## 2 System Model
### SBS Service Cost
#### 2.2.2 SBS State Switching Cost  
The switching cost for the SBS from on mode to sleep mode or from sleep mode to on mode during two adjacent time periods is mainly referred to as the energy loss and time delay. Assuming that the energy loss of switching the SBS from on mode to sleep mode is 0, and only the energy lost in waking up the sleep state SBS is considered, the energy switching cost of the SBS can be expressed as  
\[P_{i,h}^{t}=\left\{\begin{array}{cl}\beta\Big{(}b_{i}^{t}-b_{i}^{t-1}\Big{)}^ {-}+\gamma,&\quad i\in B_{on}^{t},b_{i}^{t-1}\in(0,-1)\textit{ and }b_{i}^{t-1}\
eq b_{i}^{t}\\ 0,&\quad i\in B_{on}^{t},b_{j}^{t-1}=1\textit{ or }b_{i}^{t-1}=b_{i}^{t}\end{array}\right. \tag{10}\]  
where \(\beta\) denotes the energy consumption required to switch between two adjacent states of SBS, \(\gamma\) is the energy consumption bias, which is to make the SBS state switching cost in line with the circuit energy consumption characteristics. That is, the direct switching between two SBS states is lower than the energy consumption of an additional intermediate state, i.e., the energy consumption of the SBS switching from the Sleep state to the On state is lower than the energy consumption of the SBS first switching from the Sleep state to the Standby state and then switching from the Standby state to the On state. Function \((x)^{-}\) is defined as  
\[(x)^{-}=\left\{\begin{matrix}x,x>0\\ 0,x\leqslant 0\end{matrix}\right. \tag{11}\]  
Since the wake-up time required for sleep state SBSs of different depths varies greatly, the user service of a SBS that needs to be woken up is affected by a large delay during the  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Mode** & **Wake-Up Time (s)** & **Power Consumption** \\ \hline On & N/A & 100\% \\ Standby & 0.5 & 50\% \\ Sleep & 10 & 15\% \\ \hline \hline \end{tabular}
\end{table}
Table 1: Wake-up time and power consumption of SBS in different states [20].  
wake-up period of the SBS. In this paper, we focus on the impact on users within a period, so we define the average wake-up delay of a SBS that needs to be woken up in a period as  
\[d_{i,\mu}^{t}(\rho_{i}^{t})=\frac{T_{w}}{e^{2\rho_{i}^{t}}} \tag{12}\]  
This indicates that the wake-up delay of the SBS is negatively related to the load of the SBS, where \(T_{w}\) indicates the wake-up time. A higher load indicates that more users access the awake SBS in the \(t\) period. For the SBS, the wake-up cost can be spread among more users, so that the average wake-up delay can be lower and the SBS can be woken up more easily. On the contrary, if the relevant SBS has a low load in the \(t\) period, its wake-up delay will be high and the difficulty of wake-up will increase, in line with the actual application scenario. Combined with the state of the SBS in the previous period, Equation (12) can be expressed more specifically as  
\[d_{i,\mu}^{t}(\rho_{i}^{t})=\left\{\begin{array}{ll}\frac{0.1}{\rho_{i}^{2} \rho_{i}^{t}},&b_{i}^{t-1}=0\\ 0,&b_{i}^{t-1}=1\\ \frac{10}{\rho_{i}^{2}\rho_{i}^{t}},&b_{i}^{t-1}=-1\end{array}\right. \tag{13}\]  
Thus, the total time delay of the system in period \(t\) can be expressed as  
\[D^{t}(\mathbf{\rho}^{t})=\sum_{i\in B_{in}^{t}}d_{i,s}^{t}(\rho_{i}^{t})+d_{i,\mu} ^{t}(\rho_{i}^{t}) \tag{14}\]