## 3 System Throughput and Packet Loss Rate
### Throughput  
A packet is only successfully received at the \(k\)th satellite if it has not been erased by the channel fading realization and if there is no collision of other non-erased packets transmitted by other users at the same time-slot. Thus, given that \(u\) packets were simultaneously transmitted by different users in a given time slot, the event of a successful packet reception at the \(k\)th satellite occurs with probability \(q_{k}=u(1-\varepsilon_{k})\varepsilon_{k}^{u-1}\). Then, the throughput, defined as the number of packets successfully received per time slot, at the \(k\)th satellite, is:  
\[\mathcal{T}_{k}(G) = \sum_{u=0}^{\infty}q_{k}\,\mathbb{P}[U=u] \tag{3}\] \[= \sum_{u=0}^{\infty}\left[u(1-\varepsilon_{k})\varepsilon_{k}^{(u -1)}\frac{G^{u}e^{-G}}{u!}\right]\] \[= G(1-\varepsilon_{k})e^{-G(1-\varepsilon_{k})}.\]  
The system throughput (\(\mathcal{T}\)) is defined as the number of packets successfully received by at least one of the satellites per time slot. Therefore, multiplicities must be discarded, so that the system throughput is not only the sum of the throughput experienced by each of the satellites. For that sake, the inclusion-exclusion principle [42] can be utilized to determine the cardinality of the union of the sets of packets successfully received by each satellite, thereby discounting for the intersections. For instance, for the case of three sets \(D_{1}\), \(D_{2}\), and \(D_{3}\) (in our case each set would contain the data packets successfully received by a different satellite) via the principle of the inclusion-exclusion we can determine the number of different received packets as: \(|D_{1}\cup D_{2}\cup D_{3}|=|D_{1}|+|D_{2}|+|D_{3}|-|D_{1}\cap D_{2}|-|D_{1} \cap D_{3}|-|D_{2}\cap D_{3}|+|D_{1}\cap D_{2}\cap D_{3}|\). The elements within the \(D_{1}\), \(D_{2}\), and \(D_{3}\) sets that are double-counted are removed. This can be generalized to the case of \(K\) sets as [42; 43]  
\[\left|\bigcup_{k=1}^{K}D_{k}\right|=\sum_{\mathbb{J}\
eq\emptyset,\,\,\mathbb{ J}\subseteq\{1,\ldots,K\}}(-1)^{|\mathbb{J}|+1}\left|\bigcap_{j\in\mathbb{J}}D_{j} \right|, \tag{4}\]  
where \(\mathbb{J}\) is a set with indexes of the subsets whose intersection must be evaluated. Note that the above expression does not admit a simplification step applied in [36], which requires the setup to be symmetrical. In our particular case, as the erasure probabilities are different, the ordering (or numbering) of the satellites is relevant. Then, it is fundamental to know \(\left|\bigcap_{j\in\mathbb{J}}D_{j}\right|\), the cardinality of the intersection of the sets of packets successfully received by a subset \(\mathbb{J}\subseteq\{1,\ldots,K\}\) of satellites with cardinality \(|\mathbb{J}|\). Consequently, considering the traffic model described in Section 2, given that \(u\) packets were simultaneously transmitted by different users in a given time slot, the event that the same packet is successfully received by the \(|\mathbb{J}|\) satellites occurs with probability  
\[q_{\mathbb{J}}=u\prod_{k\in\mathbb{J}}(1-\varepsilon_{k})\varepsilon_{k}^{(u -1)}. \tag{5}\]  
After the realization of many time-slots \(N\), the average number of packets jointly received by \(|\mathbb{J}|\) satellites, for all \(u\), is \(\left|\bigcap_{j\in\mathbb{J}}D_{j}\right|=N\sum_{u=0}^{\infty}q_{\mathbb{J}} \mathbb{P}[U=u]\).