## 4 Model Design
### The Proposed ATS-GRU Model  
Let \(B_{n-1:n}\) be the average traffic loads from a satellite network at time slot \(\Delta t_{n-1}\), \(n=t_{n}-t_{n-1}\). The conventional approach conducts traffic sampling at a constant rate, as follows:  
\[\Delta t_{0,1}=\Delta t_{1,2}=\cdots=\Delta t_{n-1,n}=\cdots,\ (n=0,1,2,\cdots)\]  
The management layer of the satellite network, upon accurately receiving the load value of \(B_{n-1:n}\), can dynamically adjust routing strategies to better allocate network resources and manage service quality. At the next time slot \(\Delta t_{n}\), satellite traffic load may change to a new value of \(B_{n:n+1}\), and the management layer will have to make adjustments to the routing strategy once again. It is, nevertheless, impossible for the manager to receive the value of \(B_{n-1:n}\) before the time of \(t_{n}\). Thus, at time \(t_{n-1}\), we can estimate the approximate value \(B_{n-1:n}\), which is denoted by \(B^{\prime}_{n-1:n}\). If \(B^{\prime}_{n-1:n}<B_{n-1:n}\), the satellite link or node may reach its capacity limit and suffer from a shortage of bandwidth resource to accommodate all streaming sessions. As a solution, it is possible to optimize routing strategies to ensure that traffic is transmitted via the shortest or optimal paths or to evenly distribute traffic across multiple satellite nodes. On the other hand, if \(B^{\prime}_{n-1:n}>B_{n-1:n}\), part of the bandwidth resource is simply overlooked and useless. In this sense, the traffic prediction is critical to ensure a good performance of our proposed scheme. The 6G satellite network traffic load prediction model requires two key components:  
1. a way to accurately calculate and predict network traffic load;
2. a method to determine and predict the appropriate length of time stepping at any given time, whether long or short step sizes are required. For a 6G satellite network, the traffic load varies in such a dynamic way that a fixed step size can hardly achieve a good performance all the time. For this reason, the adaptive time-stepping method combined with GRU neural networks is more suitable for predicting the dynamic traffic load of a satellite network. The ATS-GRU predictor adjusts the step size dynamically according to the square of prediction errors. Generally speaking, a greater squared value will cause ATS-GRU to increase the time step size to provide faster tracking, while a smaller squared value will cause ATS-GRU to decrease the time step size to obtain a smaller misadjustment. The GRU model takes as input a time window of traffic load data, represented as \(X(n)=\left[Y_{n-m},Y_{n-m+1},\cdots,Y_{n}\right]\), with \(m\) representing the size of the time window. These data are processed by the GRU model to generate a traffic load prediction for the next time \(t_{n+1}\) point, denoted as \(\hat{Y}_{n+1}\). Subsequently, we can assess the prediction accuracy using various evaluation metrics, resulting in the prediction error \(e(n)\). ATS-GRU rules the time-stepping size at time \(t_{n}\) as \(T_{n}\), and then updates it in integration by  
\[S^{\prime}_{s}(n+1)=S_{s}(n)\bigg{[}\gamma+(1-\gamma)\frac{|e(n)|}{E_{b}} \bigg{]} \tag{7}\]  
with \(0<\gamma<1,\ E_{b}>0\), and  
\[S_{s}(n+1)=\left\{ \begin{array}{ll}S_{s}^{\max},&\quad\text{if }S^{\prime}_{s}(n+1)>S_{s}^{\max}\\ \text{gin}_{s}^{\min},&\quad\text{if }S^{\prime}_{s}(n+1)<S_{s}^{\min}\\ S^{\prime}_{s}(n+1),&\quad\text{otherwise}\end{array} \right. \tag{8}\]  
where \(\gamma\) is the remembering factor and is usually chosen in the range (0, 1) to provide exponential forgetting, \(E_{b}\) is the targeted prediction error bound, and \(S_{s}^{\max}\) and \(S_{s}^{\min}\) are the upper and lower bounds for time-stepping size. We will give a more detailed explanation of the parameters next. The scheme of ATS-GRU optimizes the sampling interval by the following principle. The larger the prediction error is, the sharper the traffic load curve changes. As a result,we must increase the time-stepping size for satisfying prediction accuracy.