# Spectral Coexistence of QoS-Constrained and IoT Traffic in Satellite Systems
## 4 Finite Blocklength Analysis
### Orthogonal Allocation  
Let us first focus on the orthogonal configuration, and denote by \(n\) the (finite) number of channel uses available over the whole transmission frame. Following the notation of Section 2, the transmission of \(\text{s}_{\text{a}}\) spans \(n_{\text{a}}:=an\) channel uses, and takes place over an AWGN channel of SNR \(\gamma_{\text{a}}\) defined in (1), i.e., we implicitly focus on values of \(\alpha\) such that \(\alpha n\in\mathbb{N}\). In this case, a vanishingly small error probability cannot be granted, even in the absence of interference, and the performance of \(\mathscr{F}_{\text{a}}\) is properly described by the couple \((\text{R}_{\text{a}}^{*},\text{P}_{e})\), where \(\text{R}_{\text{a}}^{*}\) denotes the maximum rate that can be supported for a codeword error probability \(\text{P}_{e}\). Specifically, the two quantities are related as follows [31]:  
\[\text{R}_{\text{a}}^{*}(\text{P}_{e})=\mathcal{C}_{\text{a}}-\sqrt{\frac{ \gamma_{\text{a}}}{n_{\text{a}}}}\ Q^{-1}(\text{P}_{e})+\mathcal{O}\bigg{(} \frac{\log n_{\text{a}}}{n_{\text{a}}}\bigg{)} \tag{19}\]  
where the channel capacity \(\mathcal{C}_{\text{a}}\) and _channel dispersion_\(\mathcal{V}_{\text{a}}\) are defined by the following expressions:  
\[\mathcal{C}_{\text{a}} =\log_{2}(1+\gamma_{\text{a}})\] \[\mathcal{V}_{\text{a}} =\frac{\gamma_{\text{a}}(2+\gamma_{\text{a}})}{2(1+\gamma_{\text {a}})^{2}}(\log_{2}e)^{2}\]  
and \(Q^{-1}(\cdot)\) denotes the inverse \(Q\) function. If we now recall that the user \(\text{s}_{\text{a}}\) transmits only for a fraction \(\alpha\) of the time, and approximates (19) by disregarding the terms of order \(\mathcal{O}(\log n_{\text{a}}/n_{\text{a}})\), the set of rates achievable by \(\mathscr{F}_{\text{a}}\) for a target error probability \(\text{P}_{e}^{*}\) can be characterized as follows:  
\[\text{R}_{\text{a}}<\alpha\bigg{(}\mathcal{C}_{\text{a}}-\sqrt{\frac{\mathcal{ V}_{\text{a}}}{n_{\text{a}}}}\ Q^{-1}(\text{P}_{e}^{*})\bigg{)}. \tag{20}\]  
Consider now service \(\mathscr{F}_{\text{b}}\). In line with what was discussed in Section 3, collisions are once more regarded as destructive, so slots in which more than one user transmits lead to the loss of all the packets sent. In the setup under study, however, a non-vanishing error probability is to be expected for any transmission rate \(\text{R}_{\text{b}}\)  even over a _singleton_ slot, in view of the finite number \(n_{\text{s}}\) of available channel uses. Accordingly, introducing the channel dispersion,  
\[\mathcal{V}_{\text{b}}=\frac{\frac{\rho_{\text{b}}}{\sigma^{2}}\Big{(}2+\frac {\rho_{\text{b}}}{\sigma^{2}}\Big{)}^{2}}{2\Big{(}1+\frac{\rho_{\text{b}}}{ \sigma^{2}}\Big{)}^{2}}(\log_{2}e)^{2}.\]and recalling the channel capacity \(\mathcal{C}_{\text{b}}\) in (5), the maximum rate \(\mathsf{R}_{\text{b}}^{*}\) that can be granted to \(\mathcal{S}_{\text{b}}\) under a codeword error probability \(\epsilon\) can be approximated as follows:  
\[\mathsf{R}_{\text{b}}^{*}(\epsilon)\cong\mathcal{C}_{\text{b}}-\sqrt{\frac{ \mathcal{V}_{\text{b}}}{n_{s}}}\ Q^{-1}(\epsilon). \tag{21}\]  
In terms of system design, (21) allows then to tune the rate-reliability requirements of \(\mathcal{S}_{\text{b}}\). From this standpoint, we assume within our study mMTC traffic to be of the best-effort nature, i.e., without stringent requirements in terms of reliability, and select the transmission rate in order to maximize the attained spectral efficiency.