# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 2 System Model and Preliminaries
### Overlay Allocation  
When the system is operated in overlay mode, both services are allowed to access the channel concurrently and without time restrictions, as illustrated in Figure 0(b). Such a policy allows \(\mathcal{S}_{\mathsf{a}}\) and \(\mathcal{S}_{\mathsf{b}}\) to enjoy the whole share of resources at the cost of suffering from mutual interference, due to the loss of orthogonality. In this case, the whole frame is assumed to be split in \(\mathsf{M}\) slots, each consisting of \(n_{s}\) channel uses. Using all the resources, \(\mathsf{s}_{\mathsf{a}}\) transmits then a codeword \(\mathbf{X}^{(\mathsf{a})}\) of size \(n_{s}\mathsf{M}\) channel uses. In turn, users of service \(\mathcal{S}_{\mathsf{b}}\) can send a packet over any of the \(\mathsf{M}\) available slots. To properly capture the signal model, although \(\mathsf{s}_{\mathsf{a}}\) encodes its message across all the available channel uses, it is convenient to express \(\mathbf{X}^{(\mathsf{a})}\) as the concatenation of the \(\mathsf{M}\) sub-codewords transmitted over the slots as follows:  
\[\mathbf{X}^{(\mathsf{a})}=\left[\mathbf{X}_{1}^{(\mathsf{a})},\ldots,\mathbf{ X}_{\mathsf{M}}^{(\mathsf{a})}\right]\]  
Figure 1: Reference timeline for the orthogonal and overlay resource allocations.  
where \(\mathbf{X}_{\ell}^{(\mathsf{a})}\sim\mathcal{CN}(\mathbf{0},\rho_{\mathsf{a}}\mathsf{I}_{n_ {\mathsf{a}}})\), \(\ell=1,\ldots,\mathsf{M}\). Following this notation, the channel output at the receiver over the \(\ell\)-th slot conditioned on having \(U=u\) users of \(\mathcal{I}_{\mathsf{b}}\) access the channel takes the following form:  
\[\mathbf{Y}_{\ell}=\mathbf{X}_{\ell}^{(\mathsf{a})}+\sum_{j=1}^{u}\mathbf{X}_{\ell}^{( \mathsf{b},j)}+\mathbf{W}_{\ell}. \tag{3}\]  
A packet of a best-effort user is thus always affected by the interference coming from \(\mathcal{I}_{\mathsf{a}}\), and possibly by other transmissions of \(\mathcal{I}_{\mathsf{b}}\). Conversely, any sub-codeword sent by \(\mathsf{s}_{\mathsf{a}}\) is received interference-free with probability \(e^{-\lambda}\), i.e., if no mMTC user has transmitted over the corresponding slot. It is worth stressing that the number of interfering packets from \(\mathcal{I}_{\mathsf{b}}\) is not known a priori, so the coding rate of \(\mathsf{s}_{\mathsf{a}}\) cannot be dynamically adapted on a slot-by-slot basis.  
For both the orthogonal and overlay configurations, we are interested in the maximum sum rate that the system can offer to the best-effort service while granting the QoS requirements of \(\mathcal{I}_{\mathsf{a}}\). Specifically, denoting by \(\mathsf{R}_{\mathsf{a}}\) the rate in bits per channel use of \(\mathsf{s}_{\mathsf{a}}\) and by \(\mathsf{R}_{\mathsf{b}}\) the average aggregate number of bits per channel use decoded at the receiver for \(\mathcal{I}_{\mathsf{b}}\), we aim at deriving the rate region given by all pairs \((\mathsf{R}_{\mathsf{a}},\mathsf{R}_{\mathsf{b}})\) that allow \(\mathsf{s}_{\mathsf{a}}\) to experience an error probability that is less than or equal to a target value.