## 3 Problem Formulation
### Problem Formulation  
The slot occupancy of the FlexE frames transmitted on the link \(e_{ij}\) is represented by an array \(U_{e_{ij}}=\left[u_{1},u_{2},\ldots,u_{N_{frame}^{TS_{max}}}\right]\), where each binary bit of the array represents the occupancy of the slot of the corresponding frame, where the value \"0\" means that the slot is available in the idle state, and the value \"1\" means the slot has been occupied, and \(N_{frame}^{TS_{max}}\) denotes the total number of slots in each FlexE frame. In addition, we assume that there are already some background services in the network occupying the slot resources.  
Importantly, the current cellular network faces challenges in propagating the high-capacity data with improved speed, QoS, latency, and efficient HO and mobility management [37]. And ref. [38] proposed an overall prediction method of QoE parameters of users and telecommunication networks based on the QoS indicators' values prediction, discussed four normalization techniques, and finally proposed a normalization method of the index scale. However, the problem presented in this paper is oriented to scenarios with high requirements for low latency, and delay is one of the most important factors in QoS. Therefore, we decided to set the optimization objective to minimize the end-to-end delay, where the objective function is defined as follows:  
\[Min\sum_{e_{ij}\in P_{k}}\tau_{e_{ij}}^{\text{prop}}+\sum_{v_{j}\in V}\tau_{e_{ ij}}^{forward}+\sum_{e_{ij}\in P_{k}}\tau_{e_{ij}}^{sche} \tag{2}\]  
where \(\tau_{e_{ij}}^{\text{prop}}\) indicates the propagation delay of service \(r_{k}\) from \(s_{k}\) to \(d_{k}\), \(\tau_{e_{ij}}^{forward}\) is the forwarding delay generated by the service at the intermediate nodes in the path, and \(\tau_{e_{ij}}^{sche}\) is the scheduling delay caused by the service due to slot allocation.  
The above objective function should be minimized under the following constraints.  
Routing Continuity Constraint: (3) means there is a routing continuity constraint between \(s_{k}\) to \(d_{k}\), where \(\varphi_{p,l}^{n_{k}}\) is a binary variable that is equal to \(1\) if the \(P_{r_{k}}\) occupies \(e_{pl}\) and is equal to \(0\) for otherwise.  
\[\sum_{e_{pl}\in P_{k}}\varphi_{p,l}^{r_{k}}-\sum_{e_{ql}\in P_{k}}\varphi_{l,q }^{r_{k}}=\left\{\begin{array}{ll}1,p=s_{k}\\ \\ 0,others\end{array},\forall p,q\in V,p\
eq q,\forall l\in V\right. \tag{3}\]  
Bandwidth Continuity Constraint: (4) means the service \(r_{k}\) occupies the same size of bandwidth resources on different links \(e_{pl}\) and \(e_{lq}\) in the route.  
\[b_{p,l}^{r_{k}}=b_{l,q}^{r_{k}},\forall p,q\in V,p\
eq q,\forall l\in V \tag{4}\]  
Capacity Constraint: (5) ensures that the number of PHYS allocated to the services has a sufficient calendar slot.  
\[\sum_{e_{ij}\in P_{k}}n_{r_{k},e_{ij}}^{TS}\leq N_{frame}^{TS_{max}} \tag{5}\]  
Slot Uniqueness Constraint: (6) indicates that the slot of each link is allowed to carry only one data code block at a certain moment, and the variable \(\delta_{e_{j},p_{j_{j}}}\) is equal to \(1\) if the service \(r_{k}\) contains link \(e_{ij}\) and is equal to 0 for otherwise, while the variable \(U_{e_{ij}}(t)\) is equal to \(1\) if the \(t\)th slot is occupied and is equal to 0 for otherwise.  
\[\sum_{n_{k}\in R}\delta_{e_{ij},p_{r_{k}}}\cdot U_{e_{ij}}(t)\leq 1,\forall i,j \in V,\forall e_{ij}\in P_{r_{k}},\forall t\in\left[1,n_{r_{k},e_{ij}}^{TS}\right] \tag{6}\]