# An Efficient Dynamic Load Balancing Scheme Based on Nash Bargaining in SDN
## 2 Analysis and Modeling
### Model Construction  
The multi-controller network generally adopts the in-band communication mode [21]--that is, the switch and the controller are deployed at the same node, and the control flow and the data flow are transmitted using the same channel. According to the relevant knowledge of graph theory, SDN is modeled as an undirected graph \(G=\{V,E\}\). In network \(G\), \(V\) represents the set of controllers and switch nodes, and \(E\) represents the set of links between the nodes. Suppose there are \(M\) controllers in the network; the controller set is represented as \(C=\{c_{1},c_{2},\ldots,c_{M}\}\), and the capacity is \(\omega_{i}\). There are \(N\) switches, and the switch set is represented as \(S=\{s_{1},s_{2},\ldots,s_{N}\}\). Each controller and its managed switches form a subdomain, and controllers between different domains communicate with each other and share the same network view. Moreover, each switch in the network has only one master controller and any slave controller. In order to clearly describe the network model, the network topology corresponding to Figure 1 is given below, as shown in Figure 2.  
Figure 1: Multi-controller network architecture.  
Detailed descriptions of the main parameters are shown in Table 1.  
In order to explain the migration process of the switch in detail, the related parameters in the network are defined as follows.  
**Definition 1**.: _Mapping matrix. The connection relationship between the controller and the switch in the domain is defined as the mapping matrix--i.e., \(X=[x_{ki}]_{\text{MON}}\), where \(x_{ki}=1\) means that switch \(s_{k}\) is managed by the controller \(c_{i}\), \(x_{ki}=0\) means that it is not managed by \(c_{i}\); moreover, a switch can only be managed by one controller, so it satisfies \(\sum\limits_{i=1}^{M}x_{ki}=1\)_  
**Definition 2**.: _Node connectivity. The connection between two nodes is denoted as \(\Lambda=\left[a_{ij}\right]_{\text{MON}}\), where \(a_{ij}=1\) means that there is a connection relationship between nodes \(s_{i}\) and \(s_{j}\). If \(a_{ij}=0\), the two nodes do not have a connection relationship._  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Network Parameters** & **Description** \\ \hline \(X\) & Mapping matrix \\ \(\Lambda\) & Connected matrix \\ \(d_{ki}\) & Hops between devices \(k\) and \(i\) \\ \(\Psi_{c_{i}}\) & Controller \(c_{i}\) maintains the traffic required by the management domain \\ \(\Upsilon_{c_{i}}\) & Flow table installation overhead generated by controller \(c_{i}\) \\ \(\Gamma_{c_{i}}\) & Inter-domain controller synchronization overhead of controller \(c_{i}\) \\ \(L_{c_{i}}\) & Load of controller \(c_{i}\) \\ \(\overline{L}\) & The average load of the controller \\ \(LBD\) & Load balancing degree of the controller \\ \(P_{y_{\text{R}}}^{com}\) & The communication cost of migrating switch \(s_{k}\) \\ \(P_{s_{k}}\) & Migration request cost of switch \(s_{k}\) \\ \(Cost\) & Total switch migration cost \\ \(g\) & The game, containing three elements: players, strategy, and revenue \\ \(a_{LBD}\) & Bargaining breakdown point of load balance degree \\ \(\text{a}_{Cost}\) & Bargaining breakdown point of migration cost \\ \hline \hline \end{tabular}
\end{table}
Table 1: Network parameters and description.  
Figure 2: Network topological structure.  
The purpose of switch migration is to balance the load on the control plane effectively. However, switch migration can also produce high migration costs. Therefore, this paper focuses on load balancing and migration cost issues after network remapping. Next, the two performance indicators are described in detail.