## 3 Experiments and Results
### Satellite Network Delay Prediction Based on GMACC
\(v_{k}\) is the disturbance term of time \(k\) and represents the unpredictable part of the network delay time series. In the autoregressive modeling of time series delay, there is a certain error between the measured delay and the real delay. Moreover, because the queuing delay is closely related to the unpredictable network traffic, the change of queuing delay also causes noise in the overall network delay modeling. The above two unpredictable signals are regarded as the disturbance term in the autoregressive model. In the ordinary autoregressive model, the disturbance term is assumed to be Gaussian distributed. However, due to the burst of network traffic and the queue features of network nodes, the network traffic shows explosive and asymmetric characteristics. When a node suddenly receives a large number of data packets, the queuing delay rises abruptly in a short time. However, due to the limited processing capacity of nodes, the speed of the queuing delay reducing is limited. Thus, the probability distribution of the sudden rise and fall of the queuing delay are quite different, showing an obvious asymmetry. To make the discussion more convincing, this paper simulated the queue delay in MATLAB software (version 9.3) [41] and described its specific asymmetric characteristics inspired by the idea of queuing theory. According to the previous discussion, the arrival packet traffic in a network node queue is sudden and unpredictable, and the speed of packets leaving the node queue is constant. Suppose that there are two kinds of service packets passing through one certain network node in a period of time. The packet arrival process of one service is relatively stable, while the arrival process of the other service is very sudden and explosive. The speed of the network node processing a data packet is constant. According to the above settings, the number of packets arriving at each time  
Figure 5: All kinds of delay in a satellite network. and the number of packets processed can be simulated and calculated in MATLAB. Then, a time series can be generated to represent the change of the number of packets in the node queue over time. It should be noted that this paper assumed that the upper limit of the simulated queue length was large enough, so there was no packet loss. In the case of a constant processing speed, the queue length is proportional to the queue delay, so the change of the queue length can also represent the change of the queue delay. In this simulation, the queue length of nodes, the first-order difference of the queue length (i.e., the change of queue length), and the statistical distribution of the first-order difference are shown in Figure 6, Figure 7 and Figure 8, respectively. It can be observed from Figure 6 that the rising slope of the queue length time series is usually larger than the falling slope, which indicates that in the packet queuing simulation, the queue growth rate is faster than the queue reduction rate. From Figure 7, one can observe that the distribution range of the first-order difference of the node queue length on the positive axis and the negative axis are quite different. The maximum value on the positive axis can reach dozens, while the minimum value on the negative axis is not less than \(-5\). Figure 8 shows that the distribution of the first-order difference of the node queue length is unimodal, and the mean value is approximately equal to 0. In addition, there are some outliers in the right tail, which is thicker than the tails of an exponential distribution. Thus, it is a heavy-tail distribution. Because the right tail is much longer than the left tail, the distribution can be described as a right-skewed distribution. Next, we simulated the satellite network delay prediction in MATLAB. In the simulation, all criteria related to the GMACC were included in the comparison algorithms. Because the optimization method was a gradient descent method, only the names of the criteria were used to represent the algorithms in order to simplify the expression.