# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 4 Finite Blocklength Analysis
### Overlay Allocation  
As discussed in Section 2, when the system is operated in overlay mode, the total number of channel uses, \(n\), available over a frame is split into \(\mathsf{M}\) time slots of \(n_{s}\) channel uses each, i.e., \(n=\mathsf{M}n_{s}\). Let us denote by \(\mathbf{U}=[U_{1},\dots,U_{\mathsf{M}}]\) the random vector whose components describe the number of users of \(\mathcal{S}_{\text{b}}\) that transmit over each of the slots. Recalling the i.i.d. Poisson distribution of the r.v. \(U_{\ell}\), the joint PMF of \(\mathbf{U}\) can be expressed for any arrival intensity \(\lambda\) as follows:  
\[p_{\mathbf{U}}(\mathbf{u})=\prod_{\ell=1}^{\mathsf{M}}\frac{\lambda^{u_{\ell} }e^{-\lambda}}{u_{\ell}!}. \tag{23}\]  
Consider now the transmission of \(\mathsf{s}_{\text{a}}\), and condition on a specific realization \(\mathbf{U}=\mathbf{u}\). In this case, each portion of the sent message experiences a distinct (and independent) interference level, allowing to model the problem as the transmission of an \(n\) channel uses codeword over \(\mathsf{M}\) parallel AWGN channels. Following the approach presented in [35] in the context of block-fading, the relation between maximum sustainable rate \(\mathsf{R}_{\text{a}}^{*}\) and codeword error probability \(\mathsf{P}_{e}\) we can then be approximated as follows:  
\[\mathsf{R}_{\text{a}}^{*}(\mathsf{P}_{e},\mathbf{u})\cong\frac{1}{\mathsf{M} }\mathcal{C}_{\text{a,ov}}(\mathbf{u})-\sqrt{\frac{\mathcal{V}_{\text{a,ov}} (\mathbf{u})}{n_{s}\mathsf{M}^{2}}}Q^{-1}(\mathsf{P}_{e}) \tag{24}\]  
where  
\[\mathcal{C}_{\text{a,ov}}(\mathbf{u}) =\sum_{\ell=1}^{\mathsf{M}}\log_{2}(1+\gamma_{\text{a,\ell}})\] \[\mathcal{V}_{\text{a,ov}}(\mathbf{u}) =\sum_{\ell=1}^{\mathsf{M}}\frac{\gamma_{\text{a,\ell}}(2+\gamma_{ \text{a,\ell}})}{2(1+\gamma_{\text{a,\ell}})^{2}}(\log_{2}e)^{2}\]  
and we introduce for convenience the SINR experienced by \(\mathsf{s}_{\text{a}}\) over the \(\ell\)-th slot as follows:  
\[\gamma_{\text{a,\ell}}:=\frac{\rho_{\text{a}}}{\sigma^{2}+u_{\ell}\rho_{\text{ b}}}.\]Solving (24) with respect to \(\mathsf{P}_{e}\), we can easily obtain the error probability when a given rate \(\mathsf{R}_{\mathsf{a}}\) is employed by \(\mathsf{s}_{\mathsf{a}}\) and a specific realization of the interference pattern \(\mathbf{U}=\mathbf{u}\) is experienced:  
\[\mathsf{P}_{e}(\mathbf{u})=Q\left(\frac{\frac{\mathcal{C}_{\mathsf{a},\mathsf{ o}^{\prime}}(\mathbf{u})}{\mathsf{M}}-\mathsf{R}_{\mathsf{a}}}{\sqrt{ \frac{\mathcal{V}_{\mathsf{a},\mathsf{o}^{\prime}}(\mathbf{u})}{n_{\mathsf{e }}M^{2}}}}\right). \tag{25}\]  
We now observe that, in view of the random nature of the interference of service \(\mathcal{S}_{\mathsf{b}}\), it is more meaningful to evaluate the _average_ error probability \(\overline{\mathsf{P}}_{e}\) as follows:  
\[\overline{\mathsf{P}}_{e}=\mathbb{E}[\,\mathsf{P}_{e}(\mathbf{u})\,]=\sum_{ \mathbf{u}}\mathsf{P}_{e}(\mathbf{u})\,p_{\mathbf{U}}(\mathbf{u}) \tag{26}\]  
by employing (23) and (25). The QoS requirements of \(\mathcal{S}_{\mathsf{a}}\) are properly specified in this case by the pair \((\mathsf{R}_{\mathsf{a}},\overline{\mathsf{P}}_{e})\). In this perspective, the result in (26) offers a relevant system design tool. Indeed, for any targeted rate \(\mathsf{R}_{\mathsf{a}}\), an inspection of the equation allows to derive the maximum traffic intensity \(\lambda_{m}\) of \(\mathcal{S}_{\mathsf{b}}\) that can be supported without violating the average error probability \(\overline{\mathsf{P}}_{e}\). Taking the lead from this, let us then focus on \(\mathcal{S}_{\mathsf{b}}\)for which we need to compute the maximum aggregate rate that can be achieved under the constraint on \(\lambda_{m}\). In this case, three conditions have to be met for a mMTC message to be retrieved: (i) the message of \(\mathsf{s}_{\mathsf{a}}\) has to be decoded and its interference canceled; (ii) the packet of \(\mathcal{S}_{\mathsf{b}}\) has to be sent over a singleton slot; and (iii) the \(n_{s}\)-channel use codeword has to be decoded.