# Online Distributed User Association for Heterogeneous Radio Access Network
## 3 System Model and Problem Formulation  
We consider a \(\mathbf{K}\)-tier HetNet, and denote the set of tiers as \(\mathrm{K}=\{1,2,...,\mathbf{K}\}\), the set of all BSs as \(\mathrm{B}=\{1,2,...,\mathbf{B}\}\), and the set of UEs as \(\mathrm{U}=\{1,2,...,\mathbf{U}\}\). The cardinality of B and U give the total number of BSs and UEs in the system, respectively. Each small BS uses a wireless backhaul to connect with the macro BS, either directly or indirectly. We adopt a universal frequency reuse scheme among the different tiers. This model can easily be extended to a multi-RAT scenario. We assume that at a given point of time, a fraction of the total UEs, say \(\mathbf{U}\), will get access to the network. Without the loss of generality, the UEs are indexed as their arrival in the system--that is, where UE 1 arrives first and UE \(\mathbf{U}\) arrives last. The maximum capacity, as well as maximum transmission power varies from one tier to another. Each BS has a positive integer capacity, also termed the size of a BS. We denote \(\lambda_{j,k}\) and \(P_{j,k}\) as the current load and the transmission power respectively of BS \(j\) at tier \(k\), where \(j\in\mathrm{B}\) and \(k\in\mathrm{K}\). To calculate the load of a BS, we consider two different parameters--the number of UEs and the SINR \begin{table}
\begin{tabular}{l l l l l} \hline \hline
**Ref.** & **Control** & **Environment** & **HetNet** & **Fairness** \\ \hline  
entralized & Dynamic & Yes & High \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of user association schemes. experienced from a BS. While considering the number of UEs as a load parameter, we consider the following assumptions. Each UE in the network consumes a portion of the time-frequency resource from the BS. Here, we consider an orthogonal frequency-division multiple access (OFDMA)-based medium access control (MAC) protocol with fair sub-carrier scheduling, as given in Ref. [27]; hence, the load of the BS could be the total number of UEs associated with it [17]. The average data rate from a BS also signifies the load, as the highly loaded BS's average data rate will be lower, as compared with other BSs. We present \(load_{1}\) and \(load_{2}\) as the loads occurring due to the number of UEs and the average data rate of the BS, respectively. In a HetNet environment, BSs of different tiers can provide services to a varying number of UEs, and come with varying capacities. For example, a macro BS can offer services to a considerably large number of UEs, as compared to a pico BS. If we consider the same number of UEs attached to both macro BS and pico BS, then the load is not same for both the BSs; in contrast, the macro BS is lightly loaded as compared to that of pico BS. One important parameter, SINR, plays a key role in accessing the network. At first, a UE scans all the possible BSs it can access, and then it calculates the SINR, \(\eta_{i,j,k}\), of these BSs:  
\[\eta_{i,j,k}=\frac{P_{i,j}h_{i,j,k}}{\sum_{l\in\text{B},l\
eq j}P_{l,k}h_{i,l, k}+\sigma^{2}}\, \tag{1}\]  
where \(P_{j,k}\) is the transmission power of BS \(j\) at tier \(k\), \(h_{i,j,k}\) is the channel gain between UE \(i\), and BS \(j\) at tier \(k\). \(\sigma^{2}\) denotes the noise power level, and \(\sum_{l\in\text{B},l\
eq j}\eta_{l,k}h_{i,l,k}\) is the interference received from other BSs of the same tiers. The instantaneous data rate, \(r_{i,j,k}\), associated with the SINR, \(\eta_{i,j,k}\), is given by:  
\[r_{i,j,k}=W\log_{2}(1+\eta_{i,j,k}) \tag{2}\]  
In a conventional network, the UE will associate with that BS whose SINR is the maximum, among all the BSs in range. In a HetNet environment this type of association scheme will lead to load imbalance among BSs. UEs attached to macro BS will not get the desired data rate, though the instantaneous data rate is quite high. This is because the total resources will be divided among all the associated UEs, and the per-user data will be significantly low. Let \(N_{j,k}\) be the set of UEs attached to the BS \(j\) of tier \(k\).