## 3 System Model and Problem Formulation
### System Model  
#### 3.1.1 Network Model  
We define the underlying network as an undirected graph composed of \(G(N,E)\), where \(N\) represents the set of SDN switches and \(E\) represents the set of physical links. The number of controllers that will be placed on the network is defined as \(k\), and the set of controllers is defined as \(C=\{c_{1},c_{2},\ldots,c_{k}\}\). Each controller manages a certain number of switches. Importantly, each controller is located on a switch in the network. That is, some nodes are responsible for both control and forwarding functions. Moreover, two communication models exist for southbound interfaces in SDN, in-band and out-of-band [34]. In this paper, we adopt the switch-to-controller communication model with in-band connections. In a real network scenario, the servers are more widely distributed geographically, resulting in a higher cost of building the network. For the out-of-band communication model, it is necessary to add dedicated links for controlling traffic transmission and available physical ports for adding dedicated links to achieve network scalability. As a result, to reduce the cost of network deployment and efficiently utilize resources such as links and ports, it is more reasonable to adopt the in-band communication model between switches and controllers for practical network scenarios. Similarly, this paper adopts in-band communication between controllers. A competitive relationship exists between the switch-controller delay and the inter-controller delay. When reducing the switch-controller latency, the controllers are deployed relatively far from each other, increasing the inter-controller delay. Conversely, controllers are deployed relatively close together to reduce inter-control communication latency, resulting in an increase in switch-controller latency. If the controllers are required to be deployed geographically close to each other, then it is more reasonable to use out-of-band link interconnections between \begin{table}
\begin{tabular}{c c} \hline \hline
**Notation** & **Description** \\ \hline \(G(N,E)\) & The network with \(|N|\) switches and \(|E|\) physical links. \\ \(C=\{c_{1},c_{2},\ldots,c_{k}\}\) & The set of controllers, where \(k\) is the number of controllers. \\ \(\lambda_{i}\) & The average flow requests generated by switch \(i\in N\). \\ \(Ca\) & The capacity of the controller. \\ \(p_{sync}\) & The frequency of synchronization between controllers. \\ \(d_{ij}^{nc}\) & The shortest transmission path length between switch \(i\) and controller \(j\). \\ \(d_{ij}^{cc}\) & The shortest transmission path length between controller \(i\) and controller \(j\). \\ \(v\) & The propagation rate of electrical signals in cables. \\ \(\sigma_{c}\), \(\sigma_{n}\) & The degree centrality of controller \(c\) and switch \(n\), respectively. \\ \(\sigma_{ij}\), \(\sigma_{ij}(e)\) & The total number of shortest paths and the number of shortest paths through link \(e\) from switch \(i\) to switch \(j\), respectively. \\ \(l_{c}\), \(l_{opt}\) & The actual load and the ideal load of the controller \(c\). \\ \(\rho_{ne}\), \(\rho_{ce}\) & Link consumption for switch-controller communication and link consumption for inter-controller communication. \\ \(A_{c}\) & The set of switches managed by controller \(c\). \\ \(P_{ij}\) & The set of control paths from switch \(i\) to switch \(j\). \\ \(B_{e}\) & The bandwidth of link \(e\). \\ \(y_{j}\) & Binary variable whose value is 1 if switch \(j\) has a controller placed \\  & and 0 otherwise. \\ \(x_{ij}\) & Binary variable whose value is 1 if switch \(i\) is linked to controller \(j\) and 0 otherwise. \\ \(z_{e_{ij}}\) & Binary variable with value 1 if link \(e\)is in \(P_{ij}\), 0 otherwise. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of symbols. controllers. However, considering the actual network environment, the switch-controller delay is as significant as the inter-controller latency. Thus, controllers adopt in-band communication to balance switch-controller delay with inter-controller delay.