# Lossless and High-Throughput Congestion Control in Satellite-Based Cloud Platforms +
## 3 Integrated System and Algorithm Design
### Throughput Optimization Model
#### 3.3.2 Average Bandwidth Comparison
\begin{split}&\int_{\lambda_{N}}^{+\infty}\Big{[}1-\Phi^{N}(z) \Big{]}\mathrm{d}z\sim\int_{\lambda_{N}}^{+\infty}\Bigg{[}1-\Bigg{(}1-\frac{ \mathrm{e}^{-z^{2}/2}}{\sqrt{2\pi z}}\Bigg{)}^{N}\Bigg{]}\mathrm{d}z\sim\int_{ \lambda_{N}}^{+\infty}\frac{N\mathrm{e}^{-z^{2}/2}}{\sqrt{2\pi z}}\mathrm{d}z\\ &\leq\frac{N}{\sqrt{2\pi}}\int_{\lambda_{N}}^{+\infty}\frac{z \mathrm{e}^{-z^{2}/2}}{(\lambda_{N})^{2}}\mathrm{d}z=\frac{1}{\sqrt{2\pi( \lambda_{N})}^{2}}=\frac{1}{2\sqrt{2\pi\ln N}}\approx 0\end{split}  \tag{18}\]  
Note the conditions that  
\[\Phi(\lambda_{N}-\epsilon)\sim 1-\frac{\mathrm{e}^{-\ln N+\epsilon\sqrt{2\ln N }-\frac{\epsilon^{2}}{2}}}{\sqrt{2\pi}\Big{(}\sqrt{2\ln N}-\epsilon\Big{)}}= 1-\frac{\mathrm{e}^{-\frac{\epsilon^{2}}{2}}\mathrm{e}^{\epsilon\sqrt{2\ln N} }}{\sqrt{2\pi N}\Big{(}\sqrt{2\ln N}-\epsilon\Big{)}} \tag{19}\]  
\[\lim_{N\rightarrow\infty}\frac{\mathrm{e}^{-\frac{\epsilon^{2}}{2}}\mathrm{e} ^{\epsilon\sqrt{2\ln N}}}{\sqrt{2\pi}\Big{(}\sqrt{2\ln N}-\epsilon\Big{)}}=+\infty \tag{20}\]  
For \(\epsilon\to 0,N\rightarrow+\infty\), we can derive that  
\[ \begin{split}&\int_{0}^{\lambda_{N}}\Phi^{N}(z)\mathrm{d}z=\int_{ \lambda^{N}-\epsilon}^{\lambda_{N}-\epsilon}\Phi^{N}(z)\mathrm{d}z+\int_{ \lambda_{N}-\epsilon}^{\lambda_{N}}\Phi^{N}(z)\mathrm{d}z\\ &\precsim\int_{0}^{\lambda_{N}-\epsilon}\Bigg{(}1-\frac{\mathrm{ e}^{-\frac{\epsilon^{2}}{2}}\mathrm{e}^{\epsilon\sqrt{2\ln N}}}{\sqrt{2\pi}N \Big{(}\sqrt{2\ln N}-\epsilon\Big{)}}\Bigg{)}^{N}\mathrm{d}z+\epsilon\\ &\sim\lambda_{N}\exp\left(-\frac{\mathrm{e}^{-\frac{\epsilon^{2} }{2}}\mathrm{e}^{\epsilon\sqrt{2\ln N}}}{\sqrt{2\pi}\Big{(}\sqrt{2\ln N}- \epsilon\Big{)}}\right)+\epsilon\approx 0\end{split}  \tag{21}\]  
Finally, for \(N\rightarrow+\infty\),  
\[ \begin{split}&-\int_{-\infty}^{0}\Big{[}1-\[PERSON]^{N}(-z)\Big{]} \mathrm{d}z=-\int_{0}^{+\infty}\Big{[}1-\[PERSON]^{N}(z)\Big{]}\mathrm{d}z\\ &=-\int_{0}^{\lambda_{N}}\mathrm{d}z+\int_{0}^{\lambda_{N}}\[PERSON]^{ N}(z)\mathrm{d}z-\int_{\lambda_{N}}^{+\infty}\Big{[}1-\[PERSON]^{N}(z)\Big{]} \mathrm{d}z\sim-\lambda_{N}=-\sqrt{2\ln N}\end{split}  \tag{22}\]  
Therefore, for \(N\rightarrow+\infty\), it can be concluded that  
\[E[Z_{N}]=-\int_{-\infty}^{0}\Big{[}1-\[PERSON]^{N}(-z)\Big{]}\mathrm{d}z+\int_{0}^ {+\infty}[1-\Phi(z)]^{N}\mathrm{d}z\sim-\sqrt{2\ln N} \tag{23}\]  
Note that \(\bar{b}_{i}(t)=\frac{b(t)-\mu}{\sigma}\), so the mathematical expectation of \(\min_{i}\{b_{i}(t)\}\) is approximately equal to \(\mu+E[Z_{N}]\sigma\). \(E[Z_{1}]=0\), \(E[Z_{2}]=-1/\sqrt{\pi}\), \(E[Z_{3}]=-1.5/\sqrt{\pi}\) when \(N\rightarrow+\infty\), \(E[Z_{N}]\) reaches its extremum:  
\[\lim_{N\rightarrow\infty}\frac{E[Z_{N}]}{-\sqrt{2\ln N}}=1 \tag{24}\]  
In summary, using the CCLF method, the equivalent average available bandwidth of the network is \(\mu\). However, using existing methods, the equivalent bandwidth is approximately equal to \(\mu+E[Z_{N}]\sigma\). Since, when \(N\geq 1\), \(E[Z_{N}]\leq 0\), \(\mu\geq\mu+E[Z_{N}]\sigma\). Therefore, the following conclusion is drawn:  
\[b_{\text{CCLF}}(t)\geq\overline{b}_{\text{exist}}(t) \tag{25}\]  
To summarize, the CCLF method optimizes the average bandwidth, which represents the enhancement of network throughput.