# Joint Cell Association and User Scheduling in Carrier Aggregated Heterogeneous Networks
## 2 Problem Statement
### System Overview  
A cellular network with multiple low power femtocell BSs underlaid macrocells is considered in this paper, where carrier aggregation is deployed to improve the overall network date rate and coverage ability on the cell edge areas (Figure 1).  
In this paper, a downlink scenario is considered in a multi-layer HetNet with \(N\) available frequency bands denoted as a set of \(\mathcal{N}=\{1,...,N\}\) for carrier aggregation, and for each carrier band \(n\in\mathcal{N}\), the bandwidth is denoted as \(B^{n}\). Assuming a set of \(\mathcal{K}=\{1,...,K\}\) base stations are normally scattered in the HetNet, where massive femtocell BSs and the macrocell BS are layered by their transmit power level. A set of \(\mathcal{M}=\{1,...,M\}\) normally distributed users are considered active in the two-layer HetNets.  
Figure 1: Carrier aggregation (CA) based Heterogenous Networks.  
Assume a typical downlink transmission link from BS \(k\in\mathcal{K}\) to its intended user \(m\in\mathcal{M}\) occupying radio resource carrier \(n\in\mathcal{N}\), where \(x_{k}^{n}\) is the transmit symbol. Then, the received signal at user \(m\) could be denoted as  
\[y_{k,m}^{n}=\underbrace{H_{k,m}^{n}x_{k}^{n}}_{Signal}+\underbrace{\sum_{j=1,j \
eq k}^{K}H_{j,m}^{n}x_{j}^{n}}_{Interferences}+\underbrace{\sigma_{n,m}}_{Noise}\, \tag{1}\]  
where \(H_{k,m}^{n}\) is the complex channel state information from BS \(k\) to user \(m\) on carrier band \(n\). The received signal at user \(m\) is demonstrated as the combination of the intended signal, co-channel interferences from the neighboring cells within the HetNets and the additive white Gaussian noise (AWGN) with zero mean and variance \(N_{0}\). Then, the achievable rate for user \(m\) in cell \(k\) scheduled on carrier \(n\) should be  
\[C_{k,m}^{n}=\mathbb{E}\{B^{n}\log_{2}(1+SINR_{k,m}^{n})\}, \tag{2}\]  
where \(SINR_{k,m}^{n}\) is the signal to interference plus noise ratio accordingly. Under the assumption of signal model (1), it is denoted as  
\[SINR_{k,m}^{n}=\frac{P_{k}^{n}G_{k,m}^{n}}{N_{0}B^{n}+\sum\limits_{j=1,j\
eq k }^{K}P_{j}^{n}G_{j,m}^{n}}. \tag{3}\]  
\(P_{k}^{n}\) is the transmit power level of BS \(k\) on carrier band \(n\), which is denoted from \(P_{k}^{n}=\mathbb{E}\{\mid x_{k}^{n}\mid^{2}\}\). \(G_{k,m}^{n}=\mid H_{k,m}^{n}\mid^{2}\) is the channel power gain which captures both large and small scale fading characteristics. And \(N_{0}B^{n}\) is the power level of the white Gaussian noise of bandwidth \(B^{n}\).  
To balance the traffic load among different tiers in the carrier aggregated HetNets, the cell association schemes are developed to determine the optimal user scheduling policy with binary indicators \(\psi_{k,m}^{n}\), where  
\[\psi_{k,m}^{n}=\left\{\begin{array}{ll}1,&\text{user $m$ is associated with BS $k$ on band $n$,}\\ 0,&\text{otherwise.}\end{array}\right. \tag{4}\]  
Therefore, on each carrier band \(n\), the traffic load for each BS \(k\) is determined as  
\[L_{k}^{n}=\sum\limits_{m=1}^{M}\psi_{k,m}^{n}\ \ \forall k,n, \tag{6}\]  
where equal and orthogonal radio resource and equal power allocation is assumed among users associated with the same BS on the same carrier band in this paper.