# A Multiple Access Protocol Based on Gray Forecast for Satellite Network
## 3 Grey Forecast Model for Network Access Reservation  
In the CFDAMA-GF protocol, a forecast method is used in the access procedure so that the number of reservation slots currently sent by a user is greater than the number of packets in the current queue to ensure that the packet arriving at the next moment can be sent in time without queuing. In this paper, the gray forecast method is utilized to predict the user's access request. In gray system theory, an uncertain system with known information is considered as a research object [18]. Although the known information is extracted and developed, this theory makes a scientific forecast through some inherent laws of developments. In summary, the gray system contains both known and unknown information systems or uncertain information systems. Gray forecast is one of the main contents of gray system theory, and it can be understood as a forecast of a range of time-related processes within a certain range. Although this process appears to be random and complex, it has a certain order or boundary, so the process of data collection shows potential for law development. The most commonly-used model of the gray process is the grey model GM(1,1) forecast model. With this model, the differential equations of the original sequence are solved, and some sequences that can approximate differential equation models are built for other sequences. A certain inherent quality exists because of self-similarity in network data flow, and GM(1,1) can be based on a small amount of known information modeling and prediction and be suitable for the network source forecast in this paper. The modeling process of the gray forecasting model is as follows:  
If \(X^{(0)}=\{X^{(0)}(1),X^{(0)}(2),\cdots,X^{(0)}(n)\}\) is the original sequence, then the original form of the GM (1,1) model can be written as Equation (1). \[X^{(0)}(k)+aX^{(1)}(k)=b \tag{1}\]  
where \(a\) and \(b\) are the development coefficients and gray actuating quantity, respectively. An accumulation generation of the original quantity is shown in Equation (2). \begin{table}
\begin{tabular}{c c} \hline \hline
**User ID** & **Number of User Free Slots** \\ \hline User 0 & Number of User 0 timeslots \\ User 1 & Number of User 1 timeslots \\ User 2 & Number of User 2 timeslots \\ \(\cdots\) & \(\cdots\) \\ User \(n\) & Number of user \(n\) slots \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Free allocation table. \[X^{(1)}=\{X^{(1)}(1),X^{(1)}(2),\cdots,X^{(1)}(n)\} \tag{2}\]  
where \(X^{(1)}(t)=\sum\limits_{k=1}^{t}X^{(0)}(k)\). The mean generation sequence with consecutive neighbors of \(X^{(1)}\) is depicted in Equation (3). \[Z^{(1)}=\{Z^{(1)}(2),Z^{(1)}(3),\cdots,Z^{(1)}(n)\} \tag{3}\]  
where \(Z^{(1)}(k)=\frac{1}{2}(X^{(1)}(k)+X^{(1)}(k-1)),\;k=1,2,3,\cdots,n\). An accumulative matrix **B** and a constant vector **Y** are illustrated in Equations (4) and (5), respectively. \[\textbf{B}= \begin{bmatrix}-Z^{(1)}(2)&1\\ -Z^{(1)}(3)&1\\ \cdots&\ldots\\ -Z^{(1)}(n)&1\end{bmatrix}  \tag{4}\]  
\[\textbf{Y}= \begin{bmatrix}X^{(0)}(2)\\ X^{(0)}(3)\\ \cdots\\ X^{(0)}(n)\end{bmatrix}  \tag{5}\]  
If \(a\) is a gray parameter, then the least squares estimate of the gray parameter \(\widehat{a}\) of Equation (1) is satisfied with Equation (6). \[\widehat{a}= \begin{bmatrix}a\\ u\end{bmatrix} =(\textbf{B}^{T}\textbf{B})^{-1}\textbf{B}^{T}\textbf{Y} \tag{6}\]  
Solving the time response function of Equation (1), we obtain the following Equation (7). \[\widehat{X}^{(1)}(t+1)=(\widehat{X}^{(0)}(1)-u/a)e^{-at}+u/a \tag{7}\]  
Source forecasting is executed. The reducing value \(\widehat{X}^{(0)}(t+1)\) shown in Equation (8) is the predicted value of the \(t+1\) moment. \[\widehat{X}^{(0)}(t+1)=\widehat{X}^{(1)}(t+1)-\widehat{X}^{(1)}(t) \tag{8}\]  
In predicting the bursts of service in this paper, the relevant parameters applied to the gray forecast are as follows:  
1. The number of user application slots: \(N\). 2. The number of burst service packet arrivals in each cycle: \(w\). 3.