}=r)\\ &=\mathbb{P}\left(\beta\sum_{i\in M_{v}}P_{i}^{rx}\leq P^{rx}(r) \bigm{|}N,m,R_{u}=r\right)\\ &=CP_{N,m,r},\end{split} \tag{14}\]  
where \(CP_{N,m,r}\) is the Msg3 capture probability, which means the probability that device \(u\) whose ramping number is \(r\) is captured by the eNB when \(m\) devices among \(N\) devices collide with \(u\). Note that \(CP_{N,0,r}\) is the capture probability when \(u\) selects the singleton preamble, and therefore, \(CP_{N,0,r}=1\) for any \(N\) and \(r\). Because the PDF of \(P_{i}^{rx}\) is defined, and is independent and identically distributed, _i.i.d._, Equation (14) can be solved using _the m-fold convolution_ of the PDF of \(P_{i}^{rx}\) for given values of \(N,m\), and \(r\)[16].  
Then, the following conditional probability can be obtained from Equations (8), (10) and (14), and _the chain rule_:  
\[\begin{split}&\mathbb{P}(S_{u},R_{u}=r,Z_{u}=k|N,M_{v}=m,V_{u}=v)\\  
Consequently, we can obtain the one-shot RA throughput when \(N\) devices participate in RA as  
\[E[S|N] =N\mathbb{P}(S_{u}|N)\] \[=N\sum_{m=0}^{N-1}\binom{N-1}{m}\left(\frac{1}{V}\right)^{m}\left(1 -\frac{1}{V}\right)^{N-m-1}\!\!CP_{N,m}. \tag{18}\]