## 3 Interference Coordination Scheme Analysis
### Optimization Strategy for User Fairness  
Let, \(m=1,2,\ldots,N_{\text{M}}\) represent the set of macrocells; the number of macrocells is denoted by \(N_{\text{M}}\). The set of all edge users of the macrocell \(m\) is presented by \(U_{m}^{\text{Edge}}\) and the number of edge users of the macrocell \(m\) is denoted by \(N_{m}^{\text{U,Edge}}\). Moreover, the set of all edge users includes the set of macro edge users, the set of general pico edge users and the set of CRE edge users which are presented by \(U_{m}^{\text{M,Edge}}\), \(U_{m}^{\text{P,Edge}}\) and \(U_{m}^{\text{CRE,Edge}}\), respectively. The number of users in the above set is denoted by \(N_{m}^{\text{M,Edge}}\), \(N_{m}^{\text{P,Edge}}\) and \(N_{m}^{\text{CRE,Edge}}\), respectively, and the total number of edge users in the networks is denoted by \(N_{\text{U}}^{\text{Edge}}\). For the HetNets, the average throughput of edge users is an important metric [9] for judging the advantage of the performance of the networks. To a certain extent, the performance of edge users can reflect the user fairness of the network. Therefore, the joint optimization strategy of the adaptive CRE and ABS designed in this paper considers the performance requirements of the edge users of the networks. The adaptive mode means that the ABS ratio value of each macrocell can be different while the CRE bias value of each picocell can also be different and the optimal result is achieved according to the load situation in the networks. Therefore, the optimal ABS ratio value is denoted by  
Figure 2: Basic principle of ABS. \(\mathbf{\xi}=\left[\xi_{1},\xi_{2},\dots,\xi_{N_{\text{M}}}\right]\) and the optimal CRE bias value is denoted by \(\eta=\left[\eta_{2},\eta_{3},\dots,\eta_{N_{\text{B}}}\right]\). The average throughput function of edge users of the HetNets is given by:  
\[T(\mathbf{\xi},\mathbf{\eta})=\frac{\sum\limits_{m=1}^{N_{\text{M}}}\sum\limits_{u=1}^{ N_{\text{U},\text{Edge}}}C_{m,u}^{\text{Edge}}}{N_{\text{U}}^{\text{Edge}}}, \tag{8}\]  
where  
\[N_{\text{U}}^{\text{Edge}}=\sum\limits_{m=1}^{N_{\text{M}}}\left(N_{m}^{\text{ M,Edge}}+N_{m}^{\text{P,Edge}}+N_{m}^{\text{CRE,Edge}}\right), \tag{9}\]  
\[C_{m,u}^{\text{Edge}}=\left\{ \begin{array}{l}(1-\xi_{m})R_{m,u}^{\text{Edge} }\,,u\in\left(U_{m}^{\text{M,Edge}}\cup U_{m}^{\text{P,Edge}}\right)\\ \xi_{m}R_{m,u}^{\text{Edge}}\,,u\in N_{m}^{\text{CRE,Edge}}\end{array} \right.. \tag{10}\]  
In (10), \(\xi_{m}\) represents the ABS ratio value of the macrocell \(m\), while the \(R_{m,u}^{\text{Edge}}\) represents the transmission rate of the user \(u\) of the macrocell \(m\) and can be found by (3)-(5) according to the transmission mode and type of user. \(N_{m}^{\text{M,Edge}}\), \(N_{m}^{\text{P,Edge}}\) and \(N_{m}^{\text{CRE,Edge}}\) depends on the CRE bias value \(\eta\) because more and more macro users are off-loaded to picocells as the CRE bias values increase and that will lead to less available resources distributed by the CRE edge users. Thus, the group of network edge users will mainly consist of a part of CRE edge users and \(N_{m}^{\text{M,Edge}}\) reduces and \(N_{m}^{\text{CRE,Edge}}\) increases at this time. The objective function and the maximization problem can be expressed as:  
\[ \begin{array}{rl}\text{max }T(\mathbf{\xi},\mathbf{\eta})\\ \text{s.t.}\mathbf{C}_{1}&:0\leq\xi_{m}\leq 1,m=1,2,\dots,N_{\text{M}},\\ \mathbf{C}_{2}&:\eta_{b}\geq 0,b=2,3,\dots,N_{\text{B}},\\ \mathbf{C}_{3}&:\sum\limits_{n=1}^{N_{\text{SC}}}p_{b}^{n}\leq\left\{\begin{array} []{l}\text{Pmax},\,b=1\\ \text{Pmax},\,b\
eq 1\\ \text{P}_{\text{P}}^{\text{max}},b\
eq 1\end{array}\right.,\\ \mathbf{C}_{4}&:0.5<\alpha_{b}^{n}<1,b=1,2,\dots,N_{\text{B}},n=1,2,\dots,N_{\text{ SC}}^{b},\end{array}  \tag{11}\]  
where \(\mathbf{C}_{1}\) and \(\mathbf{C}_{2}\) can guarantee the range of the ABS ratio value and CRE bias value.