## 2 Multi-Connectivity Model Formulation  
### Problem Definition  
Let us consider a heterogeneous RAN where different UEs with multi-connectivity capabilities are camping. A given \(u\)-th UE considers \(M\) different RATs and \(N\) different cells per RAT as candidates for the multi-connectivity. Then, let us denote as \(A_{u}=\{C_{m,n}\}\) the set of candidate cells detected by the \(u\)-th UE, where \(C_{m,n}\) denotes the \(n\)-th cell of the \(m\)-th RAT with \(n=1,\,\ldots,\,N\) and \(m=1,\,\ldots,\,M\). It is worth mentioning that, due to the mobility of the UE, the specific detected cells in a given RAT may change with time. In this respect,it is assumed that the \(N\) candidate cells of a RAT correspond to the best \(N\) cells detected by the UE at a certain time based on measurements averaged during a time window of duration \(\Delta T\) s.  
Through the use of multi-connectivity, the traffic of the \(u\)-th UE is split across multiple RATs/cells of the set \(A_{u}\). It is assumed that, at a certain time, the UE can be simultaneously connected to a maximum of \(N_{max}\) cells among the \(M\)\(\cdot\)\(N\) candidates. The multi-connectivity configuration of the \(u\)-th UE can be expressed as the \(M\)\(\times\)\(N\) matrix \(\mathbf{B_{u}}=\{\beta_{m,n}\}\), where \(\beta_{m,n}\in[0,1]\) defines the fraction of total traffic of the \(u\)-th UE that is delivered through the \(n\)-th cell of the \(m\)-th RAT. Then, the objective is to find the optimal configuration \(\mathbf{B_{u}}=\{\beta_{m,n}\}\) to be applied in a time window of \(\Delta T\) that allows ensuring the Quality of Service (QoS) requirements with minimum resource consumption and avoiding overload situations in the different RATs/cells. In this respect, it is assumed that the QoS requirement of the user \(u\) is expressed in terms of a required bit rate \(R_{u}\) (b/s) to be provided. To formalize the problem, let us denote as \(T_{u}\)(\(\mathbf{B_{u}}\)) the total throughput or bit rate obtained by user \(u\) during the last time window period \(\Delta T\) as a result of the multi-connectivity configuration \(\mathbf{B_{u}}\). Let us also denote \(a_{m,n}(\beta_{m,n})\) as the number of physical resources in the \(m\)-th cell and \(n\)-th RAT assigned to the \(u\)-th UE to transmit the traffic corresponding to fraction \(\beta_{m,n}\). Assuming LTE and 5G NR-based RATs, the physical resources correspond to the Physical Resource Blocks (PRBs), each one defined as a set of 12 contiguous subcarriers in an OFDMA access [3]. Then, considering that \(b_{m,n}\) is the bandwidth of one PRB in the \(m\)-th cell and \(n\)-th RAT, the bandwidth allocated to the user \(u\) in this RAT, denoted as \(\gamma(\beta_{m,n})\), is given by:  
\[\gamma(\beta_{m,n})=a_{m,n}(\beta_{m,n})\cdot b_{m,n} \tag{1}\]  
In addition, the total fraction of occupied PRBs in a RAT/cell accounting for all the UEs connected to that cell is denoted as \(\rho_{m,n}(\beta_{m,n})\). With all the above, the considered problem to be solved for the \(u\)-th UE is formally defined  
\[\mathbf{B_{u}}=arg\,min\left[\frac{1}{w_{max}}\sum_{m=1}^{M}\sum_{n=1}^{N} \gamma(\beta_{m,n})\right]\text{s.t. }T_{u}(\mathbf{B_{u}})\geq R_{u}\,\ \rho_{m,n}(\beta_{m,n})\leq\rho_{max}\ \forall m,n\sum_{m=1}^{M}\sum_{n=1}^{N}\beta_{m,n}=1 \tag{2}\]  
where \(w_{max}\) is the maximum possible bandwidth to be assigned to the user \(u\) and \(\rho_{max}\in[0,\ 1]\) is the maximum threshold established to avoid overload situations in a cell. Then, the problem in Equation (2) intends to find the multi-connectivity configuration \(\mathbf{B_{u}}\) that minimizes the fraction of total bandwidth allocated to the \(u\)-th UE while at the same time ensuring that the provided throughput is above the required bit rate, i.e., \(T_{u}(\mathbf{B_{u}})\geq R_{u}\) and that the total fraction of occupied PRBs in the involved RATs/cells is lower than a threshold \(\rho_{max}\).