# Online Distributed User Association for Heterogeneous Radio Access Network
## 3 System Model and Problem Formulation
### Problem Formulation  
Taking a utility function perspective, we assume that UE \(i\) obtains utility \(U_{i}(R_{i,j,k})\) when the receiving data rate is \(R_{i,j,k}\), which is a function of the actual data rate \(R_{i,j,k}\). Let \(x_{i,j,k}\) be the association indicator, \(x_{i,j,k}=1\) if user \(i\) is connected to the BS \(j\) of tier \(k\) and \(0\) otherwise. We formulate the user association problem for \(load_{1}\), that maximizes the aggregate utility function (denoted as \(\mathbf{Q}1\)):  
\[\underset{x_{i,j,k}}{\text{maximize}} \sum_{i\in\text{U}}\sum_{j\in\text{B}}\sum_{k\in\text{K}}x_{i,j,k} \mathcal{U}_{i}\Big{(}\frac{r_{i,j,k}}{\sum_{i\in\text{U}}x_{i,j,k}}\Big{)} \tag{6}\] \[s.t. \sum_{j\in\text{B}}\sum_{k\in\text{K}}x_{i,j,k}=1,\;\;i\in\text{U}\] (7) \[\eta_{i,j,k}\geq\tau,\;\;i\in\text{U},\;\;j\in\text{B},\;\;k\in \text{K}\] (8) \[x_{i,j,k}\in\{0,1\},\;\;i\in\text{U},\;\;j\in\text{B},\;\;k\in \text{K}, \tag{9}\]  
where the constraints ensure that a UE can only be associated with a single BS, and the SINR should be greater than some threshold \(\tau\). The SINR threshold \(\tau\) is the minimum SINR level required to decode the received signal from the BS. For example, the SINR required to decode a Quadrature Phase Shift Keying  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Description** \\ \hline K & Set of tiers \\ B & Set of BSs \\ U & Set of UEs \\ \(\mathbf{B}\) & Number of BSs in the network \\ \(\mathbf{U}\) & Number of UEs in the network \\ \(\lambda_{j,k}^{load1}\) & Current \(load_{1}\) of BS \(j\) belongs to \(k\)th tier \\ \(\lambda_{j,k}^{load2}\) & Current \(load_{2}\) of BS \(j\) belongs to \(k\)th tier \\ \(P_{j,k}\) & Transmission power of a BS \(j\) belongs to the \(k\)th tier \\ \(\eta_{j,j,k}\) & SINR from BS \(j\) of tier \(k\) to UE \(i\) \\ \(\sigma^{2}\) & Noise power level \\ \(W\) & Bandwidth \\ \(r_{i,j,k}\) & Instantaneous rate from BS \(j\) of tier \(k\) to UE \(i\) \\ \(R_{i,j,k}\) & Actual experienced data rate from BS \(j\) of tier \(k\) to UE \(i\) \\ \(x_{i,j,k}\) & UE and BS association indicator \\ \(N_{j,k}\) & Set of UEs attached to BS \(j\) of tier \(k\) \\ \(B_{i}\) & Set of BSs the UE \(i\) can associate with \\ \(\mathcal{U}_{i,j,k}(.)\) & Utility of UE \(i\) from BS \(j\) of tier \(k\) \\ \(\mathcal{V}_{j,k}(.)\) & Utility of BS \(j\) of tier \(k\) \\ \(\Omega_{j,k}\) & Set of UEs that the BS \(j\) of tier \(k\) can be associated with \\ \(\alpha_{i,j,k}\) & Association parameter calculated by UE \(i\) considering BS \(j\) of tier \(k\) \\ \(\delta_{i,j,k}\) & Probability of association calculated by UE \(i\) considering BS \(j\) of tier \(k\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Brief description of acronyms.  
(QPSK)-modulated signal is comparatively lower than that of a Quadrature amplitude modulation (QAM)-modulated signal. The association indicator, \(x_{i,j,k}\), ensures that a UE can only connect to a single BS. This problem can be easily extended to multiple BS association. Since the above problem **Q1** is an integer program, it is generally difficult to find the optimal solution. The problem **Q1** can easily be converted to a convex optimization problem by simply making the utility function as a logarithmic utility and relaxing the constraint in Equation (9) as \(x_{i,j,k}\geq 0.\) This converted convex optimization problem can be solved effectively. The above optimization problem, **Q1**, can be reconfigured, which maximizes the average utility function (denoted as **Q2**) as follows:  
\[\underset{x_{i,j,k}}{\text{maximize}}\ \ \sum_{i\in\textbf{U}}\sum_{j\in \textbf{B}}\sum_{k\in\mathds{K}}x_{i,j,k}\frac{\mathcal{U}_{i}\Big{(}\frac{r_{ i,j,k}}{\sum_{i\in\mathds{U}}x_{i,j,k}}\Big{)}}{|N_{j,k}|} \tag{10}\]  
The problem **Q2** maximizes the average UE throughput, which ensures fairness in the network. The problem **Q2** has same set of constraints as that of **Q1**.