# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 4 Finite Blocklength Analysis
### Overlay Allocation
On the other hand, it is relevant to remark that, also in the non-ergodic setting, there exists a significant range of rate values for service \(\mathcal{G}_{\mathsf{a}}\) that allows to operate \(\mathcal{G}_{\mathsf{b}}\) at the maximum aggregate rate. Such a result confirms the potential of the overlay approach. We also investigate the trends under finite codeword length in Figure 4. In this case, not only the time slots, but also the number of channel uses per time slot are finite and we thus rely on the analysis provided in Section 4. As for the previous scenario, we set the target average codeword error probability of service \(\mathcal{I}_{\text{a}}\) to \(10^{-3}\), and consider \(\rho_{\text{a}}/\sigma^{2}=\rho_{\text{b}}/\sigma^{2}=0\) dB. The set of time slots is \(\mathsf{M}\in\{25,100,200\}\) and the set of channel uses per time slot \(n_{s}\in\{100,1000\}\). The latter choice is representative of practical mMTC applications, e.g., LoRa [5] and SigFox [8], which enable the transmission of payloads of up to 96 and 2000 bits, respectively. Accordingly, the set of total channel uses in the orthogonal allocation evaluates to \(n\in\{2500,10,000,20,000,25,000\}\). In the main plot of Figure 4, the orthogonal allocation and the overlay allocation for \(\mathsf{M}\in\{25,100,200\}\) and \(n_{s}=100\) channel uses per time slot are compared. As we can observe, increasing the number of time slots has a more beneficial impact on the overlay allocation than on the orthogonal one. In the former case, the increase in the number of time slots allows supporting a larger channel load for service \(\mathcal{I}_{\text{b}}\) for the same target outage probability. In the latter case instead, it only impacts the correction term in (20) with respect to capacity, and thus results in a minor benefit on the achievable rate pairs. It is also worth noting that, as expected, the finite number of channel uses strongly affects the maximum achievable rate of service \(\mathcal{I}_{\text{b}}\), reducing it by \(\sim\)18% with respect to the asymptotic case. More interestingly, in contrast with what was discussed in the asymptotic setting, a region where an orthogonal allocation is superior to the overlay configuration emerges. Focusing on the \(\mathsf{M}=25\) case, such a region is well highlighted in the subplot, where both scenarios with \(n_{s}\in\{100,1000\}\) are depicted. Although rather limited and only for rates of service \(\mathcal{I}_{\text{a}}\) larger than 0.8 (bit/ch. use), such inversion suggests that reserving a portion of time for the IoT service alone is beneficial instead of letting the two services compete completely. In order to further investigate this aspect, we provide in Figure 5a different angle on the considered results. In particular, we analyze the ratio of the maximum rate achievable by service \(\mathcal{F}_{\text{b}}\) with an overlay allocation to the one obtained in the orthogonal case as a function of \(\mathsf{R}_{\text{a}}\). Values larger than 1 thus identify regions of \(\mathsf{R}_{\text{a}}\) where an overlay allocation can outperform the orthogonal one, so that a more aggregate mMTC rate is achieved while granting the same performance to the QoS-constrained service. To highlight the impact of short-packet transmissions, we compare the asymptotic, ergodic setting with finite length scenarios. In the latter case, we set the target average codeword error probability of service \(\mathcal{F}_{\text{a}}\) to \(10^{-3}\). Let us focus first on the setting \(\rho_{\text{a}}/\sigma^{2}=\rho_{\text{b}}/\sigma^{2}=0\) dB. As already shown in Figure 2, under the asymptotic, ergodic setting, the overlay allocation is always beneficial. However, Figure 5a reveals the presence of an optimal operating point at \(\mathsf{R}_{\text{a}}\cong 0.92\) (bit/ch. use) for which the benefit achieved by the overlay allocation is maximized. A similar trend can also be observed for both \(\mathsf{M}=100\) and \(\mathsf{M}=25\), although for progressively smaller values of rate \(\mathsf{R}_{\text{a}}\).