## 4 Problem Formulation
### MILP Optimization Model
\begin{table}
\begin{tabular}{l l} \hline \hline \(u_{\text{\tiny{AP}}}\) & binary, \(u_{\text{\tiny{AP}}}=1\) if cluster \(c\) has assigned PP node \(v\) for DU processing \\ \(u_{\text{\tiny{AP}}}^{\text{DU}}\) & binary, \(u_{\text{\tiny{AP}}}^{\text{DU}}=1\) if DU processing of demand \(d\) is performed in PP node \(v\) \\ \(u_{\text{\tiny{AP}}}^{\text{CU}}\) & binary, \(u_{\text{\tiny{AP}}}^{\text{CU}}=1\) if CU processing of demand \(d\) is performed in PP node \(v\) \\ \(u_{\text{\tiny{AP}}}^{\text{DUCU}}\) & binary, \(u_{\text{\tiny{AP}}}^{\text{DUCU}}=1\) if both CU and DU processing of demand \(d\) is performed in PP node \(v\) \\ \(u_{\text{\tiny{AP}}}^{\text{CU-URLLC}}\) & binary, \(u_{\text{\tiny{AP}}}^{\text{CU-URLLC}}=1\) if CU processing of URLLC slice \(s\) is performed in PP node \(v\) \\ \(z_{v}\) & binary, \(z_{v}=1\) if PP node \(v\) is active \\ \(x_{dfp}\) & binary, \(x_{dfp}=1\) if path \(p\) is selected to realize flow \(f\) of demand \(d\) \\ \(y_{dfe}\) & binary, \(y_{dfe}=1\) if flow \(f\) of demand \(d\) is routed over link \(e\) \\ \(y_{dfd}^{\text{LP}}\) & binary, \(y_{dfd}^{\text{LP}}=1\) if at least one of the LP flows belonging to \(\mathcal{L}(d,f,e,t)\) is active \\ \(w_{df}\) & continuous, latency of flow \(f\) belonging to demand \(d\) \\ \(w_{dfe}^{\text{HP}}\) & continuous, latency in link \(e\) for flow \(f\) of demand \(d\) caused by higher/equal-priority flows \\ \(w_{dfe}^{\text{LP}}\) & continuous, latency in link \(e\) for flow \(f\) of demand \(d\) caused by lower priority flows \\ \(w_{dfe}^{\text{DM}}\) & continuous, dynamic latency in link \(e\) for flow \(f\) belonging to demand \(d\) \\ \(w_{dfe}^{\text{DM}}\) & continuous, static latency in link \(e\) for flow \(f\) belonging to demand \(d\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Problem variables. The MILP model is presented in Table 3 and explained in the following. The optimization objective, expressed by (2), is to minimize the number of active PP nodes in the network. Constraints (3)-(9) concern the placement of DU and CU entities in PP nodes. Constraint (3) ensures the selection of a PP node for the DU processing for a cluster of demands. Concurrently, constraint (4) ensures that each demand in a cluster uses the PP assigned to this cluster for DU processing. Constraint (5) activates the PP node where a DU is placed. Constraint (6) ensures the selection of a PP node for CU processing for each URLLC slice. Constraint (7) ensures that each demand belonging to the URLLC slice uses the PP assigned to this slice for CU processing. Constraint (8) activates the PP node where a CU is placed. Constraint (9) ensures that the overall load at a PP, related to the DU and CU processing, does not exceed its capacity. Constraints (10)-(14) concern the selection of paths for routing of FH and MH flows in the packet-switched transport network. Constraint (10) ensures the selection of a single path for each FH flow. Constraint (11) ensures that the FH path selected for a demand terminates at the PP node processing the DU of this demand. Constraint (12) ensures the selection of an MH path for a demand belonging to an eMBB slice. Moreover, it guarantees that this path terminates at the PP node processing the DU of this demand. Note that the other end of the path should be at the hub node, where the CU is placed for eMBB. Therefore, the set of candidate MH paths for slice eMBB contains only such paths. Constraints (13) and (14) ensure that the MH flow of a demand belonging to an URRLC slice either is absent in the network, if both DU and CU are processed at the same PP (\(u_{dq}^{\text{DUCU}}=1\)), or is routed over a single path terminating at the PP nodes processing the DU and CU entities. Constraints (15) and (16) are used to control and determine the usage of network links. Constraint (15) ensures that the volume (bit rate) of traffic routed over a link does not exceed its capacity. Constraint (16) determines if a network flow is routed over a given link.