## 2 System Model  
A downlink heterogeneous network model consisting of an MBS and multiple SBSs was considered. The MBS is at the center of region \(\mathcal{L}\subseteq\mathbb{R}^{2}\), and a large number of SBSs are deployed around it to meet the peak traffic demand. \(B_{M}=\{1\}\) and \(B_{S}=\{2,3,\cdots,M\}\) denote the index sets of the MBS and SBSs, respectively, and \(B=B_{M}\bigcup B_{S}\) denotes the index set of all BSs. SBSs are distributed independently and randomly around MBS, and there are overlapping coverage areas among SBSs. In addition, since SBSs generally use digital subscriber line (DSL) or cable modem (CM) as backhaul links, their capacity is much smaller than the backhaul links of the MBS; therefore, cache units are configured for all SBSs to cache certain contents at SBSs. In contrast, MBSs do not, thus all contents are obtained from the core network through the backhaul link for the MBS. We assume that the set of all possible user requests for content in the region at a given time is \(\mathcal{F}=\{1,2,\cdots,F\}\) and \(S_{f}\) denotes the size of content \(f\). For simplicity, it is assumed that all the contents are of the same size, \(S_{f}=S\), because even if the content is of different sizes, it can be divided into blocks of the same size or encoded into blocks of the same size by encoding techniques. Therefore, the results of this paper can be applied to the scenarios with different content sizes as well. Numerous studies have shown that only a small fraction of content is frequently requested by users and the probability of users requesting content follows Zipf distribution. Assuming that the contents of \(\mathcal{F}\) are ordered in descending order of requested probability, the request probability of the \(f\)th content is expressed as  
\[q_{f}=\frac{f^{-z}}{\sum\limits_{l=1}^{F}l^{-z}} \tag{1}\]  
where \(z\) is the skew factor, and a larger \(z\) indicates that the content requested by users is more concentrated and a large number of users request only a small part of the content located in the header \(\mathcal{F}\). We assume that the requests for content \(f\) from users located at \(x\) in the region in period \(t\) obey a Poisson process with density \(\lambda_{f}^{t}(x)\), and that the requests for different contents are Poisson processes which are independent of each other. According to the superposition property of the Poisson process, the user's request for any content at \(x\) in period \(t\) obeys the Poisson process with density \(\lambda^{t}(x)=\sum\limits_{f=1}^{F}\lambda_{f}^{t}(x)\). The traffic load density at \(x\) in region \(\mathcal{L}\) in period \(t\) is defined [14] as \(\gamma^{t}(x)\doteq\sum\limits_{f=1}^{F}\lambda_{f}^{t}(x)S_{f}\), which represents the average content traffic per unit area per unit time, and the unit time in this paper refers to the duration of a period \(t\). In addition, the probability that a user located at \(x\) requests content \(f\) can be expressed as  
\[q_{f}=\frac{\int_{\mathcal{L}}\lambda_{f}(x)dx}{\int_{\mathcal{L}}\lambda(x) dx} \tag{2}\]  
We assume that all SBSs have the same caching policy, which is to cache some of the content in order of content popularity in the region from high to low, until the cache capacity is full. Let the capacity of the cache unit of the SBS be \(F_{c}S\), i.e., the SBS caches \(F_{c}\) contents and the set of cached contents is \(\mathcal{F}_{c}=\{1,2,\cdots,F_{c}\}\).