## 3 Problem Formulation  
In this work, we consider a practical RRM problem in terms of both spectrum and power in a CoMP-based HetNet modeled in the last section. To improve the synthesis performance, the objective involves data rates, power consumption and fairness among UEs at the same time.  
The optimal data rate of the network can be achieved if resources are allocated to UEs with better channel conditions, regardless of those in \"poor\" condition. A side-effect of this scheme is unfavorable fairness of UEs' data rates. In order to enhance the fairness, we introduce the concept of proportional fairness into the objective of the RRM problem. As in [13], we weighted UE's data rate by its average data rate determined by:  
\[\bar{R}_{k}=\alpha\bar{R}_{k}^{\text{before}}+\left(1-\alpha\right)R_{k} \tag{4}\]  
where \(0<\alpha<1\) is the forgetting factor. The introduced weights bring UEs with worse channel conditions more of a possibility to occupy resources and, therefore, increase the UEs' data rates. The fairness of UEs' data rates in the network will be improved in this way. To quantify the degree of this fairness, [14] introduces a fairness factor, defined as:  
\[F=\left.\left(\sum\limits_{k=1}^{K}R_{k}\right)^{2}\middle/K\sum\limits_{k=1}^ {K}R_{k}^{2}\right.. \tag{5}\]  
Additionally, for the purpose of conserving energy, energy efficiency should be thoughtfully considered. The energy efficiency is defined by the ratio of the obtained data rate to the total power consumed correspondingly. Combining with proportional fairness principle, the objective of the RRM problem is formulated as,  
\[\max_{\beta_{m,k}^{n},p_{m}^{m}}\quad\sum\limits_{k=1}^{K}\sum\limits_{n=1}^{N _{\text{RB}}}\frac{R_{k}^{n}}{\bar{R}_{k}}\left/\sum\limits_{m=1}^{M}\sum \limits_{n=1}^{N_{\text{RB}}}\sum\limits_{k=1}^{K}\beta_{m,k}^{n}p_{m}^{n}\right. \tag{6}\]  
Equation (6) can be considered to maximize the weighted energy efficiency of the network.  
In a practical network, system performance is restricted by several factors. In addition to limited transmit power at each TP, the finite capacity of backhaul links defines the upper limit of throughput achieved by a TP during a TTI. Furthermore, to guarantee the quality of transmissions, we impose a threshold to the data rate of each transmission. In summary, the RRM problem of the considered system can be formulated as:\[\max_{\beta_{m,k}^{n},p_{m}^{n}} \sum_{k=1}^{K}\sum_{n=1}^{N_{\text{RB}}}\frac{R_{k}^{n}}{\overline{R }_{k}}\Bigg{/}\sum_{m=1}^{M}\sum_{n=1}^{N_{\text{RB}}}\sum_{k=1}^{K}\beta_{m,k}^ {n}p_{m}^{n}\] s.t. \[\text{C1:}\ 0\leq p_{m}^{n}\leq S,\forall m,n\] \[\text{C2:}\ \beta_{m,k}^{n}\in\{0,1\},\sum_{k=1}^{K}\beta_{m,k}^{n} \leq 1,\ \ \forall m,n,k \tag{7}\] \[\text{C3:}\ \sum_{n=1}^{N_{\text{RB}}}\sum_{k=1}^{K}\beta_{m,k}^{n}R_{k }^{n}\leq C_{m},\forall m\] \[\text{C4:}\ R_{k}^{n}\geq R_{\text{thres}},\forall k,n\]  
In Equation (7), C1 shows the power constraint at each transmission where \(S\) is the largest transmit power allowed by the system; C2 ensures each index \(\beta_{m,k}^{n}\) to be a bit number, so that a TP can serve no more than one UE on each RB; C3 demonstrates the constrained throughput of a TP caused by the limited capacity of backhaul connections to the CU, where \(C_{m}\) represents the capacity of the backhaul link connecting TP \(m\) and the CU; and C4 guarantees the data rate of each ongoing transmission, where \(R_{\text{thres}}\) is the given threshold of the data rate.