# Caching Placement Optimization Strategy Based on Comprehensive Utility in Edge Computing
## 2 Related Work
### The Cache Value of the Data Block
#### 3.1.1 The Popularity of the Data Block  
The popularization of the data block is determined by various factors, including its access frequency, the average time interval, and the recency of access. Access frequency means the proportion of the total number of users accessing the data block in the entire set of data blocks. Therefore, the access frequency \(f_{i}^{req}\) for the data block \(i\) can be modeled in Equation (1).  
\[f_{i}^{req}=\frac{V_{i}}{T_{i}^{now}-T_{i}^{first}} \tag{1}\]  
In Equation (1), \(V_{i}\) is the number of times when the block is being accessed. \(T_{i}^{first}\) is the time of the first access and \(T_{i}^{now}\) is the currently accessed time. The average time interval, \(T_{i}^{inter}\), represents the average time between two consecutive accesses, which can be expressed in Equation (2).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters** & **Definition** \\ \hline \(i\) & Data block \(i\) \\ \(j\) & Cache server \(j\) \\ \(x_{ij}\) & Whether a data block \(i\) is placed on the edge server \(j\) \\ \(f_{i_{j}}^{req}\) & Access frequency of data block \(i\) \\ \(T_{i_{j}}^{inter}\) & The average access time interval of the data block \(i\) \\ \(T_{i}^{st}\) & The recency of the data block \(i\) \\ \(Pop_{i}\) & The popularity of the data block \(i\) \\ \(re_{i}\) & The remaining validity ratio of the data block \(i\) \\ \(Rep_{j}\) & The replacement rate of the edge server \(j\) \\ \(hit_{i,j}\) & The hit rate of the cached data block \(i\) on the server \(j\) \\ \(Value_{i}^{i}\) & The value of the data block \(i\) cached on the \(j\)-th server \\ \(Repc_{i}^{j}\) & The replacement cost incurred to cache data blocks \(i\) on the edge server \(j\). \\ \(b_{j\prime}\) & The network bandwidth between the edge server \(j\) and the physical location \(j\) \\ \(sc_{j}\) & The resource usage load is caused by placing a data block on the edge server \(j\). \\ \(BC\) & The bandwidth cost resulting from data transmission from the edge server to users \\ \(rate_{i,j}\) & The requested rate of users requesting data blocks from the edge server \(j\). \\ \(request_{i,j}\) & The total number of times a user accesses the data block \(i\) from the edge server \(j\) \\ \(C_{i}^{j}\) & The acquisition cost of the data block \(i\) to be cached from the physical location \(j\) \\ \(UL\) & the edge server \(j\) \\ \(TC\) & The placement time cost of placing the set of data blocks to be cached on the edge server. \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of the main parameters.  
\[T_{i}^{\text{inter}}=\frac{T_{i}^{\text{{last}}}-T_{i}^{\text{{first}}}}{V_{i}} \tag{2}\]  
In Equation (2), \(T_{i}^{\text{{last}}}\) denotes the time when the data block \(i\) was last accessed, \(T_{i}^{\text{{first}}}\) indicates the time when the data block \(i\) was first accessed. The size of the average access interval \(T_{i}^{\text{{inter}}}\) represents the prevalence of the data block.  
The recency of access \(T_{i}^{\text{{st}}}\) represents the time interval between the current access time and the last access time of the data block, which can be expressed as:  
\[T_{i}^{\text{{st}}}=\frac{1}{T_{i}^{\text{{now}}}-T_{i}^{\text{{last}}}} \tag{3}\]  
Therefore, combining Equations (1)-(3), the cached data block population is shown in Equation (4).  
\[Pop_{i}=\frac{f_{i}^{\text{{req}}}*T_{i}^{\text{{inter}}}}{T_{i}^{\text{{st}}} }=\frac{\frac{V_{i}}{T_{i}^{\text{{new}}}-T_{i}^{\text{{first}}}}*\frac{1}{T_ {i}^{\text{{new}}}-T_{i}^{\text{{last}}}}}{\frac{T_{i}^{\text{{last}}}-T_{i} ^{\text{{first}}}}{V_{i}}} \tag{4}\]