# NOMA or OMA in Delay-QoS Limited Satellite Communications: Effective Capacity Analysis
## 3 Effective Capacity
### Sum Capacity Difference between NOMA and OMA Schemes  
Finally, the sum capacity achieved with the NOMA and TDMA scheme can be written as \(E^{N}=E_{c}^{N}+E_{t}^{N}\) and \(E^{T}=E_{c}^{T}+E_{t}^{T}\), respectively. Thus, the sum capacity difference of the considered system can be given as  
\[E^{N}-E^{T}=E_{c}^{N}+E_{t}^{N}-E_{c}^{T}-E_{t}^{T}. \tag{30}\]  
Since the exact closed-form expression can be obtained by simply inserting (19) and (21)-(23) into (30); here, we just study the sum capacity difference in some extreme SNR cases. Such as when \(\rho\to 0\), according to the results given in (15) and (16), we can obtain that \(\lim\limits_{\rho\to 0}E^{N}-E^{T}=0\), due to \(\lim\limits_{\rho\to 0}E_{j}^{N/T}\big{(}\theta_{j}\big{)}=0,(j=c,t)\). Moreover, since the sum capacity difference with respect to \(\rho\) can be derived as  
\[\frac{\partial\big{(}E^{N}-E^{T}\big{)}}{\partial\rho}=\frac{\partial\big{(}E _{c}^{N}-E_{c}^{T}\big{)}}{\partial\rho}+\frac{\partial\big{(}E_{t}^{N}-E_{t }^{T}\big{)}}{\partial\rho}. \tag{31}\]  
When \(\rho\to 0\), based on the expressions given in (24) and (25), we can obtain  
\[\lim\limits_{\rho\to 0}\frac{\partial\big{(}E^{N}-E^{T}\big{)}}{ \partial\rho} \approx \frac{1-\alpha}{\ln 2}\mathcal{E}\Big{[}\big{|}g_{c}\big{|}^{2} \Big{]}+\frac{\alpha}{\ln 2}\mathcal{E}\Big{[}\big{|}g_{t}\big{|}^{2}\Big{]}- \frac{1}{2\ln 2}\mathcal{E}\Big{[}\big{|}g_{c}\big{|}^{2}\Big{]}-\frac{1}{2\ln 2} \mathcal{E}\Big{[}\big{|}g_{t}\big{|}^{2}\Big{]} \tag{32}\] \[\approx \frac{1-2\alpha}{2\ln 2}\Big{(}\mathcal{E}\Big{[}\big{|}g_{c} \big{|}^{2}\Big{]}-\mathcal{E}\Big{[}\big{|}g_{t}\big{|}^{2}\Big{]}\Big{)},\]  
which is negative since \(\alpha\leq 0.5\) and the assumption that User \(t\) experiences a better fading severity than that User \(t\) undergoes.  
When \(\rho\gg 0\), according to (11)-(14), we have  
\[\lim\limits_{\rho\gg 0}\Big{(}E^{N}-E^{T}\Big{)} \tag{33}\] \[\approx-\frac{1}{\sigma_{c}}\ln\mathcal{E}\Big{[}\alpha\frac{ \frac{\sigma_{c}}{\ln 2}}{\rho}\Big{]}+\frac{1}{\sigma_{c}}\ln\bigg{(}\mathcal{E} \bigg{[}\big{(}\rho\big{|}g_{c}\big{|}^{2}\Big{)}^{\frac{-0.5\sigma_{c}}{\ln 2 }}\bigg{]}\bigg{)}-\frac{1}{\sigma_{t}}\ln\bigg{(}\mathcal{E}\bigg{[}\big{(} \sigma\big{|}g_{t}\big{|}^{2}\Big{)}^{\frac{-0.5\sigma_{t}}{\ln 2}}\bigg{]}\bigg{)}+ \frac{1}{\sigma_{t}}\ln\bigg{(}\mathcal{E}\bigg{[}\big{(}\rho\big{|}g_{t}\big{|} ^{2}\Big{)}^{\frac{-0.5\sigma_{t}}{\ln 2}}\bigg{]}\bigg{)}\] \[\approx\frac{1}{\sigma_{c}}\ln\frac{\mathcal{E}\bigg{[}\big{(} \big{|}g_{c}\big{|}^{2}\big{)}^{\frac{-0.5\sigma_{c}}{\ln 2}}\bigg{]}}{\alpha\frac{ \sigma_{c}}{\ln 2}\rho\frac{\sigma_{c}}{\ln 2}}+\frac{1}{\sigma_{t}}\ln\frac{\rho \frac{\sigma_{c}\sigma_{t}}{\ln 2}\mathcal{E}\bigg{[}\big{(}\big{|}g_{t}\big{|}^{2}\Big{)}^{\frac{-0.5 \sigma_{t}}{\ln 2}}\bigg{]}}{\alpha\frac{-\sigma_{t}}{\ln 2}\mathcal{E}\bigg{[}\big{(} \big{|}g_{t}\big{|}^{2}\Big{)}^{\frac{-0.5\sigma_{t}}{\ln 2}}\bigg{]}},\]whose negative or positive is closely related to the values of power allocation factor \(\alpha\), transmit SNR \(\rho\), delay-QoS exponents, and fading qualities of Users \(c\) and \(t\). For simplicity, here, we only consider \(\theta_{c}=\theta_{t}\), then we have  
\[\lim_{\rho\gg 0}\left(E^{N}\!-\!E^{T}\right)\approx\frac{1}{\varphi_{c}}\left\{ \ln\mathcal{E}\left[\left(\left|g_{c}\right|^{2}\right)^{\frac{-0.5\mathrm{g_{ c}}}{\ln 2}}\right]+\ln\mathcal{E}\left[\left(\left|g_{t}\right|^{2}\right)^{ \frac{-0.5\mathrm{g_{c}}}{\ln 2}}\right]-\ln\mathcal{E}\left[\left(\left|g_{t} \right|^{2}\right)^{-\frac{\mathrm{g_{c}}}{\ln 2}}\right]\right\}. \tag{34}\]