# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 3 Asymptotic Analysis
### Overlay Allocation
#### 3.2.2 Non-Ergodic Case  
Albeit insightful in pinpointing a fundamental trade-off between the achievable rate for \(\mathcal{I}_{\mathsf{a}}\) and the amount of mMTC traffic that can be served, the ergodic setup fails to capture some important aspects of practical relevance. In particular, the assumption of having an asymptotically large number of slots over which user \(\mathsf{s_{a}}\) can encode its message can become highly inaccurate for typical systems, characterized by frames of a length of few hundreds of slots, e.g., [21]. In such conditions, each codeword of service \(\mathscr{S}_{\mathsf{a}}\) experiences a finite number \(\mathsf{M}\) of interference realizations, and a vanishingly small error probability can no longer be granted. The QoS requirements of the service are therefore more appropriately characterized by a rate-reliability pair, where \(\mathsf{R_{a}}\) is complemented by a tolerable error probability for the sent codeword. In the non-ergodic setting, the latter is captured using information theory tools by the outage probability \(\mathsf{P_{out}}\). Specifically, for the block-interference channel under study, the two quantities are related as follows:  
\[\mathsf{P_{out}}=\mathbb{P}\Bigg{\{}\frac{1}{\mathsf{M}}\sum_{\ell=1}^{ \mathsf{M}}C_{\mathsf{a},\ell}<\mathsf{R_{a}}\Bigg{\}} \tag{10}\]  
where \(C_{\mathsf{a},\ell}\) denotes the instantaneous capacity for \(\mathsf{s_{a}}\) over the \(\ell\)-th slot, and can be expressed as a function of the number \(U_{\ell}\) of \(\mathscr{S}_{\mathsf{b}}\) users transmitting over the slot as follows:  
\[C_{\mathsf{a},\ell}:=\log_{2}\!\left(1+\frac{\rho_{\mathsf{a}}}{\sigma^{2}+U_ {\ell}\rho_{\mathsf{b}}}\right)\!. \tag{11}\]  
Due to the discrete levels of interference that can be experienced, the r.v. \(C_{\mathsf{a},\ell}\) takes values in the alphabet as follows:  
\[\mathcal{A}=\bigg{\{}c_{\ell}\in\mathbb{R}\,\Big{|}\,c_{\ell}=\log_{2}\!\left( 1+\frac{\rho_{\mathsf{a}}}{\sigma^{2}+u\rho_{\mathsf{b}}}\right)\!,u\in \mathbb{N}\bigg{\}}\]  
and its PMF can readily be derived from (11), leading to the following:  
\[p_{C_{\mathsf{a},\ell}}(c_{\ell})=\mathbb{P}\bigg{\{}u=\frac{\rho_{\mathsf{a} }}{\rho_{\mathsf{b}}(2^{c_{\ell}}-1)}-\frac{\sigma^{2}}{\rho_{\mathsf{b}}} \bigg{\}}=\frac{e^{-\lambda}\Lambda^{\beta(c_{\ell})}}{\beta(c_{\ell})!}\qquad \forall c_{\ell}\in\mathcal{A} \tag{12}\]  
where the final expression resorts to the ancillary quantity as follows:  
\[\beta(c_{\ell}):=\frac{\rho_{\mathsf{a}}}{\rho_{\mathsf{b}}(2^{c_{\ell}}-1)}- \frac{\sigma^{2}}{\rho_{\mathsf{b}}}.\]  
Leaning on this, let us define for compactness the r.v. as follows:  
\[V:=\sum_{\ell=1}^{\mathsf{M}}C_{\mathsf{a},\ell}.\]  
taking values in the set \(\mathcal{A}_{V}=\Big{\{}v\in\mathbb{R}\,|\,\,\,v=\sum_{\ell=1}^{\mathsf{M}}c _{\ell},\,c_{\ell}\in\mathcal{A}\Big{\}}\). Observing that the instantaneous capacity values over different slots are i.i.d., the PMF of \(V\) can simply be computed as the M-fold convolution of (12), to obtain for any \(v\in\mathcal{A}_{V}\) the following:  
\[p_{V}(v)=\underbrace{\Big{(}p_{C_{\mathsf{a},\ell}}*p_{C_{\mathsf{a},\ell}}* \cdots*p_{C_{\mathsf{a},\ell}}\Big{)}}_{\mathsf{M}}(v).\]  
Accordingly, an exact expression of the outage probability in (10) can finally be derived as follows:  
\[\mathsf{P_{out}}=\mathbb{P}\{V<\mathsf{MR_{a}}\}=\sum_{ \begin{subarray}{c}v\in \mathcal{A}_{V}\\ v<\mathsf{MR_{a}}\end{subarray} }p_{V}(v). \tag{13}\]  
This result offers a useful system design tool. Indeed, for any pair \((\mathsf{R_{a}},\mathsf{P_{out}})\), (12) and (13) provide an exact characterization of the maximum channel load \(\lambda_{m}\) that can be granted to mMTC traffic without violating the QoS requirements of \(\mathscr{S}_{\mathsf{a}}\). Taking such a constraint into account, let us now focus on \(\mathcal{H}_{\text{b}}\), aiming to derive the aggregate rate \(\mathsf{R}_{\text{b}}\) that can be achieved for the maximum arrival intensity sustainable by \(\mathcal{F}_{\text{a}}\).