# Edge Server Deployment Strategy Based on Queueing Search Meta-Heuristic Algorithm
## 3 Problem Modeling
### Latency Modeling  
Since this paper addresses the server deployment problem, not the network transmission problem, we assume that the request sent by the user will not be affected by the channel delay and the processing time of the base station during the transmission process; this assumption is made to simplify the model and focus on analyzing the request's hopping path and efficiency in the base station network. The hop count schematic is shown in Figure 2.  
Figure 1: Network model diagram.  
Specifically, when a request is made by an end device j, the request is first transmitted to the base station k covering the end device, i.e., \(C_{jk}=1\), a process defined as a hop count. The calculation of the hopping number is based on the transmission path of the request from the user device to the base station; we assume that the channel delay and the base station processing time are ignored, so the hopping number mainly depends on the physical distance between base stations and the network topology. If the base station is an MEC base station, then the request from the end device will be considered to have been successfully forwarded to the edge server without additional hops; assuming that the end device is not covered by an MEC base station, the number of hops for the user can be expressed as:  
\[H_{u}=1+H_{b_{u},e_{u}}, \tag{2}\]  
where \(H_{u}\) denotes the number of hops of the terminal device u, \(b_{u}\) denotes the base station covering the terminal device u, \(e_{u}\) denotes the base station mounted on the edge server nearest to the base station \(b_{u}\), and \(H_{b_{u},e_{u}}\) denotes the number of hops required between the base station \(b_{u}\) and the MEC station \(e_{u}\), and for a non-MEC base station \(b_{u}\
otin E\), the shortest number of hops to the nearest MEC base station is defined as follows:  
\[H_{b_{u},e_{u}}=\underset{e_{u}\in E}{\min}D(b_{u},e_{u}), \tag{3}\]  
\(D(b_{u},e_{u})\) denotes the iterative computation by [PERSON]'s algorithm:  
\[D(b_{u},e_{u})=\underset{k}{\arg\min}[M_{b_{u},b_{k}}+M_{b_{k},e_{u}}],\forall k \in B, \tag{4}\]  
\(D(b_{u},e_{u})\) denotes the shortest path calculated by [PERSON]'s algorithm, \(M_{b_{u},b_{k}}\) denotes the number of hops from base station \(b_{u}\) to base station \(b_{k}\), and \(M_{b_{k},e_{u}}\) denotes the number of hops from base station \(b_{k}\) to MEC station \(e_{u}\). We can derive the total number of hops for a single user u, and \(H_{u}\) can be expressed by the following segmentation function:  
\[H_{u}=\left\{\begin{array}{l}1,\phi(u)\in e\\ 1+H_{\phi(u),e},\phi(u)\
otin e\end{array}\right., \tag{5}\]  
Figure 2: Diagram illustrating the number of hops.  
In Equation (1), \(\phi(u)\in E\) indicates that user u is covered by the MEC base station, and \(\phi(u)\
otin E\) indicates that user u is not covered by the MEC base station and a hopping operation is required. The total number of hops can be expressed as the following:  
\[hop_{t}=\mathrm{H_{total}}=\sum_{u\in U}H_{u}=\mid\{u|\phi(u)\in E\}\mid+\sum_{ \phi(u)\
otin E}(1+H_{\phi(u),e}), \tag{6}\]  
The average number of hops per user terminal can then be expressed as follows:  
\[T_{ave}=\overline{H}=\frac{1}{|u|}H_{total}=\frac{U_{direct}}{|u|}+\frac{1}{|u |}\sum_{b\in B}U_{b}(1+H_{b,e}), \tag{7}\]  
where \(U_{direct}\) denotes the number of users directly connected to the MEC base station, \(U_{b}\) denotes the number of users covered by base station b, and the average number of hops is the total number of hops divided by the total number of terminal devices.  
The problem is transformed into a single-objective multi-constrained optimization problem, which can be expressed as follows:  
\[\min T_{ave}, \tag{8}\]  
\[s.t.(1)-(7)\]