# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 3 Asymptotic Analysis
### Overlay Allocation
#### 3.2.2 Non-Ergodic Case
In this perspective, we observe that two conditions have to be met for an mMTC packet to be retrieved over a slot in a non-ergodic setup: (i) the packet of \(\mathsf{s}_{\text{a}}\) is decoded, and (ii) no other user of \(\mathcal{F}_{\text{b}}\) concurrently accesses the channel over the slot of interest. Note that condition (i) is consistent with the system model assumptions. Indeed, should the message of \(\mathsf{s}_{\text{a}}\) not be decoded, its interference contribution cannot be removed, affecting any underlying mMTC packet. Recalling the discussion of Remark 1, this would prevent the successful retrieval of information for users of \(\mathcal{H}_{\text{b}}\). Furthermore, let us denote by \(W\) the r.v. taking values in \(\{0,\dots,\mathsf{M}\}\) and counting the number of slots characterized by the transmission of a single user (named also _singleton slots_) of \(\mathcal{F}_{\text{b}}\), which, for the Poisson traffic under study, follows a binomial distribution of parameters \(\mathsf{M}\) and \(\lambda e^{-\lambda}\). Following this notation, for any arrival intensity \(\lambda\), the aggregate rate of \(\mathcal{F}_{\text{b}}\) can be conveniently expressed as follows:  
\[\mathsf{R}_{\text{b}}=\frac{1}{\mathsf{M}}\sum_{w=0}^{\mathsf{M}}\left[w\log_{ 2}\Bigl{(}1+\frac{\rho_{\text{b}}}{\sigma^{2}}\Bigr{)}\cdot\left(1-\mathbb{P} \Biggl{\{}\sum_{\ell=1}^{\mathsf{M}}C_{\text{a},\ell}<\mathsf{MR}_{\text{a}} \Bigm{|}W=w\Biggr{\}}\right)\right]\cdot p_{W}(w) \tag{14}\]  
where, recalling (10), the term within square brackets takes into account how all \(w\) singleton slots bring a contribution of \(\log_{2}(1+\rho_{\text{b}}/\sigma^{2})\) bits per channel use if, and only if, \(\mathsf{s}_{\text{a}}\) is not in outage. To complete the calculation, we therefore need to derive the outage probability for \(\mathcal{F}_{\text{a}}\) conditioned on the r.v. \(W\). To this aim, it is useful to rearrange the conditional expression as follows:  
\[\mathbb{P}\Biggl{\{}\sum_{\ell=1}^{\mathsf{M}}C_{\text{a},\ell}< \mathsf{MR}_{\text{a}}\Bigm{|}W=w\Biggr{\}}=\] \[\mathbb{P}\Biggl{\{}w\log_{2}\biggl{(}1+\frac{\rho_{\text{a}}}{ \sigma^{2}+\rho_{\text{b}}}\biggr{)}+\sum_{\ell,U_{\ell}\
eq 1}C_{\text{a},\ell}< \mathsf{R}_{\text{a}}\mathsf{M}\Bigm{|}W=w\Biggr{\}}= \tag{15}\] \[\mathbb{P}\Biggl{\{}\sum_{\ell,U_{\ell}\
eq 1}C_{\text{a},\ell}< \mathsf{R}_{\text{a}}\mathsf{M}-w\log_{2}\biggl{(}1+\frac{\rho_{\text{a}}}{ \sigma^{2}+\rho_{\text{b}}}\biggr{)}\Bigm{|}W=w\Biggr{\}} \tag{16}\]  
where the first addend on the right-hand side of (15) accounts for the sum of the instantaneous capacity of all \(w\) slots with a single transmission of service \(\mathcal{H}_{\text{b}}\), whereas the summation takes into consideration the contributions of all other \(\mathsf{M}-w\) slots (i.e., those with \(U_{\ell}\
eq 1\)). In turn, the PMF of the instantaneous capacity \(C_{\text{a},\ell}\) over a _non-singleton_ slot can be directly derived from (12), to obtain for any value in the alphabet \(\Bigl{\{}c_{\ell}\in\mathbb{R}\,|\,c_{\ell}=\log_{2}\Bigl{(}1+\frac{\rho_{ \text{b}}}{\sigma^{2}+\rho_{\text{b}}}\Bigr{)},\,u\in\mathbb{N}\setminus\{1 \}\Bigr{\}}\),  
\[\mathbb{P}\{C_{\text{a},\ell}=c_{\ell}\,|\,U_{\ell}\
eq 1\}=\frac{e^{- \lambda}\lambda^{\beta(c_{\ell})}}{\beta(c_{\ell})!}\frac{1}{1-\lambda e^{- \lambda}}. \tag{17}\]  
where the normalization factor readily follows from the Poisson distribution of the mMTC traffic. Recalling now that also the r.v. \(C_{\text{a},\ell}\) over non-singleton slots are i.i.d., the PMF of the sum of \(\mathsf{M}-w\) such values can be once more derived by taking the \((\mathsf{M}-w)\)-fold convolution of (17), allowing to compute (16).