## 3 LDCP Problem
### MILP Formulation
\begin{table}
\begin{tabular}{l l} \hline \hline
**Sets** & \\ \hline \(\mathcal{V}\) & network nodes \\ \(\mathcal{V}^{P}\) & PP nodes; where \(\mathcal{V}^{P}\subset\mathcal{V}\) \\ \(\mathcal{E}\) & network links \\ \(\mathcal{E}^{Sout}\) & switch output links \\ \(\mathcal{D}\) & demands \\ \(\mathcal{D}^{U}\) & uplink demands; \(\mathcal{D}^{U}\subset\mathcal{D}\) \\ \(\mathcal{D}^{D}\) & downlink demands; \(\mathcal{D}^{D}\subset\mathcal{D}\) \\ \(\mathcal{F}\) & types of flows; \(\mathcal{F}=\{\text{FH},\text{MH},\text{BH}\}\) \\ \(\mathcal{Q}^{HEP}(d,f)\) & demand-flow pairs of an equal/higher priority than flow \(f\) of demand \(d\) \\ \(\mathcal{Q}^{LP}(d,f)\) & demand-flow pairs of a lower priority than flow \(f\) of demand \(d\) \\ \(\mathcal{V}^{src}(d,f)\) & allowable source nodes of flow \(f\) of demand \(d\); \(\mathcal{V}^{src}(d,f)\subset\mathcal{V}\) \\ \(\mathcal{V}^{dest}(d,f)\) & allowable destination nodes of flow \(f\) of demand \(d\); \(\mathcal{V}^{dest}(d,f)\subset\mathcal{V}\) \\ \(\mathcal{C}\) & clusters of RUs \\ \hline
**Parameters** & \\ \hline \(\alpha(d,f,i,j,e)\) & \(=1\) if flow \(f\) of demand \(d\) originated in node \(i\) and terminated in node \(j\) is routed through link \(e\) \\ \(C(d)\) & cluster to which the RU of demand \(d\) belongs \\ \(\rho^{D}\) & processing load of a DU \\ \(\rho^{C}\) & processing load of a CU \\ \(\rho(v)\) & processing capacity of PP node \(v\in\mathcal{V}^{P}\) \\ \(H(d,f)\) & bit-rate of flow \(f\) of demand \(d\) \\ \(K(e)\) & capacity (bit-rate) of link \(e\) \\ \(L^{P}(e)\) & propagation delay of link \(e\) \\ \(L^{SF}(e)\) & store-and-forward delay produced in the origin node (switch) of link \(e\) \\ \(L(d,f,e)\) & delay produced by transmission of the burst of frames of flow \(f\) of demand \(d\) at link \(e\) \\ \(L^{max}(f)\) & maximum one-way latency of flow \(f\in\mathcal{F}\) \\ \hline
**Variables** & \\ \hline \(x_{dfij}\) & binary, \(x_{dfj}=1\) if flow \(f\) of demand \(d\) is realized between nodes \(i\) and \(j\) \\ \(x_{dfe}\) & binary, \(x_{dfe}=1\) if flow \(f\) of demand \(d\) is routed over link \(e\) \\ \(x_{dfdfe}\) & binary, \(x_{dfe}=1\) if flow \(f\) of demand \(d\) and flow \(\bar{f}\) of demand \(\bar{d}\) are routed over link \(e\) \\ \(u^{D}_{de}\) & binary, \(u^{D}_{de}=1\) if DU processing of demand \(d\) is performed in PP node \(v\) \\ \(u^{CD}_{de}\) & binary, \(u^{CD}_{de}=1\) if dU and DU processing of demand \(d\) is performed in PP node \(v\) \\ \(y_{cv}\) & binary, \(y_{cv}=1\) if cluster \(c\) has assigned PP node \(v\) for DU processing \\ \(y_{v}\) & binary, \(y_{v}=1\) if PP node \(v\) is active \\ \(w_{df}\) & continuous, latency of flow \(f\) belonging to demand \(d\) \\ \(w^{slat}_{dfe}\) & continuous, static latency in link \(e\) for flow \(f\) belonging to demand \(d\) \\ \(d^{dm}_{dfe}\) & continuous, dynamic latency in link \(e\) for flow \(f\) belonging to demand \(d\) \\ \(w^{HEP}_{dfe}\) & continuous, latency in link \(e\) for flow \(f\) of demand \(d\) caused by higher/equal priority flows \\ \(u^{LP}_{dfe}\) & continuous, latency in link \(e\) for flow \(f\) of demand \(d\) caused by lower priority flows \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Notation.