## 4 Problem Formulation
### MILP Optimization Model
Constraints (17)-(22) are used to estimate and guarantee flow latencies in the network. The constraints have been formulated in the efficient MILP model that we developed in [20]. Here, we repeat the main reasoning behind these constraints. Constraint (17) estimates the worst case buffering latency (\(u_{dfe}^{HEP}\)) of a packet belonging to a flow at a switch output link caused by the packets belonging to higher and equal-priority flows. Namely, if link \(e\) is not used by flow \(f\) of demand \(d\) (\(y_{dfe}=0\)), then the left-hand side (LHS) of inequality (17) is equal to or lower than 0. As a consequence, it has no impact on variable \(w_{dfe}^{HEP}\) (it can take a value of 0 or any value allowable by other constraints), and the latency of flow \(f\) is not constrained by the queuing of other flows in link \(e\). Otherwise, if link \(e\) is used by flow \(f\) (\(y_{dfe}=1\)), then variable \(w_{dfe}^{HEP}\) takes the value of buffering latency caused by the interfering higher/equal-priority flows (from set \(\mathcal{H}(d,f,e)\)) or any higher value allowable by other constraints. Constraints (18) and (19) estimate the worst case buffering latency (\(w_{dfe}^{LP}\)) of a packet belonging to a flow at a switch output link caused by the packets belonging to lower priority flows. In particular, constraint (18) ensures the activation of variable \(y_{dfe}^{LP}\), whenever there is a lower priority flow from subset \(\mathcal{L}(d,f,e,t)\) realized in the network. Constraint (19) ensures that latency \(L(t)\) produced by a lower priority flow is included in the estimation of buffering latency of flow \(f\) of demand \(d\) in link \(e\) if both flows are present in the network. Constraint (20) estimates the overall worst case buffering latency for a flow at a switch output link. Constraint (21) estimates the static latency of a flow produced in a network link, which accounts for the link propagation delay, the store-and-forward delay in the origin node of this link (if the node is a switch), and the burst transmission delay in the link. Constraint (22) estimates the latency of a flow and ensures that it is within the allowable limit for this flow. Eventually, constraints (23) and (24) define the range of the problem variables. On the LHS of constraints (10)-(16), additional symbols \(\lambda_{d}^{\rm FH}\), \(\tau_{d\sigma}^{\rm FH}\), \(\mu_{d\sigma}^{\rm DU-eMB}\), \(\mu_{d\sigma}^{\rm DU-URLLC}\), \(\mu_{d\sigma}^{\rm CU-URLLC}\), \(\kappa_{e}\), and \(\sigma_{d\varepsilon}\) are included, which denote the dual variables associated with these constraints in the linear programming (LP) relaxation of problem (2)-(24). Note that these dual variables are related to the constraints in which path selection variable \(x_{dfp}\) appears, and they are used in a column-generation (CG) procedure presented in Section 5.3. For the presentation clarity, we omit the dual variables related to the rest of the constraints that are irrelevant for CG. **Remark 1**. : _MILP optimization problem (2)-(24) comprises, among others, two \(\mathcal{NP}\)-hard subproblems: the single-path allocation problem [18], represented by constraints (10) and (15), and the 0-1 knapsack problem, represented by constraints (9) and (15). Therefore, the problem is also \(\mathcal{NP}\)-hard, indicating that it may be challenging to solve._