# 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  
In this section, we compare the average bandwidths \(b_{\text{CCLF}}\) and \(\overline{b}_{\text{exist}}\) derived from the preceding analysis. Initially, we assume that the available bandwidth of each link follows a Gaussian distribution. Utilizing the squeeze theorem and asymptotic analysis, we compute the mathematical expectations of both \(b_{\text{CCLF}}\) and \(\overline{b}_{\text{exist}}\). Through rigorous derivation and analytical examination, we demonstrate that when the number of forwarding hops \(N\geq 1\), it holds true that \(b_{\text{CCLF}}(t)\geq\overline{b}_{\text{exist}}(t)\). Assuming that the available bandwidth \(b_{i}(t)\) follows a Gaussian distribution with a mean of \(\mu\) and a variance of \(\sigma^{2}\), the bandwidth for each link and each moment is independent. For the CCLF method,  
\[b_{\text{CCLF}}(t)=\min_{i}\left\{\frac{1}{t}\int_{0}^{t}b_{i}(\tau)\text{d}\tau \right\}=\mu \tag{11}\]  
In comparison, for the existing congestion control methods, let \(\tilde{b}_{i}(t)=\frac{b_{i}(t)-\mu}{\sigma}\), and then \(\tilde{b}_{i}(t)\) is a standard normal distribution with a mean of \(0\) and a variance of \(1\). Its Probability Density Function (PDF) is \(\phi(x)=(2\pi)^{-1/2}\text{e}^{-x^{2}/2}\), and its Cumulative Distribution Function (CDF) is \(\Phi(x)=\int_{-\infty}^{x}\phi(t)\text{d}t=\frac{1+\text{erf}\left(x/\sqrt{2} \right)}{2}\), where \(\text{erf}(x)=\frac{2\pi}{\sqrt{\pi}}\int_{0}^{x}\text{e}^{-t^{2}}\text{d}t\). First, we calculate the mathematical expectation for the bandwidth \(Z_{N}=\min_{i}\left\{\tilde{b}_{i}(t)\right\}\) of the bottleneck link, where the CDF of \(Z_{N}\) is equal to  
\[F_{Z_{N}}(z)=1-\prod_{i=1}^{N}P\left(\tilde{b}_{i}(t)\geq z\right)=1-\left[1- \Phi(z)\right]^{N} \tag{12}\]  
So its mathematical expectation is  
\[ \begin{split} E[Z_{N}]&=\int_{-\infty}^{\infty}z \left(\frac{\text{d}}{\text{d}z}F_{Z_{N}}(z)\right)\text{d}z=\int_{-\infty}^{0 }z\left(\frac{\text{d}}{\text{d}z}F_{Z_{N}}(z)\right)\text{d}z-\int_{0}^{+ \infty}z\left(\frac{\text{d}}{\text{d}z}(1-F_{Z_{N}}(z))\right)\text{d}z\\ &=-\int_{-\infty}^{0}\left[1-\Phi^{N}(-z)\right]\text{d}z+\int_{0 }^{+\infty}\left[1-\Phi(z)\right]^{N}\text{d}z\end{split}  \tag{13}\]  
We use the formulas \(\Phi(-z)+\Phi(z)=1\), \(\lim_{z\rightarrow+\infty}zF_{Z_{N}}(z)=\lim_{z\rightarrow+\infty}z\left(1-F_ {Z_{N}}(z)\right)=0\). Note that following the conditions \(-\frac{(t-1)^{2}}{2}\leq 0\), this means that \(-\frac{t^{2}}{2}\leq-t+\frac{1}{2}\). Therefore, it can be inferred that  
\[0\leq 1-\Phi(z)=\frac{1}{\sqrt{2\pi}}\int_{z}^{+\infty}\text{e}^{-t^{2}/2} \text{d}t\leq\frac{1}{\sqrt{2\pi}}\int_{z}^{+\infty}\text{e}^{-t+1/2}\text{d} t=\sqrt{\frac{\text{e}}{2\pi}}\text{e}^{-z} \tag{14}\]  
Then, the following conditions are met:  
\[0\leq\int_{0}^{+\infty}\left[1-\Phi(z)\right]^{N}\text{d}z\leq\left(\sqrt{ \frac{\text{e}}{2\pi}}\right)^{N}\int_{0}^{+\infty}\text{e}^{-Nz}\text{d}z= \frac{1}{N}\left(\sqrt{\frac{\text{e}}{2\pi}}\right)^{N} \tag{15}\]  
According to the squeeze theorem, \(\lim_{N\rightarrow\infty}\frac{1}{N}\left(\sqrt{\frac{\text{e}}{2\pi}}\right) ^{N}=0\), so it can be inferred that  
\[\lim_{N\rightarrow\infty}\int_{0}^{+\infty}\left[1-\Phi(z)\right]^{N}\text{d} z=0 \tag{16}\]  
According to the asymptotic theory presented in Chapter 10 of reference [36], as substantiated on page 302, it has been established that  
\[1-\Phi(z)\sim\frac{\phi(z)}{z}=\frac{\text{e}^{-z^{2}/2}}{\sqrt{2\pi}z} \tag{17}\]Let \(\lambda_{N}=\sqrt{2\ln N}\); then, \(z>\lambda_{N}\), \(e^{-z^{2}/2}<\frac{1}{N}\). When \(N\rightarrow+\infty\), there is  
\[