# Edge Server Deployment Strategy Based on Queueing Search Meta-Heuristic Algorithm
## 3 Problem Modeling
### Network Modeling  
In this paper, we consider the edge server (ES) deployment problem. To reduce costs, we aim to select a subset of the base stations (BSs) that have already been deployed and use them as potential sites for deploying edge servers. In this paper, the average number of hops is used as a proxy for average latency to evaluate the effectiveness of deployment strategies. As shown in [20; 21], replacing average latency with hop count has been demonstrated as a feasible approach. Moreover, the proposed method is not restricted to this specific latency model; by replacing the delay model, it can be adapted to a wider range of scenarios. Designing more realistic network latency models and evaluation methods to further extend the applicability of our approach will be a key direction of our future work. The ES deployment problem can be modeled as a connected undirected graph \(\mathrm{G}=(B\cup\mathrm{S},\ E)\), as shown in Figure 1, consisting of mobile terminal devices, base stations, and potential deployment locations for edge servers. Here, \(B=\{b_{1},\dots,b_{2},\dots,b_{n}\}\) \begin{table}
\begin{tabular}{c c} \hline \hline
**Symbols** & **Meanings** \\ \hline \(G\) & Network diagram \\ \hline \(B,U,E\) & Base station aggregation, user aggregation, edge server aggregation \\ \hline \(b,u,e\) & Number of base stations, number of users, number of edge Servers \\ \hline \(B_{don}\) & Base station network density \\ \hline \(C\) & Matrix of information about users covered by base stations \\ \hline \(C_{jk}\) & Whether user \(j\) is covered by base station \(k\) \\ \hline \(B_{jk}\) & Number of hops between base station \(j\) and base station \(k\) \\ \hline \(B_{conn}\) & Base station interconnection information matrix \\ \hline \(S\) & The shortest path between base stations’ information matrix \\ \hline \(hop_{t}\) & Total number of hops \\ \hline \(H_{ave}\) & Number of hops per capita \\ \hline business1,2,3 & Business 1, 2, 3 \\ \hline \(N\) & Total number of customers \\ \hline \(X_{i}\) & Status of customer \(i\) \\ \hline \(X_{ij}\) & Customer \(i\) for operation \(j\) \\ \hline \(X_{i}^{new}\) & Status update pattern of customer \(i\) \\ \hline \(q_{in}\) & Number of people in queue \(n\) for operation \(i\) \\ \hline \(Q\) & Number of search agents \\ \hline \(T_{mm}\) & Service time for employee \(n\) to process operation m \\ \hline \(A_{mn}\) & Capacity of employee \(n\) to process operation m \\ \hline \(F_{11},F_{12}\) & Fluctuation \\ \hline \(\alpha\) & Random numbers generated between \([-1,1]\) that control the direction of fluctuations \\ \hline \(E\) & D-dimensional vector generated based on Erlang distribution \\ \hline \(f_{i}\) & State complexity of customer \(i\) \\ \hline \(rank(f_{i})\) & State complexity of customers in descending order \\ \hline \(\mathrm{Pr}_{i}\) & Probability of handling operation \(i\) \\ \hline \(e\) & Random numbers based on Erlang distribution \\ \hline \(N_{i}\) & Number of customers that need to process operation \(i\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Important symbols and significance. represents the set of base stations, \(|B|=\mathsf{b}\), and \(E=\{e_{1},\ldots,e_{i},\ldots,e_{k}\}\) represents the set of ES deployment locations, \(|E|=e\). The edge computing network model diagram is shown in Figure 1.  
In the considered edge computing scenario, the number of users is denoted by \(\mathsf{u}\). The locations of users and base stations are generated randomly, and \(\mathsf{B}_{den}\) is used to represent the base station network density. The network density between base stations is given by the following equation:  
\[B_{den}=\frac{b(b-1)}{2}, \tag{1}\]  
where \(\mathsf{b}\) denotes the number of base stations. Base station information is generated according to the network density between base stations, and a matrix is used to represent the interconnection information between the base stations.