## 4 Problem Formulation
### Notation  
The following notions and notation are used in the network and problem modeling. Network: The packet-switched Xhaul network is represented by directed and connected graph \(\mathcal{G}=(\mathcal{V},\mathcal{E})\), where \(\mathcal{V}\) is the set of network nodes and \(\mathcal{E}\) is the set of network links. Set \(\mathcal{V}\) includes the RU, PP, switch, and hub nodes. Subset \(\mathcal{V}^{\text{PP}}\subset\mathcal{V}\) comprises the PP nodes, and subset \(\mathcal{E}^{\text{S}}\subset\mathcal{E}\) comprises the output links of the switches. \(C(v)\) is the processing capacity of PP node \(v\in\mathcal{V}^{\text{PP}}\), and \(H(e)\) is the transmission capacity of link \(e\in\mathcal{E}\). \(L^{\text{P}}(e)\) is the propagation delay in link \(e\). \(L^{\text{SF}}(e)\) is the store-and-forward delay of the switch being the origin of link \(e\in\mathcal{E}^{\mathcal{S}}\); \(L^{\text{SF}}(e)=0\) if \(e\) is not outgoing from a switch. Traffic demands: The set of traffic demands is denoted as \(\mathcal{D}\). Demand \(d\in\mathcal{D}\) represents the radio traffic to be carried from/to an RU over the network and processed at the PP or hub nodes performing the DU and CU functions for this demand. The demands are realized in both directions: uplink (RU\(\rightarrow\)DU\(\rightarrow\)CU) and downlink (CU\(\rightarrow\)DU\(\rightarrow\)RU). The processing loads of the DU and CU entities of demand \(d\) are denoted, respectively, as \(\rho^{\text{DU}}(d)\) and \(\rho^{\text{CU}}(d)\). Network slicing: The set of types of network slices is denoted as \(\mathcal{N}\). In this work, set \(\mathcal{N}\) includes two types of slices: eMBB and URLLC; \(\mathcal{N}=\{\text{eMBB},\text{URLLC}\}\). The set of slices realized in the network is denoted as \(\mathcal{S}\). Each network slice \(s\in\mathcal{S}\) represents a subset of demands \(\mathcal{D}(s)\subseteq\mathcal{D}\). The slices are disjoint, and a pair of uplink and downlink demands belong to the same slice. Subsets \(\mathcal{S}^{\text{eMBB}}\subseteq\mathcal{S}\) and \(\mathcal{S}^{\text{URLLC}}\subseteq\mathcal{S}\) comprise the slices of type eMBB and URLLC, respectively. DU and CU placement: The set of clusters of RUs is denoted as \(\mathcal{C}\). Cluster \(c\in\mathcal{C}\) comprises a subset of RUs that requires the placement of their DUs in the same PP node. The demands associated with a certain RU belong to the same cluster. The set of all demands associated with cluster \(c\) is denoted as \(\mathcal{D}(c)\). In this work, for all demands of a network slice of type URLLC, their CU entities are placed in the same PP node. Concurrently, the CU processing is performed in the hub node for all demands of the network slices of type eMBB. Network flows: The set of types of traffic flows is denoted as \(\mathcal{F}\). In this work, two types of flows are considered: (a) fronthaul flows between RUs and the PP nodes in which their DU entities are placed and (b) midhaul flows between the PP/hub nodes performing the DU and CU functions; \(\mathcal{F}=\{\text{FH},\text{MH}\}\). Both flows are realized in the network for each demand unless the DU and CU are placed in the same PP; in such a case, the MH flow is absent. The bit rate of flow \(f\) of demand \(d\) is denoted as \(H(d,f)\). The maximum one-way latency limit for flow \(f\) of demand \(d\) is denoted as \(L^{\text{max}}(d,f)\). The delay produced by the transmission of the burst of frames of flow \(f\) of demand \(d\) in link \(e\) is denoted as \(L(d,f,e)\). Let \(\mathcal{H}(d,f,e)\) and \(\mathcal{L}(d,f,e)\) denote the sets of demand-flow pairs \((\bar{d},\bar{f})\), \(\bar{d}\in\mathcal{D}\) and \(\bar{f}\in\mathcal{F}\), which may cause buffering latencies of flow \(f\) of demand \(d\) in link \(e\in\mathcal{E}^{\mathcal{S}}\). Set \(\mathcal{H}(d,f,e)\) comprises the flows of either equal or higher priority than flow \(f\) of demand \(d\), whereas set \(\mathcal{L}(d,f,e)\) comprises lower priority flows.