## 4 Mixed-Flow Load-Balanced Scheduling (MFLBS)
### Mixed-Flow Load-Balanced Scheduled Network
#### 4.3.1 Important Parameters  
**Definition 1**.: _Set a time scalar \(\lambda\). When the flow in the network survives longer than \(\lambda\) in the dynamic routing network, it will be transferred to the static routing network for subsequent transmission._  
\[\lambda=\frac{\sum_{K=1}^{K}t_{f_{k}}}{K} \tag{7}\]  
_where Equation (7) counts the data in the past period. \(K\) is the sum of the number of flows scheduled during this period, and \(t_{f_{k}}\) represents the duration of the flow \(f_{k}\) from arriving at the network to leaving the network. So, \(\lambda\) represents the average survival time of the scheduled flow in the past period. In this way, we can have a convenient and simple judgment on the determination of short flow and long flow in the future network. When \(t_{f_{k}}<\lambda\), it is judged to be a short flow and scheduled in a dynamic routing network; as the survival time increases, when \(t_{f_{k}}>\lambda\), it is considered to be a long flow and enters the static network for transmission._  
**Definition 2**.: _When the number of times a flow that changes routes in a dynamic routing network exceeds \(\mu\), regardless of whether its survival time exceeds \(\lambda\), it is transferred to the static routing network for the remaining transmission._  
Unlimited rerouting not only fails to solve the congestion but also increases additional overhead. So, we record the number of rerouting of each flow; when it exceeds \(\mu\), the path is fixed. In this article, we set \(\mu\) as 3.  
**Definition 3**.: _V(\(\alpha\)) is a collection of bandwidth coefficients on different links. Different \(\alpha\langle i,j\rangle\) values in the set divide the bandwidth of dynamic and static routing networks on different shortest links \(\langle i,j\rangle\)._  
\[RC_{i,j}=\alpha<i,j>\times C_{i,j} \tag{8}\]  
\[SC_{i,j}=C_{i,j}-RC_{i,j} \tag{9}\]  
Equation (8) shows that the bandwidth of the dynamic routing network on \(l_{i,j}\) is equal to the total bandwidth of the network multiplied by the coefficient ratio \(\alpha\) on the dependent links. Therefore, the static routing network in Equation (9) occupies the remaining bandwidth on the link.  
Then, we discuss how to determine the value of a. After the above analysis, the short flow will complete the transmission in the dynamic routing network in an ideal state, and the long flow will be transmitted through the two networks. So, a dynamic routing network not only calculates the sum of short-flow traffic but also adds the amount of traffic that is allocated during the period when a long flow stays in the dynamic network. In this way, we know that the dynamic routing network occupies all the short-flow traffic and part of the long-flow traffic.  
\[\alpha<i,j>=\frac{\sum_{k=1}^{K}\alpha_{i,j}^{f_{k}}+\lambda\sum_{k^{\prime}=1 }^{K^{\prime}}\frac{\alpha_{i,j}^{f_{k^{\prime}}}}{\tau_{f_{k^{\prime}}}}}{C_ {i,j}} \tag{10}\]  
Combining Equations (7) and (8), we can deduce the value of a in Equation (10). Among them, the short flow \(f_{k}\) is the flow that completes the transmission in the dynamic routing network, and the long flow \(f_{k^{\prime}}\) is the flow that completes the transmission in the static routing network. In addition, not only \(\lambda\), but \(\alpha\) can also be calculated and adjusted based on a period of historical data.