# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 3 Asymptotic Analysis
### Orthogonal Allocation  
When resources are orthogonally split among services, \(\mathsf{s}_{\mathsf{a}}\) delivers data over an additive white gaussian noise (AWGN) channel with SNR \(\gamma_{\mathsf{a}}\) defined in (1), yet is allowed to transmit only for a fraction \(\alpha\) of the time. In this setting, the user can then communicate with vanishing small error probability for any rate as follows:  
\[\mathsf{R}_{\mathsf{a}}<\alpha\log_{2}(1+\gamma_{\mathsf{a}}). \tag{4}\]  
Let us focus instead on the share granted to \(\mathcal{I}_{\mathsf{b}}\), and indicate as \(\mathsf{R}_{\mathsf{b},\ell}\) the communication rate achieved by the service over the \(\ell\)-th slot. If no user accesses the channel (i.e., \(U=0\)), then clearly \(\mathsf{R}_{\mathsf{b},\ell}=0\). Conversely, if a single transmission is performed (\(U=1\), with probability \(\lambda e^{-\lambda}\)), data are sent over an AWGN channel with capacity as follows:  
\[\mathcal{C}_{\mathsf{b}}=\log_{2}\Bigl{(}1+\frac{\rho_{\mathsf{b}}}{\sigma^{ 2}}\Bigr{)}. \tag{5}\]  
Finally, when more than one user becomes active over the slot, a _collision_ takes place, and the channel output at the receiver takes the general form of (2). As will be discussed later (see Remark 1), we assume terminals of \(\mathcal{I}_{\mathsf{b}}\) to encode information at a rate approaching \(\mathcal{C}_{\mathsf{b}}\), as would be done in the absence of interference. Accordingly, in the event of a collision, the actual channel capacity falls below the employed rate, and we regard all packets sent over the slot of interest to be lost. In other words, following a common modeling approach for random access schemes (e.g., [20]), collisions are of a _destructive_ nature, and \(\mathsf{R}_{\mathsf{b},\ell}=0\) whenever more than one user accesses the channel concurrently (\(U>1\)). In summary, (i) for \(U\
eq 1\), \(\mathsf{R}_{\mathsf{b},\ell}=0\), and (ii) for \(U=1\), \(\mathsf{R}_{\mathsf{b},\ell}=\mathcal{C}_{\mathsf{b}}\). Combining these observations, the average rate for \(\mathcal{I}_{\mathsf{b}}\) in bits per channel use over a slot readily follows:  
\[\mathbb{E}[\mathsf{R}_{\mathsf{b},\ell}]=\lambda e^{-\lambda}\,\mathcal{C}_{ \mathsf{b}}. \tag{6}\]  
Equation (6) clarifies how, for any given power level \(\rho_{\mathsf{b}}\), the slotted ALOHA best-effort channel shall be operated at a load \(\lambda=1\) (pkt/slot) in order to maximize the average aggregate rate. Recalling that only a fraction \((1-\alpha)\) of the resources can be leveraged, the achievable sum-rates for \(\mathcal{S}_{\text{b}}\) are then characterized as follows:  
\[\text{R}_{\text{b}}<(1-\alpha)\,e^{-1}\cdot\log_{2}\Bigl{(}1+\frac{\rho_{\text{ b}}}{\sigma^{2}}\Bigr{)}. \tag{7}\]  
**Remark 1**.: _The widely-employed assumption of destructive collisions reflected in (6) is of practical relevance for the considered mMTC setting. Indeed, given that the number of competing terminals over a slot cannot generally be predicted before transmission, a coding scheme that effectively and dynamically takes into account interfering packets may be difficult to develop and goes beyond the complexity of the available terminals. On the other hand, a backoff on the transmission rate to increase resiliency to interference may not be efficient in lightly loaded channels, where sporadic access may lead to having many slots without contention, resulting in significance performance loss for the tight link budget configurations that are typical of mMTC links. Finally, we note that many machine-type applications attempt the delivery of updates that are often repeated or very much correlated in content over time (e.g., reporting of sensed data), positively trading off higher packet loss rates due to interference for a larger aggregate channel sum rate achieved in the absence of coordination among nodes._