## 2 System Model
### Random Channel Access Model  
Communication occurs over a shared channel where multiple access is regulated by a FSA protocol. Referring to assumption \(A_{5}\), nodes always have backlogged data that must be delivered. To start the communication session, the illuminator triggers the beginning of a frame, which has a duration of \(L\) time slots, by broadcasting a _beacon_ signal. The _beacon_ is also used to communicate the frame length to the backscattering nodes. Upon receiving the beacon, the backscattering nodes randomly select a slot within the frame and transmit data in the allotted slot. This process may results in idle slots, where no node transmits, single node transmissions or multiple transmissions, where interference may cause loss of information. The dynamic of the medium access strategy is illustrated in Figure 2 and formally described in what follows. In order for the \(k\)-th node to be able to transmit data in a frame, it must correctly receive the _beacon_. This occurs if the received power \(\mathsf{P}_{f,k}\) is above the node's sensitivity threshold \(\gamma_{h}\). Therefore, due to the fading affecting the forward links, only a subset \(\mathcal{A}\) of nodes in the network can effectively communicate in a given frame. Denoting by \(\mathcal{P}_{\mathcal{A}}\) the event where the nodes in \(\mathcal{A}\) successfully receive the _beacon_ signal, that is \(\mathcal{P}_{\mathcal{A}}=\{\mathsf{P}_{f,k}\textgreater\gamma_{h},\forall k \in\mathcal{A}\}\), the probability that the set of nodes \(\mathcal{A}\) is able to communicate in a frame can be expressed as  
\[\Pr(\mathcal{P}_{\mathcal{A}}) =\prod_{i\in\mathcal{A}}\Pr\Bigl{(}\mathsf{P}_{f,i}\textgreater \gamma_{h}\Bigr{)}\prod_{j\in\mathcal{K}\setminus\mathcal{A}}\Bigl{[}1-\Pr \Bigl{(}\mathsf{P}_{f,j}\textgreater\gamma_{h}\Bigr{)}\Bigr{]}\] \[=\prod_{i\in\mathcal{A}}\Pr\Bigl{(}\mathsf{S}_{f,i}\textgreater \ln\frac{\gamma_{h}}{\alpha_{i}P_{\text{tx}}}\Bigr{)}\prod_{j\in\mathcal{K} \setminus\mathcal{A}}\Bigl{[}1-\Pr\Bigl{(}\mathsf{S}_{f,i}\textgreater \ln\frac{\gamma_{h}}{\alpha_{j}P_{\text{tx}}}\Bigr{)}\Bigr{]}\] \[=\prod_{i\in\mathcal{A}}[1-F_{\mathsf{S}}(\hat{\gamma}_{h,i})] \prod_{j\in\mathcal{K}\setminus\mathcal{A}}F_{\mathsf{S}}(\hat{\gamma}_{h,j}). \tag{4}\]  
where \(\mathcal{K}\) is the set of backscattering nodes, \(\hat{\gamma}_{h,i}=\ln\frac{\gamma_{h}}{\alpha_{i}P_{\text{tx}}}\) and \(F_{\mathsf{S}}(x)\) is the Cumulative Distribution Function (CDF) of a zero-mean Gaussian random variable with variance \(\sigma^{2}\). Due to power outage, the nodes in \(\mathcal{K}\setminus\mathcal{A}\) are not able to communicate until a new beacon is issued by the illuminator. Within a frame, the nodes in \(\mathcal{A}\) independently initializes a slot counter, whose value is sampled at random from the interval \([0,L-1]\) and decremented in each slot. Nodes associated with a null counter gain access to the channel and start transmitting. Denoting by \(\mathcal{I}\subset\mathcal{A}\) the set of transmitting nodes in a slot, the following mutually exclusive outcomes can be experimented on the receivers' side:  
1. Idle Slot: \(|\mathcal{I}|=0\), that corresponds to the case where no node has a null slot counter, and thus no transmission occurs;
2. Single transmission slot: \(|\mathcal{I}|=1\), that corresponds to the event where only one node has a slot counter equal to zero. In this case, the received signal is correctly detected by the \(m\)-th receiver if the backscattered power \(\mathsf{P}_{b,k}^{(m)}\) is larger than the receiver sensitivity threshold, which we denote as \(\gamma_{d}\).