# CSNOMA: Carrier Sense Non-Orthogonal Multiple Access
## 3 System Model
### Network Model  
Assume that a wireless network where transmitters are randomly located according to a 2-dimensional homogeneous Poisson point process (PPP) with intensity \(\lambda\). All distance units are in meters. Each transmitter \(i\) has a dedicated receiver \(k(i)\) that is situated at a distance \(r_{t}\) from the transmitter with random direction. Figure 1 shows a snapshot of the wireless network. According to the displacement theorem [40], receivers also follow a homogeneous PPP. All transmitters have infinite backlogged data to transmit. A common channel is shared by the entire network. Thus, all concurrent transmissions act as interference. All transmitters emit the packet with constant transmission power \(P\). The received signal power at a node \(j\) from a node \(i\) is denoted by \(g_{i,j}d_{i,j}^{-\alpha}P\), where \(g_{i,j}\) is an exponential random variable with unit mean and reflects the effect of Rayleigh fading. The value \(d_{i,j}\) denotes the distance between nodes \(i\) and \(j\) with the path loss exponent \(\alpha\). We assume that the time is slotted and synchronized so that transmissions start with the beginning of time slot and continue during the slot length. The transmitter/receiver pair can vary over the time, but it is assumed that the topology and channel gains are fixed during a slot duration. We assume that the network is interference-limited and the noise at receiver can be ignored. An instantaneous signal-to-interference-ratio (SIR) of a typical receiver, \(\gamma_{k(i)}\), which is a measure of the impact of the interference, is given by  
\[\gamma_{k(i)}=\frac{g_{i,k(i)}r_{t}^{-\alpha}P}{\sum\limits_{u\in T_{i}}g_{u,k (i)}d_{u,k(i)}^{-\alpha}P}, \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(\alpha\) & path loss exponent, \(\alpha>2\) \\ \hline \(\beta\) & target SIR threshold \(\beta>1\) \\ \hline \(r_{t}\) & communication distance \\ \hline \(r_{s}\) & carrier sensing distance \\ \hline \(r_{a}\) & strong interferer distance \\ \hline \(r_{c}\) & maximum cancellable distance \\ \hline \(b\left(o,r\right)\) & a ball of radius \(r\) centered at origin \\ \hline \(a\left(o,r_{1},r_{2}\right)\) & an annulus between \(r_{1}\) and \(r_{2}\) \\ \hline \(I_{r_{1}}^{r_{2}}\) & interference at typical receiver from \(a\left(o,r_{1},r_{2}\right)\), \(I_{r_{1}}=I_{r_{1}}^{\alpha}\) \\ \hline \(A(x)\) & area of region \(x\) \\ \hline \(N(x)\) & number of points in region \(x\) \\ \hline \(\lambda\) & node density \\ \hline \(\lambda_{t}\) & transmitting node density \\ \hline \(\lambda_{b}\) & additional transmitting node density \\ \hline \(\Phi\) & a Poisson point process of intensity \(\lambda\) \\ \hline \(\left\{X_{i}\right\}\) & a set of points in \(\Phi\) \\ \hline \(\left|X_{i}\right|\) & euclidian distance between \(X_{i}\) and origin \\ \hline \(p_{s}\) & transmission success probability \\ \hline \(p_{s}^{\text{IC}-k}\) & transmission success probability with \(k\)-times interference cancellation \\ \hline \hline \end{tabular}
\end{table}
Table 2: Key mathematical notations.  
where \(\mathcal{T}_{i}\) denotes the set of simultaneously transmitting nodes with the node \(i\). The MAC scheme confirms the set \(\mathcal{T}_{i}\). For the ALOHA-like random access scheme, the spatial pattern can be successfully modeled by PPP [4]. That is, \(\mathcal{T}_{i}\) is an independently thinned point process of the original node set.  
For a given target SIR \(\beta\), the transmission is successful if the SIR at the receiver \(k(i)\) is greater than or equal to \(\beta\). The instantaneous data rate of the transmitter \(i\) is a function of the target SIR. We use the [PERSON]'s formula \(\log_{2}\left(1+\beta\right)\) with a unit bandwidth. Assume that all receivers are able to fulfill the successive interference cancellation (SIC). In next subsection, we will describe the interference cancellation model.