# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 4 Finite Blocklength Analysis
### Overlay Allocation
To account for all these conditions, we follow an approach similar to the one discussed in Section 3.2.2, and condition on the number of singleton slots \(W\) that take place over the observed frame, writing the aggregate average rate achieved for a traffic intensity \(\lambda\) as follows:  
\[\mathsf{R}_{\mathsf{b}}=\frac{1}{\mathsf{M}}\,\sum_{w=0}^{\mathsf{M}}w\, \mathsf{R}_{\mathsf{b}}^{*}(\epsilon^{*})(1-\epsilon^{*})\left(\sum_{ \mathbf{u}}(1-\mathsf{P}_{e}(\mathbf{u}))\,p_{\mathbf{U}}(\mathbf{u}\,|\,w) \right)\,p_{\mathsf{W}}(w). \tag{27}\]  
where we assume that the coding scheme employed by \(\mathcal{S}_{\mathsf{b}}\) has been tuned following (22), so that each of the \(w\) singleton slots brings a contribution of \(\mathsf{R}_{\mathsf{b}}^{*}(\epsilon^{*})(1-\epsilon^{*})\) information bits per channel use. Recalling that \(W\) follows a binomial distribution of parameters \((\mathsf{M},\lambda e^{-\lambda})\), the complete evaluation of (27) simply requires to specify the joint PMF of the number of users of \(\mathcal{S}_{\mathsf{b}}\) that transmitted over each of the \(\mathsf{M}\) slots, conditioned on having \(W=w\) singleton ones. The distribution can be computed by considering two cases. First, for any vector \(\mathbf{u}\) whose number of components with value \(1\) (i.e., the number of singleton slots in the considered frame realization) is different from \(w\), we clearly have \(p_{\mathbf{U}}(\mathbf{u}\,|\,w)=0\). For any other \(\mathbf{u}\), instead. \[p_{\mathbf{U}}(\mathbf{u}\,|\,w)=\frac{\prod_{\ell=1}^{\mathsf{M}}\frac{ \lambda^{u_{\ell}}e^{-\lambda}}{u_{\ell}!}}{\left(\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!and \(\mathcal{G}_{\text{b}}\) reach the receiver with the same power level so that \(\rho_{\text{a}}/\sigma^{2}=\rho_{\text{b}}/\sigma^{2}=0\) dB. This setting reflects the coexistence of two services whose transmitters have a similar hardware (amplifier, antenna) and thus are received with comparable signal strength. In the latter, the QoS-constrained traffic of \(\mathcal{S}_{\text{a}}\) is transmitted with higher power, and we consider the configuration \(\rho_{\text{a}}/\sigma^{2}=10\) dB, \(\rho_{\text{b}}/\sigma^{2}=0\) dB. The second setting, instead, assumes that the QoS-constrained service features transmitting units equipped with a better and possibly more costly hardware, thus resulting in a more favorable link budget. The selected configurations are in line with LEO satellite systems targeting IoT applications, e.g., [36]. As a starting point, let us focus on the asymptotic, ergodic setting introduced in Section 3. In this case, a vanishingly small error probability can always be granted to \(\mathcal{S}_{\text{a}}\), whose QoS requirements are solely specified in terms of a target data rate \(\mathsf{R}_{\text{a}}\). The corresponding results are reported in Figure 2. The two solid lines represent the boundary of the rate pairs achievable by orthogonal and overlay allocations respectively, according to (4), (7) and (8). The shadowed light-red area comprises rate pairs achievable only with an overlay allocation, whereas the shadowed light-blue area denotes rates that are non-achievable by any of the two schemes. The two SNR scenarios, i.e., \(\rho_{\text{a}}/\sigma^{2}=\rho_{\text{b}}/\sigma^{2}=0\) dB and \(\rho_{\text{a}}/\sigma^{2}=10\) dB, \(\rho_{\text{b}}/\sigma^{2}=0\) dB are depicted in Figure 2a,b, respectively. In both configurations, irrespective of the target rate of user \(\mathsf{s}_{\text{a}}\), an overlay allocation is always beneficial to service \(\mathcal{S}_{\text{b}}\) in terms of maximum achievable rates. Incidentally, we note that a slightly different trend was observed in [28], where, even in the asymptotic setting, orthogonal allocation may be beneficial for a small rate region. The discrepancy stems from two main factors. First, different channel models, i.e., AWGN with perfect power control vs. Rayleigh fading, are considered. Second, distinct decoding condition on the mMTC service are assumed.