## 2 System Model
Since all the SBSs use the same frequency band, when the user at \(x\) selects a SBS to access, other turned-on SBSs will cause co-channel interference to the user, so the user's signal to interference plus noise ratio (SINR) can be expressed as  
\[\text{SINR}_{i}^{t}(x)=\frac{P_{i}^{t}H_{i}(x)}{\sum_{k\in B_{\text{nr}}^{t} \backslash\{i\}}P_{k}^{t}(x)H_{k}(x)+{\sigma_{0}}^{2}} \tag{3}\]where \(P_{i}^{t}\) denotes the transmit power of SBS, \(i\in B_{on}^{t}\), \(H_{i}(x)\) denotes the channel power gain between the user at \(x\) and access SBS, \(i\in B_{on}^{t}\), \(\sum_{k\in B_{on}^{t}\setminus\{i\}}P_{k}^{t}(x)H_{k}(x)\) denotes the sum of interference from other SBSs except access SBS \(i\), and \(\sigma_{0}^{2}\) is the noise power. The rate available to the user at \(x\) from SBS \(i\in B_{on}^{t}\) by [PERSON]'s formula is  
\[r_{i}^{t}(x)=\mathcal{W}\log_{2}\bigl{(}1+\mathrm{SINR}_{i}^{t}(x)\bigr{)} \tag{4}\]  
where \(\mathcal{W}\) indicates the bandwidth allocated by the SBS for the user communication link. We define the SBS load density [18] as \(\phi_{i}^{t}(x)=\gamma^{t}(x)/r_{i}^{t}(x)\), which represents the proportion of time required to transmit the traffic load, \(\gamma^{t}(x)\), from SBS \(i\in B_{on}^{t}\) in period \(t\). Denoted by \(a_{i}^{t}(x)  
to implement functions such as data encryption, hardware authentication, network time protocol (NTP), etc. The third part consists of radio frequency (RF) components and power amplifiers for transmitting and receiving data and transmitting high power signals to the transmitting antenna. Reference [12] analyzed the hardware power consumption model of SBSs, finding that the main power consuming parts of SBSs are mainly composed of a RF front-end and a temperature compensated crystal oscillator heating device, where the power consumption of RF front-end accounts for about 45% of the total power consumption and the power consumption of the temperature compensated crystal oscillator heating device accounts for about 7% of the total power consumption. Therefore, shutting down some of the components of the SBSs in certain areas at appropriate times to make the SBSs perform a sleep state can effectively reduce the power consumption of the network and improve the energy efficiency of the network. However, since the state of the SBSs needs to be dynamically adjusted according to the traffic changes in the region, the SBSs that are in sleep state at any one time may need to be turned back on at the next time, and it takes some time to turn on the turned-off components in the SBSs again. Considering the characteristics of caching networks, we define a three-state SBS scheme in caching networks:  
* On: the SBS is in a normal service state, and all functional modules are turned on to keep the cache content updated and provide real-time service to users. * Standby: The SBS is in a light sleep state and only some functional modules of the SBS are turned off, thus ensuring that it can be quickly woken up when necessary. Meanwhile, the SBS can receive content request information from all users in the coverage area during the standby period and update the local cache content regularly according to the preset caching algorithm, so that it can provide content services to users immediately after it is turned on. * Sleep: A SBS in deep sleep state turns off most of its functional modules and takes some time to be woken up. Moreover, its cache module does not participate in updating popular contents in the network during sleep but can receive content request information from all users in the coverage area, and immediately updates the local cache contents according to the preset caching algorithm after being woken up, and then finally provides services to users. We denote the state indicator function of the SBS as  
\[b_{i}^{t}=\left\{ \begin{array}{ll}1,&\text{On}\\ 0,&\text{Standby}\\ -1,&\text{Sleep}\end{array} \right. \tag{7}\]  
where Standby and Sleep are both two sleep modes of the SBS, only the degree differs.