## 3 Basic Model of Traffic Engineering Routing and Limiting  
We describe the network model and its structural and functional parameters according to the notations introduced in [21; 22], and the new ones used in the current research. The primary notations used in the models are presented in Table 2. \begin{table}
\begin{tabular}{c l l l} \hline \hline
**Ref.** & **Contribution** & **Underlying Approach** & **Field of Application** \\ \hline
[26] & Proposal of a reinforcement learning (RL) based switch and controller selection scheme for switch migration, switch-aware RL load balancing (SAR-LB) under the utilization ratio of various resource types in controllers & Reinforcement Learning & SDN \\  & and switches as the inputs of the neural network & & \\ \hline
[27] & QoS-aware adaptive routing protocol (SQAR) based on reinforcement learning, which can intelligently select routes to satisfy the QoS requirements of multiple Internet of Underwater Things (IoUT) services & Reinforcement Learning & IoUT \\ \hline
[28] & Development of secure and energy-aware fog computing architecture; load balancing technique improving the complete resources utilization in SDN-based fog environment; implementation of deep belief network & Heuristic algorithms & SDN, IoT, Fog Computing \\  & (DBN)-based intrusion detection method reducing workload & & \\  & communication delays in the fog layer & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: _Cont._ \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(G=(R,E)\) & Network structure graph \\ \hline \(R=\left\{R_{i};\ i=\overline{1,m}\right\}\) & Set of vertices (routers) \\ \hline \(E=\left\{E_{i,j};\ i,j=\overline{1,m};\ i\
eq j\right\}\) & Set of edges (network links) \\ \hline \(n\) & Number of links \\ \hline \(m\) & Number of nodes \\ \hline \(K\) & Set of packet flows in the network \\ \hline \(s_{k}\) & Source node of the \(k\)th packet flow (\(k\in K\)) \\ \hline \(d_{k}\) & Destination node of the \(k\)th packet flow (\(k\in K\)) \\ \hline \(\lambda^{k}\) & Average packet intensity of the \(k\)th flow (packets per second, pps) \\ \hline \(\varphi_{ij}\) & Link bandwidth (packets per second, pps) between the \(i\)th and \(j\)th nodes (\(i,j=\overline{1,m};\ i\
eq j\)) \\ \hline \(\beta^{k}\) & Proportion of the \(k\)th flow intensity that receives a denial of service when using the multipath \\ \hline \(b^{k}\) & \(k\)th flow intensity, with which packets will be limited on the border router \\ \hline \(\alpha\) & Upper bound of the network links utilization \\ \hline \(\alpha_{TH}\) & Threshold of the network links utilization upper bound \(\alpha\) \\ \hline \(\varphi^{k}\) & Weighting coefficient responsible for the packet flows limitation at the network edge \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation summary. Thus, the graph \(G=(R,E)\) describes the network structure under the sets of vertices (routers) \(R\) and edges (network links) \(E\). Then, \(K\) packet flows circulate in the network. Namely, the \(k\)th packet flow (\(k\in K\)) is transmitted from the source node \(s_{k}\) to the destination node \(d_{k}\), with an average packet intensity \(\lambda^{k}\). In general, the result of solving the routing problem is the determination of the routing variables \(x_{i,j}^{k}\) that characterize the fraction of the \(k\)th flow intensity in the communication links belonging to the multipath. When using multipath routing, the following restrictions are imposed on these variables [21; 22]:  
\[0\leq x_{i,j}^{k}\leq 1. \tag{1}\]  
The following flow conservation conditions are introduced to ensure the routes' connectivity in the whole network, which differ from the previously known ones that allow description of traffic policing at the network edge under conditions of the overload [21; 22]:  
\[\left\{