# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 3 Asymptotic Analysis
### Overlay Allocation
#### 3.2.1 Ergodic Case  
When the number of slots within a frame is sufficiently large (i.e., \(\mathsf{M}\to\infty\)), the codeword of \(\mathsf{s_{a}}\) experiences, in the limit, all the possible interference values, making it possible to define the ergodic channel capacity.  
\[\begin{split}\mathcal{C}_{\mathsf{a}}^{(e)}&=\mathbb{ E}\left[\log_{2}\!\left(1+\frac{\rho_{\mathsf{a}}}{\sigma^{2}+U\rho_{\mathsf{b}}} \right)\right]\\ &=\sum_{u=0}^{\infty}\frac{\lambda^{u}e^{-\lambda}}{u!}\log_{2}\! \left(1+\frac{\rho_{\mathsf{a}}}{\sigma^{2}+u\rho_{\mathsf{b}}}\right).\end{split} \tag{8}\]  
In this case, a vanishingly small error probability can still be granted to the user for any rate \(\mathsf{R_{a}}<\mathcal{C}_{\mathsf{a}}^{(e)}\). Clearly, from a system design perspective, the overlay operation mode becomes meaningful--and possibly convenient--if the QoS requirements of \(\mathcal{I}_{\mathsf{a}}\) in terms of data rate are met, even in the presence of interference. We are thus interested in configurations for which the achievable rate in overlay matches the one of an orthogonal allocation, i.e., for which, leaning on (4) and (8),  
\[\alpha\log_{2}(1+\gamma_{\mathsf{a}})=\sum_{u=0}^{\infty}\frac{\lambda^{u}e^{ -\lambda}}{u!}\log_{2}\!\left(1+\frac{\rho_{\mathsf{a}}}{\sigma^{2}+u\rho_{ \mathsf{b}}}\right). \tag{9}\]  
In particular, for any value of \(\alpha\) employed to configure a dedicated-resource allocation, (9) imposes a constraint on the tolerable level of interference from \(\mathcal{I}_{\mathsf{b}}\) in overlay mode, limiting the average number of transmissions per slot and thus the intensity of the mMTC traffic. Denoting this value as \(\lambda_{m}\), the message of \(\mathsf{s_{a}}\) is retrieved and its contribution on the incoming waveform is perfectly canceled with probability approaching one for any channel load \(\lambda<\lambda_{m}\) of service \(\mathcal{I}_{\mathsf{b}}\). We note that, while a closed-form expression of \(\lambda_{m}\) is elusive due to the transcendental nature of (9), its value can easily be determined numerically. If the system is operated satisfying this constraint, the receiver can then process packets sent by mMTC users via slotted ALOHA after having performed interference cancellation, and the average aggregate rate for \(\mathcal{I}_{\mathsf{b}}\) is once again captured by (6). In order to maximize performance while meeting the requirements of the QoS-driven user, \(\mathcal{I}_{\mathsf{b}}\) shall thus be operated at a channel load as follows:  
\[\lambda^{*}:=\min\{\lambda_{m},1\}\,.\]  
We recall indeed that values of \(\lambda\) larger than 1 [pkt/slot], even if tolerable by the QoS requirements of \(\mathcal{I}_{\mathsf{a}}\), would be ineffective from a throughput perspective for the mMTC traffic in view of the detrimental effect of destructive collisions.  
Combining these remarks, and recalling that both services have now access to the whole set of resources, the rate region \((\mathsf{R_{a}},\mathsf{R_{b}})\) for the ergodic case finally evaluates to the following:  
\[\begin{split}\mathsf{R_{a}}&<\sum_{u=0}^{\infty} \frac{\lambda_{m}^{u}e^{-\lambda_{m}}}{u!}\log_{2}\!\left(1+\frac{\rho_{ \mathsf{a}}}{\sigma^{2}+u\rho_{\mathsf{b}}}\right)\\ \mathsf{R_{b}}&<\lambda^{*}e^{-\lambda^{*}}\log_{2} \!\left(1+\frac{\rho_{\mathsf{b}}}{\sigma^{2}}\right).\end{split}\]