# Coverage and Rate Analysis for Location-Aware Cross-Tier Cooperation in Two-Tier HetNets
## 2 System Model
### Location-Aware Cross-Tier Cooperation Scheme  
For a typical user located at the origin, we let \(R_{i}\) denote the distance of the typical user from their nearest BS in the \(i\)th tier. \(MBS_{0}\) and \(FBS_{0}\) are denoted as the nearest BSs in each tier, which can provide the maximum long-term received power from each tier. We consider a user association scheme on the basis of each user's location information. The macro tier has an inner region \(\mathcal{A}_{in}\), which is defined as the union of locations for which the distance to the nearest MBS is no larger than a prescribed value \(D\), whereas the outer region is given by \(\mathcal{A}_{out}=\mathbb{R}^{2}\setminus\mathcal{A}_{in}\)The FBSs within \(\mathcal{A}_{in}\) are deactivated because of their poor offloading effect from the MBSs. The users in \(\mathcal{A}_{in}\), that is, macro inner users (MIUs), consequently are only served by an MBS in this region. We denote \(\mathcal{B}\) as the serving BS set of the typical user, which can be expressed as follows:  
\[\mathcal{B}=\left\{ \begin{array}{l}\left\{\left.\left\{MBS_{0}\right\},R_{1} \leq D\right.\\ \left\{MBS_{0}\right\},R_{1}>D\&P_{1}R_{1}^{-\kappa_{1}}/P_{2}R_{2}^{-\kappa_ {2}}\geq\beta\\ \left\{FBS_{0}\right\},R_{1}>D\&P_{1}R_{1}^{-\kappa_{1}}/P_{2}R_{2}^{-\kappa_ {2}}\leq 1\\ \left\{MBS_{0},FBS_{0}\right\},R_{1}>D\&1<P_{1}R_{1}^{-\kappa_{1}}/P_{2}R_{2} ^{-\kappa_{2}}<\beta\end{array} \right. \tag{1}\]  
where \(D\) is the radius of inner region \(\mathcal{A}_{in}\), and \(\beta\geq 0\) dB is the cooperation threshold for tier 2. Without any loss of generality, we let \(\beta_{i}\) denote the cooperation threshold for the \(i\)th tier. For simplicity, it is assumed that the cooperation threshold for tier 1 is the unity (\(\beta_{1}=0\) dB) in this paper, while \(\beta_{2}=\beta\geq 0\) dB for notational brevity. Hence, the typical user \(u\in U\) can lie in one of the following four disjoint sets:  
\[u\in\left\{ \begin{array}{l}U_{M,in},\;k=1;\;\text{if}\;\mathcal{B}=\left\{ MBS_{0}\right\}\\ U_{M,out},\;k=1;\;\text{if}\;\mathcal{B}=\left\{MBS_{0}\right\}\\ U_{F},\;k=2;\;\text{if}\;\mathcal{B}=\left\{FBS_{0}\right\}\\ U_{C},\;k=\left\{1,2\right\};\;\text{if}\;\mathcal{B}=\left\{MBS_{0},\;FBS_{0}\right\} \end{array} \right. \tag{2}\]  
which satisfy \(U_{M,in}\cup U_{M,out}\cup U_{F}\cup U_{C}=U\). Clearly, the set \(U_{M,in}\) is the set of MIUs and the set \(U_{M,out}\) is the set of macro users within the outer region. The set \(U_{F}\) is the set of femto non-CoMP users, which is independent of the cooperation threshold. The set \(U_{C}\) is the set of femto CoMP users who are liable to experiencing excessive cross-tier interference. For brevity, we define a mapping \(\mathcal{K}:\left\{M,in;\;M,out;\;F,\;C\right\}\rightarrow\left\{1,\;2\right\}\) from the user-set index to the serving tier index, that is, \(\mathcal{K}\left(M,in\right)=\mathcal{K}\left(M,out\right)=1\), \(\mathcal{K}\left(F\right)=2\) and \(\mathcal{K}\left(C\right)=\left\{1,\;2\right\}\). To elaborate, the proposed LA-CTC scheme is shown in Figure 1. User 1 located in the inner region \(\mathcal{A}_{in}\) of \(MBS_{0}\), and is only associated with \(MBS_{0}\). \(MBS_{0}\) is close enough to guarantee the user 1's quality of service (QoS). On the other hand, a user located in the outer region \(\mathcal{A}_{out}\) of \(MBS_{0}\) will be associated with \(MBS_{0}\), \(FBS_{0}\) or two tiers of BSs. User 2's received power from \(MBS_{0}\) is stronger than that from \(FBS_{0}\) plus the cooperation threshold, and user 3's received power from \(FBS_{0}\) is stronger than that from \(MBS_{0}\); thus user 2 and user 3 are only associated with \(MBS_{0}\) and \(FBS_{0}\), respectively. Although user 4's received power from \(MBS_{0}\) is larger than that from \(FBS_{0}\), the received power from \(FBS_{0}\) plus the cooperation threshold is larger. User 4 will be cooperatively served by \(MBS_{0}\) and \(FBS_{0}\) by means of jointly transmitting the user's data; hence the user can be referred to as the CoMP user.