# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 4 Finite Blocklength Analysis
### Overlay Allocation
Indeed, while in our case we consider destructive collisions, ref. [28] relies on capture effect and interference cancellation (IC) also for service \(\mathcal{G}_{\text{b}}\) data units. From Figure 2, we can also infer that, when the two services are operated in overlay, \(\mathcal{S}_{\text{b}}\) can achieve its maximum rate of \(e^{-1}\cong 0.368\) (bit/ch. use) up to \(\mathsf{R}_{\text{a}}\leq 0.68\) (bit/ch. use) for \(\rho_{\text{a}}/\sigma^{2}=0\) dB, and up to \(\mathsf{R}_{\text{a}}\leq 2.76\) (bit/ch. use) for \(\rho_{\text{a}}/\sigma^{2}=10\) dB. In other words, increasing the target rate for the QoS-constrained service \(\mathcal{S}_{\text{a}}\) does not impact service \(\mathcal{S}_{\text{b}}\) in this region. Especially for the setting of Figure 2b, there is a very large range of rates for service \(\mathcal{S}_{\text{a}}\), where the IoT traffic is limited by the poor performance of a slotted ALOHA access method. Indeed, the channel code protection of the data unit of user \(\mathsf{s}_{\text{a}}\) could allow a larger channel traffic of service \(\mathcal{S}_{\text{b}}\), beyond \(\lambda=1\), but cannot be reaped due to the limitation in the medium access. Such a remark hints at how advanced alternatives relying on packet repetition, e.g., [37], can be beneficial to expand the achievable rate region, fostering the need for additional research in this direction. Finally, for \(\mathsf{R}_{\text{a}}>0.68\) (bit/ch. use), or \(\mathsf{R}_{\text{a}}>2.76\) (bit/ch. use) the maximum rate achievable for service \(\mathcal{S}_{\text{b}}\) sees a larger and steeper degradation with respect to the orthogonal allocation as \(\mathsf{R}_{\text{a}}\to\log_{2}\bigl{(}1+\rho_{\text{a}}/\sigma^{2}\bigr{)}\). As a consequence, more caution on the tuning of the traffic intensity for service \(\mathcal{S}_{\text{b}}\) shall be devoted for such rates. Let us now take a further step and consider the practical constraint imposed by having a finite number of slots within the communications frames (Section 3.2.2). Under these conditions, the transmission of service \(\mathcal{G}_{\mathsf{a}}\) fails to experience all possible interference levels from service \(\mathcal{G}_{\mathsf{b}}\), and a non-ergodic setting has to be considered for the overlay allocation. The corresponding results for a number of time slots in the set \(\mathsf{M}\in\{25,100,200\}\) are shown in Figure 3 together with the ergodic benchmark as reference. The set of time slots considered is in line with the literature on random access targeting satellite uplink scenarios, e.g., [37; 38]. For our discussion, we set the target (average) codeword error probability for service \(\mathcal{G}_{\mathsf{a}}\) to \(10^{-3}\), and focus on the \(\rho_{\mathsf{a}}/\sigma^{2}=\rho_{\mathsf{b}}/\sigma^{2}=0\) dB scenario. Similar trends were also found for the unbalanced SNR scenario, and are not reported here for the sake of compactness. The impact of finite-length frames is clearly visible already for a moderately large number of slots, and the reduction of the achievable rate region becomes even more pronounced as the system operates with smaller values of \(\mathsf{M}\). For example, if user \(\mathsf{s}_{\mathsf{a}}\) targets a rate of \(\mathsf{R}_{\mathsf{a}}=0.8\) (bit/ch. use), the maximum achievable rate for service \(\mathcal{G}_{\mathsf{b}}\) is reduced by \(\sim\)51% with respect to the ergodic setting, when \(\mathsf{M}=25\). The rate \(\mathsf{R}_{\mathsf{b}}\) contraction becomes even more relevant for larger values of \(\mathsf{R}_{\mathsf{a}}\). Conversely, increasing the number of time slots to 100 or 200 mitigates the trend. For the same target rate of \(\mathsf{R}_{\mathsf{a}}=0.8\) (bit/ch. use), the maximum achievable rate for service \(\mathcal{G}_{\mathsf{b}}\) is reduced by only \(\sim\)18% with respect to the asymptotic setting, when 200 slots are considered.