## 2 System Model
### Random Channel Access Model
Then, conditioned to the event \(\mathcal{P}_{\mathcal{A}}\), the probability for the single transmitted packet to be successfully detected can be expressed as \[\Pr(\mathsf{P}_{b,k}^{(m)}\textgreater\gamma_{d}|\mathcal{P}_{ \mathcal{A}}) =\Pr(\mathsf{P}_{b,k}^{(m)}\textgreater\gamma_{d}|\mathsf{P}_{f,k} \textgreater\gamma_{h})\] \[=\Pr(\mathsf{P}_{b,k}^{(m)}\textgreater\gamma_{d}|\mathsf{P}_{f,k }\textgreater\gamma_{h})\] \[=\frac{\Pr(\mathsf{P}_{b,k}^{(m)}\textgreater\gamma_{d},\mathsf{ P}_{f,k}\textgreater\gamma_{h})}{\Pr(\mathsf{P}_{f,k}\textgreater\gamma_{h})}\] \[=\frac{\Pr(\mathsf{P}_{b,k}^{(m)}\textgreater\gamma_{d},\mathsf{ S}_{f,k}\where we exploited the independence among fading components. Then, it may happen that the power at the \(m\)-th receiver input is below the receiver sensitivity threshold, thus yielding a data packet loss.
3. Multiple transmission slot: \(|\mathcal{I}|>1\), that corresponds to the case where multiple nodes gain access to the channel in the same slot. In this case, the received backscattered signal at the generic receiver consists of the superimposition of multiple interfering transmissions. This event is typically interpreted as a collision, nevertheless, due to the so-called _capture effect_, a node's data packet can be successfully detected at the \(m\)-th receiver despite the interference. A successful packet detection can occur when the backscattered power from a node is sufficiently large if compared to the power of the interfering signals. More precisely, this occurs when the SIR measured at a given receiver for a given node's packet is larger than the so-called SIR threshold (or _capture ratio_), which we denote as \(\gamma_{\text{SIR}}\). Formally, by defining the SIR for the \(k\)-th tag at the \(m\)-th receiver as \[\text{SIR}_{k}^{(m)}{=}\frac{\text{P}_{b,k}^{(m)}}{\sum_{j\in \mathcal{I}}^{j\
eq k}\text{P}_{b,j}^{(m)}},\] (6) the \(k\)-th nodes's packet is successfully detected at the \(m\)-th receiver with probability \[p_{c,k}^{(m)}=\Pr(\text{SIR}_{k}^{(m)}{>}\gamma_{\text{SIR}}|\mathcal{P}_{ \mathcal{A}}),\] (7) then, the probability of having a successful packet detection at the \(m\)-th receiver is given as \[p_{\text{succ}}^{(m)}=\sum_{k\in\mathcal{I}}p_{c,k}^{(m)}.\] (8) A collision occurs when the SIR associated to all the replying nodes is below the SIR threshold, and the collision probability at the \(m\)-th receiver can be trivially determined as \[p_{\text{coll}}^{(m)}{=}1-p_{\text{succ}}^{(m)}.\] (9) According to assumption \(A_{6}\), if a collision occurs, the data packets associated to all interfering nodes are dropped with consequent loss of information. The characterization of the multiple access interference and the derivation of the probability in Equation (7) are deferred to Section 3. To measure the communication performance of the considered system we define the Frame Success Rate (FSR) as the number of successfully delivered packet normalized by the frame length \(L\). More precisely, conditioned on a specific set of active nodes in a frame, the FSR can be expressed as  
\[=\prod_{i\in\mathcal{A}}\Pr\Bigl{(}\mathsf{S}_{f,i}\textgreater \ln\frac{\gamma_{h}}{\alpha_{i}P_{\text{tx}}}\Bigr{)}\prod_{j\in\mathcal{K} \setminus\mathcal{A}}\Bigl{[}1-\Pr\Bigl{(}\mathsf{S}_{f,i}\textgreater \ln\frac{\gamma_{h}}{\alpha_{j}P_{\text{tx}}}\Bigr{)}\Bigr{]}\]0  
where \(n_{\text{succ}}\) is the number of delivered packets. Then, the expected FSR, \(R_{s}\) can be obtained as  
\[=\prod_{i\in\mathcal{A}}\Pr\Bigl{(}\mathsf{S}_{f,i}\textgreater \ln\frac{\gamma_{h}}{\alpha_{i}P_{\text{tx}}}\Bigr{)}\prod_{j\in\mathcal{K} \setminus\mathcal{A}}\Bigl{[}1-\Pr\Bigl{(}\mathsf{S}_{f,i}\textgreater \ln\frac{\gamma_{h}}{\alpha_{j}P_{\text{tx}}}\Bigr{)}\Bigr{]}\]1  
Observe that \(R_{s}\) is affected by the selected frame length \(L\). Assuming that the number of nodes in the network is known at the illuminator, the frame length can be set at the optimal theoretical value \(L=K\). Nevertheless, propagation channel effects induce a shift on the optimal value of the frame length.