# Smart Beamforming for Direct LEO Satellite Access of Future IoT
## 4 Resource Sharing Beamforming Access
### Throughput Analysis  
This section is devoted to conduct the throughput analysis of the RSBA scheme. To determine under what conditions the satellite can successfully collect the data, we rely on the analytic model of a time-slotted network. The duration of the time slot is equal to the packet transmission time. First, we focus on the collision probability of a specific terminal. In the scenario under study, a collision occurs when there is overlap in time and spatial domains. If we focus on the case that two users are attempting to send data, the overlap in the spatial domain can be seen in the uv-plane. In alignment with the satellite field of view, the (u,v)-coordinates \((u_{k},v_{k})\) must satisfy \(u_{k}^{2}+v_{k}^{2}\leq\sin^{2}(\theta_{\text{MAX}})\). The minimum elevation angle of the terminal and satellite orbit will determine \(\theta_{\text{MAX}}\). We declare that the \(i\)-th and the \(j\)-th users collide if \(|u_{i}-u_{j}|^{2}+|v_{i}-v_{j}|^{2}\leq\Delta_{uv}^{2}\). The parameter \(\Delta_{uv}\), which determines the minimum angular separation to successfully separate users, depends on the spatial processing performed at the receiver. Under the premise that users are uniformly located in the coverage area, the probability that the undesired user is located in the same spot as the desired user is given by \(P_{c}=\frac{\pi\Delta_{uv}^{2}}{\pi\sin^{2}(\theta_{\text{MAX}})}\). When there is no angular discrimination, \(P_{c}=1\). Thus, we can establish that the probability that a packet is successfully received in the presence of \(k\) interfering users is given by \((1-P_{c})^{k}\). Following the guidelines reported in [37] and assuming that the arrivals follow the Poisson distribution, it follows that the probability of successful transmission for the target terminal is \(P_{\rm S}=e^{-GP_{\rm c}}\), where \(G\) denotes the average number of transmission attempts by all the users during the packet transmission. Recalling that the throughput is the average number of transmission attempts per packet duration multiplied by the probability of success, the performance metric in packets/slot can be defined as:  
\[\eta_{T}(P_{\rm S})=GP_{\rm S}=Ge^{-GP_{\rm c}}. \tag{37}\]  
Adopting the same approach as [9], the number of users that the network can support for a given probability of successful transmission is approximated to:  
\[N_{\rm max}\approx\frac{\eta_{T}(P_{t})}{d_{a}}. \tag{38}\]  
In notation terms, \(\eta_{T}(P_{t})\) correspond to the maximum achievable throughput at \(P_{\rm S}=P_{t}\), which is the desired probability of successful transmission. Finally, \(0\leq d_{a}\leq 1\) is the average activity of the users. It is important to remark that \(N_{\rm max}\) is significantly higher than \(N_{\rm U}\), which corresponds to the active users at a given time slot. It is also worth mentioning that the throughput is computed under the assumption that the satellite can always direct a beam towards each active user. In a nutshell, the analysis assumes that colliding packets can be decoded as long as terminals are sufficiently separated. This is a pessimistic assumption. Indeed, the throughput could be increased by employing more advanced receivers based on iterative successive interference cancellation (SIC). In such a case, the receiver benefits from the capture effect to solve packet collisions from users that are in close proximity [9]. To model the effect of the iterative SIC procedure, it is necessary to generate the performance curves of the selected modulation and coding schemes in additive white Gaussian (AWGN) channels. As this level of detail is not provided, the study of the iterative SIC is a topic of future research. In light of the above discussion, it appears fair to assume that the performance is governed by the array size and the spatial processing. We would expect that the higher the number of antenna elements \(N_{x},N_{y}\), the lower the distance \(\Delta_{uv}\). In general, for an arbitrary beamformer, it is difficult to formulate \(\Delta_{uv}\) in a closed-form expression.