# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 3 Asymptotic Analysis
### Overlay Allocation  
When the system is operated in overlay configuration, the incoming signal at the receiver is the superposition of transmissions performed by both services. Throughout our study, we assume decoding to start from the message sent by \(\mathsf{s}_{\mathsf{a}}\), treating the interference component of \(\mathcal{S}_{\text{b}}\) packets as noise. In the case of successful decoding, ideal interference cancellation is performed, removing the contribution of \(\mathsf{s}_{\mathsf{a}}\) from the overall waveform and presenting to the receiver the M-slot frame populated by packets of \(\mathcal{S}_{\text{b}}\) alone for further processing. Ideal interference cancellation of the packet of user \(\mathsf{s}_{\mathsf{a}}\) can be regarded as a reasonable assumption, especially taking into account that its transmission can enjoy a large number of channel uses, and the data-aided channel estimation can be properly tuned. Moreover, we also note that non-ideal interference cancellation would yield a minor impact on the model by effectively increasing the noise power suffered by the transmissions of service \(\mathcal{S}_{\text{b}}\) since both the signal and the noise are modeled as Gaussian distributed r.vs..  
**Remark 2**.: _The assumption on the order of decoding for the supported services stems from two practical considerations. First, as will be further discussed in Sections 4 and 5, mMTC traffic is typically composed of very short packets--often in the order of a few hundreds of bits-- confining pilot symbols to few channel uses per transmitted message in order to avoid excessive overhead. Accurate channel estimation needed for decoding packets of \(\mathcal{S}_{\text{b}}\), especially in the presence of interference from \(\mathsf{s}_{\mathsf{a}}\), may thus not be viable. On the other hand, the longer data units that characterize \(\mathcal{S}_{\text{a}}\) render the cost of stronger pilot sequences manageable, and, together with its more regular and predictable traffic patterns, can lead to estimates of the channel parameters that are good enough to retrieve information and perform accurate interference cancellation. In addition to this, the choice of initiating decoding from \(\mathcal{S}_{\text{a}}\) is driven by QoS arguments. It is indeed reasonable to overlay best-effort mMTC traffic to an existing service only as long as the latter does not experience a significant loss in performance. Along this line of reasoning, the receiver shall be able to decode \(\mathcal{S}_{\text{a}}\) even in the presence of underlying interference, i.e., treating the whole of \(\mathcal{S}_{\text{b}}\) traffic as noise._  
Under this assumption, \(\mathsf{s}_{\mathsf{a}}\) experiences the block-interference channel described in (3), with the interference level remaining constant over a coherence period of one slot duration, and changing to an independent realization during the subsequent \(n_{s}\) channel uses. Specifically, conditioned on the number \(U_{\ell}\) of users of \(\mathcal{S}_{\text{b}}\) accessing the channel over the \(\ell\)-th slot, the r.v. describing the signal-to-interference and noise ratio (SINR) seen at the receiver for \(\mathsf{s}_{\mathsf{a}}\) takes the following form:  
\[\mathsf{SINR}_{\ell}^{(\mathsf{a})}=\frac{\rho_{\mathsf{a}}}{\sigma^{2}+U_{ \ell}\,\rho_{\text{b}}}.\]Within this framework, to better grasp the effect of the bursty overlaid mMTC traffic, we first analyze an ergodic setting, later extending our results to the more practically-relevant non-ergodic setup.