## 3 Initial State of SBSs Based on Relaxed Threshold for LSTM Traffic Prediction
### SBSs State Transfer Design  
Since there are two levels of sleep in the SBS, Standby and Sleep, we need to set two traffic thresholds, \(\phi_{b}\) and \(\phi_{s}\), for Standby and Sleep, respectively. The traffic volume of the SBS in time period \(t\) is denoted by \(T_{i}^{t}\), and we have  
\[b_{i}^{t}=\left\{ \begin{array}{ll}1,&T_{i}^{t}\geq\phi_{b},i\in B_{S}\\ 0,&\phi_{s}\leq T_{i}^{t}<\phi_{b},b_{i}^{t-1}\in\{0,1\},i\in B_{S}\\ -1,&0\leq T_{i}^{t}<\phi_{s}\text{ }or\left\{\phi_{s}\leq T_{i}^{t}<\phi_{b},b_{i}^{t-1}=-1 \right\},i\in B_{S}\end{array} \right. \tag{15}\]  
where \(b_{i}^{t-1}\in\{0,1\}\) indicates that only in the \(t-1\) period the SBS is in On or Standby state, and in the \(t\) period the SBS may be in Standby. The set of SBSs in different states can be denoted as \(B_{on}^{t}=\left\{i\mid b_{i}^{t}=1,\forall i\in B\right\}\), \(B_{stand}^{t}=\left\{i\mid b_{i}^{t}=0,\forall i\in B\right\}\) and \(B_{sleep}^{t}=\left\{i\mid b_{i}^{t}=-1,\forall i\in B\right\}\), respectively. The state transfer of the SBS is shown in Figure 2. In order to reduce the loss of the SBS hardware due to frequent switching between different states, it is assumed that the state of the SBS cannot be changed from the Sleep state to the Standby state, and we will now explain the rationality of this assumption. If only based on the determination result of the traffic threshold, the SBS needs to be switched from Sleep to Standby in period \(t\). There is energy loss in this process, and the SBS in the Standby state does not provide services to users, but only resumes the function of synchronizing the network cache contents and can be quickly woken up in the next period. For this SBS, three states are possible in period \(t+1\); (1) Switching back to the Sleep state makes switching the SBS from Sleep to Standby in period \(t\) a negative operation, with no gain except more energy consumption for the SBS; (2) Keeping the Standby state, its SBS state transfer process is Sleep \(\rightarrow\) Standby \(\rightarrow\) Standby, compared with the state transfer process of keeping the Sleep state unchanged in period \(t\): Sleep \(\rightarrow\) Sleep \(\rightarrow\) Standby, the former obviously consumes more energy and the final state is also Standby, so it does not reduce the subsequent SBS state switching delay; (3) By switching to the On state, for the SBS from the period \(t-1\) of the Sleep state to the period \(t\) of the Standby state and then to the period \(t-1\) of the On state, the SBS experiences two states switching. Each switching state will have an energy loss, and the SBS switching cost formula shows that, compared to the one-time energy loss from Sleep to On, the total energy consumption of going through the two state stages is more, and during the duration of the Standby state, the energy consumed is also much larger than that of the Sleep state. On the other hand, the SBS switching delay from Standby to On is much lower than that from Sleep to On. Next, we analyze the combined cost of these two different state transfer methods. Assuming that the states of all SBSs except the target SBS under study are kept constant, and letting the duration of each period be \(\tau\), the additional costs incurred through the different states are:  
\(\text{Sleep}\rightarrow\text{Standby}\rightarrow\text{On}\)  
\[C1=\alpha\Big{(}P_{sleep}\tau+P_{stand}\tau+2(\beta+\gamma)\Big{)}+\frac{0.1} {e^{2\rho_{i}^{t}}} \tag{16}\]  
\(\text{Sleep}\rightarrow\text{Sleep}\rightarrow\text{On}\)  
\[C2=\alpha\Big{(}2P_{sleep}\tau+2\beta+\gamma\Big{)}+\frac{10}{e^{2\rho_{i}^{ t}}} \tag{17}\]  
Therefore, we have  
\[\Delta C=C1-C2=\alpha\Big{[}\Big{(}P_{stand}-P_{sleep}\Big{)}\tau+\gamma\Big{]}- \frac{9.9}{e^{2\rho_{i}^{t}}} \tag{18}\]  
where \(P_{sleep}\) and \(P_{stand}\) are fixed values, and the energy consumption bias, \(\gamma\), is generally a small positive number, so the value of \(\Delta C\) depends on the weighting factor, \(\alpha\), and the period duration, \(\tau\).