# Lossless and High-Throughput Congestion Control in Satellite-Based Cloud Platforms +
## 3 Integrated System and Algorithm Design
### Data Flow Scheduling Model
Consequently, the original configuration does not represent an optimal solution. If all Data are prioritized distinctly, that is, \(\forall i\
eq j,p_{i}\
eq p_{j}\), then there exists a unique optimal solution. This involves arranging the forwarding order of Data packets in descending order of \(p_{i}\). However, when some Data packets share the same priority, swapping the forwarding order of these Data packets does not affect the objective function value but can lead to significant fluctuations in flow completion times on the consumer side. To address this issue, we optimize the transmission rate of each flow to ensure a reasonable bandwidth allocation among flows with equal priority, thereby maintaining consistent performance. Let \(\left\|d(t)\right\|_{2}=\sqrt{\sum_{i}\left(d_{i}(t)\right)^{2}}\), where \(d_{i}(t)\) represents the average queuing time for packets in flow \(i\) waiting to be transmitted at time \(t\). It can be equivalently modeled as a stochastic process, since packet arrivals are random. We optimize the transmission rate \(x_{i}\) for each data flow \(i\) to ensure that \(\left\|d(t)\right\|_{2}\) decreases as quickly as possible, that is, minimizing \(\mathbb{E}\left[\frac{\mathrm{d}}{\mathrm{d}t}\|d(t)\|_{2}\right]\). Simultaneously, we seek to maximize the utility function \(\sum_{i}U_{i}(x_{i})\), where \(U_{i}(x_{i})=\log x_{i}\). Here, \(U_{i}(x_{i})\) is a twice-differentiable concave function, ensuring that the optimization problem has a unique global maximum. This is a multi-objective optimization problem. To solve it, we consolidate the multiple objectives into a single-objective optimization problem by introducing an adjustable weight parameter \(K\):  
\[ \begin{split}\text{maximize}&\left\{\sum_{i}U_{i} (x_{i})-K\cdot\mathbb{E}\left[\frac{\mathrm{d}}{\mathrm{d}t}\|d(t)\|_{2} \right]\right\}\\ \text{s.t.}&\sum_{i}x_{i}\leq C,x_{i}\geq 0\end{split}  \tag{32}\]  
In this paper, we use \(C\) to represent the total available bandwidth in bits per second. Let \(\mu_{i}\) denote the mean number of packets arriving per second for flow \(i\), and let \(\sigma_{i}\) represent the standard deviation of the number of packets arriving per second for flow \(i\). Let \(M_{i}\) be the length (in bits) of each packet in flow \(i\). Then, \(d_{i}(t)\) corresponds to a Brownian Motion with Drift, which can be described by the following Stochastic Differential Equation:  
\[\mathrm{d}d_{i}(t)=\left(\mu_{i}-\frac{x_{i}}{M_{i}}\right)\mathrm{d}t+\sigma _{i}\mathrm{d}B(t) \tag{33}\]  
where \(B(t)\) is a Brownian Motion. Using the well-known Ito formula [37], we can derive  
\[\mathrm{d}(d_{i}(t))^{2}=\left[2d_{i}(t)\left(\mu_{i}-\frac{x_{i}}{M_{i}} \right)+\left(\sigma_{i}\right)^{2}\right]\mathrm{d}t+2d_{i}(t)\sigma_{i} \mathrm{d}B(t) \tag{34}\]We set the \(\mu(t)\) and \(\sigma(t)\): \(\mu(t)=\sum_{i}2d_{i}(t)\Big{(}\mu_{i}-\frac{x_{i}}{M_{i}}\Big{)}+(\sigma_{i})^{2}\), \(\sigma(t)=2\sum_{i}d_{i}(t)\sigma_{i}\). From these settings, we can derive \(\mathrm{d}\|d(t)\|_{2}^{2}=\mu(t)\mathrm{d}t+\sigma(t)\mathrm{d}B(t)\). Next, by applying Ito formula and setting \(d(t)=\|d(t)\|_{2}\), we obtain  
\[\mathrm{d}[d(t)]=\mathrm{d}\sqrt{d^{2}(t)}=\frac{4\mu(t)d^{2}(t)-[\sigma(t)]^{ 2}}{8d^{3}(t)}\mathrm{d}t+\frac{\sigma(t)}{2d(t)}\mathrm{d}B(t) \tag{35}\]  
\[\mathbb{E}\bigg{[}\frac{\mathrm{d}}{\mathrm{d}t}\|d(t)\|_{2}\bigg{]}=\frac{4 \mu(t)d^{2}(t)-[\sigma(t)]^{2}}{8d^{3}(t)} \tag{36}\]  
\[\frac{\partial}{\partial x_{i}}\mathbb{E}\bigg{[}\frac{\mathrm{d}}{\mathrm{d} t}\|d(t)\|_{2}\bigg{]}=\frac{\partial}{\partial x_{i}}\frac{\mu(t)}{2d(t)}=- \frac{d_{i}(t)}{d(t)M_{i}} \tag{37}\]  
For the optimization problem, we aim to maximize the function as shown in (32). To solve this problem, we define the Lagrangian \(L\) as follows:  
\[L(x_{1},\ldots,x_{n},\lambda)=\sum_{i}U_{i}(x_{i})-K\cdot\mathbb{E}\Bigg{[} \frac{\mathrm{d}}{\mathrm{d}t}\sum_{i}\|d(t)\|_{2}\Bigg{]}+\lambda\Bigg{[}C- \sum_{i}x_{i}\Bigg{]} \tag{38}\]  
where \(\lambda\) is the Lagrange multiplier, and \(C\) is the available channel bandwidth.