# User-Cell Association for Security and Energy Efficiency in Ultra-Dense Heterogeneous Networks
## 2 System Model  
Let us assume to have an ultra dense deployment of heterogeneous cells where a macrocell layer providing basic service and coverage is overlayed by an ultra-dense layer of small base stations (SBSs) with different characteristics (Figure 1). Macrocell and small cells operate in different frequency bands, thus avoiding cross-tier interference. Conversely, SBSs reuse the same radio resources. We consider two different scenarios:  
* small-cell and macrocell layers serve two different classes of users, (i.e., users requiring secure communications are served by the small cells and not by the macrocell). Hence, the focus is only on the small-cell layer composed by micro, pico, and femto cells, generally indicated as small cells in what follows;
* users can be served by macrocells or small cells without any differentiation. In both cases, small cells are randomly distributed in the considered area following a Poisson point process (PPP) distribution. The density of the PPP is represented by \(\lambda_{S}\). Users are supposed to be randomly distributed in the area \(A\) following an independent PPP distribution with density \(\lambda_{U}\). Each user receives the reference signal broadcasted by the BSs around the mobile terminal. The user terminal can then evaluate and rank each signal-to-noise plus interference ratio (SINR) received over the sensitivity threshold. The SINR received by user \(u\) from cell \(c\) can be written as  
\[\text{SINR}_{u,c}=\frac{P_{c}^{[\text{TX}]}|h_{u,c}|^{2}\rho_{u,c}}{\sigma_{u}^ {2}+\sum_{ \begin{subarray}{c}\text{s}=1\\ \text{s}\
eq c\end{subarray} }^{\text{V}_{c}}|h_{u,s}|^{2}P_{s}^{[\text{TX}]}} \tag{1}\]  
Figure 1: Sketch of UDN: macrocell with micro, pico, and femto cells. where \(P_{s}^{[TX]}\) is the transmission power of the signal broadcasted by the \(s\)-th cell, \(\mathcal{N}_{C}\) is the total number of SBSs, \(\rho_{u,s}\) is the path loss from the \(s\)-th SBS to the \(u\)-th UE, \(\sigma_{u}^{2}\) is the noise power at the receiver, and \(|h_{u,s}|^{2}\) represents the gain of the channel between the \(s\)-th cell and the \(u\)-th UE that follows an exponential distribution with unit mean. The path loss model \(\rho_{u,c}\) is assumed to be dual slope that takes into account also the line of sight component at very short distances from the BS and better models the effects of BSs densification [37]  
\[\rho_{u,c}(d_{u,c})=\left\{ \begin{array}{ll}K_{0}d_{u,c}^{-\alpha_{1}},&d_{u,c}\leq\overline{d}\\ K_{1}d_{u,c}^{-\alpha_{2}},&d_{u,c}>\overline{d}\end{array} \right. \tag{2}\]  
where \(d_{u,c}\) is the distance between the BS of cell \(c\) and the terminal of the user \(u\), \(\overline{d}\) is the critical distance that separates the close-in and the long-range path loss zones, \(K_{0}\) is a catch-all constant that is equal to the path loss for unitary distance, \(K_{1}\) is a constant to ensure continuity between the two path-loss regions, \(\alpha_{1}\) is the close-in path loss exponent (usually equal to 2), and \(\alpha_{2}\) is the long-range path loss exponent (usually ranging from 2 to 6),  
Given the SINR expression, the capacity (normalized to the sub-band width) achievable by the \(u\)-th user if associated with the \(s\)-th SBS is  
\[C_{u,c}=log(1+SINR_{u,c}) \tag{3}\]  
In this paper we investigate the user association problem. Each user can associate with a single cell depending on a specific utility function \(\mathcal{U}\). In particular, the \(u\)-th user selects the cell \(\hat{c}_{u}\) that maximizes the utility \(U_{u,c}\) so that  
\[\hat{c}_{u}=\max_{c}\mathcal{U}_{u,c} \tag{4}\]  
The standard solution is that each user is associated with the cell providing the maximum SINR, that is \(\mathcal{U}_{u,c}=SINR_{u,c}\) defined in (1) [38]. However, in a UDN usually the UE is under the coverage of several BSs and the user-cell association may have a great impact on different performance metrics, and one of these can be the security. Association policies aiming at improving the network security have been investigated in [27].