## 2 System Model and Problem Formulation
### System Model
#### 2.1.1 Wireless Transmission and Content Caching Model  
The wireless transmission model can be simplified as follows. The transmission rate is set to be \(C_{MEC}\)\((\)Mbps\()\) if the MU can download the interested content from the MECS. Otherwise, a MU can only take the content from the core network (CN) by backhaul link at the rate of \(C_{Back}\) Mbps. Also, the backhaul capacity of BS is denoted as \(C_{bmax}(\)Mbps\()\) which is generally limited in dense cellular network scenarios. We further consider that \(C_{Back}<C_{MEC}\). In the model, we focus on a multi-user orthogonal frequency division multiple access (OFDMA) system in mobile networks, where each channel in the system is orthogonal to the others--in other words, no interference among MUs [24]. For simplicity, we assume that channel gains have the same distribution and small-scale fast fading will average out. Hence the wireless transmission rate of one MU depends on its available bandwidth and signal-to-noise ratio (SNR) [7]. In the cell, with available bandwidth W for each MU and a given BS transmission power \(P_{trans}\), the wireless transmission rate of a MU is given by the [PERSON]'s theorem:  
\[C = W\bigg{(}\log_{2}\bigg{(}1+\frac{P_{trans}\beta R^{-\epsilon}}{W \delta^{2}}\bigg{)}\bigg{)} \tag{1}\] \[\sum_{MUL}\ C_{Back}\leq C_{bmax} \tag{2}\]  
where \(\delta^{2}\) is the noise power, \(\epsilon\) is the pathloss exponent and \(\beta\) is the pathloss constant.  
Some previous studies have shown that in practical networks, the request probability of content can be fitted with some popularity distributions. In the proposed work, we assume that the MUs request content (i.e., videos, files, news, etc.) from a library \(\mathcal{F}=\{1,\cdots F\}\), where each content-\(f\) in this library has a same size of \(L(f)\) bits and different popularity. The probability content-\(f(f=1\cdots F)\) being requested is denoted as \(P_{\mathcal{F}}(f)\), i.e., \(\sum_{f=1}^{F}P_{\mathcal{F}}(f)=1\). As a matter of fact, the popularity of requested contents follows the Zipf's distribution [23; 24; 25], which can be expressed as:  
\[P_{\mathcal{F}}(f)\ =\ \frac{\Omega}{f^{\alpha}} \tag{3}\]  
where  
\[\Omega\ =\bigg{(}\sum_{i=1}^{F}\frac{1}{i^{\alpha}}\bigg{)}^{-1}\]  
The parameter \(\alpha\) in equation (3) describes the steepness of the distribution. Like the distribution of contents in the web proxies and the traffic dynamics of cellular devices, this kind of power law is used to characterize many real world phenomena [23]. The higher \(\alpha\) value corresponds to a steeper distribution, and indicates that a fraction of the content is more popular than the rest of the catalog (i.e., users have very similar interests). For another, lower values describe more uniform behavior almost as popular as the content (i.e., users have different interests) The parameter \(\alpha\) can take different values depending on the MUs' behaviour and MECS deployment strategies (i.e., campus, enterprise, urban, and rural environments), and its practical value in our experimental setup will be given in the subsequent sections.  
Without the loss of generality, the contents of the library are sorted in line with the descending order of popularity, in which content-\(I\) indicates the content with the highest downloading probability. As a consequence, MUs are considered to make independent downloading requests based on \(P_{\mathcal{F}}(f)\). Caching the most popular contents will be regarded as the optimal caching strategy for MECS, and hence the caching hit ratio (\(Q\)) of \(A_{c}\) MECSs can be written as:  
\[Q=1-\frac{\sum_{f=A_{c}*G_{c}}^{F}\frac{1}{f^{\alpha}}}{\sum_{f=1}^{F}\frac{1} {f^{\alpha}}} \tag{4}\]where \(C_{s}\triangleq\frac{C_{Z}}{L(f)}\) is defined as the number of the caching contents of MECS, the number of active MECSs is \(A_{c}\).  
\[1-Q=\frac{\sum_{f=A_{c}*C_{s}}^{F}\frac{1}{f^{n}}}{\sum_{f=1}^{F}\frac{1}{f^{n}}}\]  
Proof. The proof is presented in Appendix A.