## 3 Network Model and Problem Statement
### MFLBS Problem  
The goal of MFLBS is to balance the traffic load of the data center through the use of sub-net division and a combination of dynamic and static mechanisms to maximize the throughput of the entire network.  
Assuming that there are \(K\) flows \(K=\{f_{1},f_{2},f_{3},\cdots\cdots f_{k}\}\) in a certain time slot \(t_{0}\) in the network, then \(a^{f_{k}}(t_{0})\) represents the traffic transmitted from the source to the destination of a single flow \(f_{k}\) in this time slot \(t_{0}\). In other words, we divide the continuous time \(T\) into time slots \(t(t=1,2,\ldots,T)\), and the total network transmission traffic can be expressed as \(\sum_{t=1}^{T}\sum_{k=1}^{K}a^{f_{k}}(t)\). For any switch \(i\), its outgoing traffic and incoming traffic are the same, that is, the traffic \(\sum_{t=1}^{T}\sum_{k=1}^{K}a^{f_{k}}(t)\) from \(i\) to the neighboring switch \(s_{i}\) should be balanced with the traffic \(\sum_{t=1}^{T}\sum_{k=1}^{K}a^{f_{k}}(t)\) from the neighboring switch \(s_{i}\) to this switch \(i\). It is worth noting that \(a^{f_{k}}\) here refers to the actual allocated bandwidth. Because it is limited by the actual network environment, its value may be different from the original required bandwidth.  
To achieve network load balancing, we set up an ordered set for each single flow \(f_{k}\), such as \(f_{k}\{1,2\ldots,s\}\), to record its planned path. The values stored in the set are the numbers of the switches that pass through in the sequence during \(f_{k}\) scheduling, and the ordered pair \(<\)\(i\)\(>\) are two adjacent elements in the ordered set. The orderly assembly of the path is adjusted according to the load situation of the network by adopting dynamic and static mechanisms.  
The notations for the MFLBS problem formulation are listed in Notations.  
Based on the above analysis, we first give the target formula of the MFLBS problem:  
\[\text{Max}\frac{1}{T}\sum_{t=1}^{T}\sum_{k=1}^{K}a^{f_{k}}(t) \tag{1}\]  
\[\sum_{t=1}^{T}\sum_{k=1}^{K}\alpha_{i,j}^{f_{k}}(t)=\sum_{t=1}^{T}\sum_{k=1}^{ K}\alpha_{j,i}^{f_{k}}(t)\;i,j\in V(s) \tag{2}\]  
\[\gamma_{i,j}^{f_{k}}(t)\geq 0\;i,j\in V(s)\cap E(f_{k}) \tag{3}\]  
\[\sum_{k=1}^{K}\alpha_{i,j}^{f_{k}}\leq C_{i,j}\gamma_{i,j}^{f_{k}}(t)\geq \alpha_{i,j}^{f_{k}}(t)<i,j>\in V(s)\cap E(f_{k}) \tag{4}\]  
\[\gamma_{i,j}^{f_{k}}(t)\geq\alpha_{i,j}^{f_{k}}(t)i,j\in V(s)\cap E(f_{k}) \tag{5}\]  
Equation (1) shows that the optimization direction of the MFLBS problem is to maximize the average throughput of the network. The left side of the equal sign in Equation (2) represents the sum of the outgoing traffic of all switches in the network; the right side of the formula represents the incoming traffic; and \(V(S)\) is the set of switches, which means  
Figure 2: Partial mesh model.  
that the incoming and outgoing bandwidth needs to be balanced for the entire network. Equation (3) indicates that the required bandwidth for any flow in the survival time slot must be a positive number. Equation (4) ensures that the sum of the actual bandwidth allocated on any link cannot exceed the total bandwidth of its own. Equation (5) reveals that the actual allocated bandwidth may be less than the bandwidth requested by the client under the influence of the network environment; if the network is not overloaded, it is equal.