## 2 System Model  
### Network  
In this paper, we consider a heterogeneous Ultra-Dense Network consisting of \(N_{\text{MBS}}\) Macro BSs (MBS) and \(N_{\text{SBS}}\) Small Base Stations (SBS). The Macro BSs are uniformly distributed to provide coverage and to support capacity. The small BSs are randomly distributed within the covering area, following a Poisson Point Process (PPP) with a density of \(\lambda_{\text{SBS}}\). Let \(B^{\Omega}=\{b|b=1,2,\cdots,|B^{\Omega}|\}\) denote the set of BSs consisting of both Macro BSs and Small BSs, where \(|B^{\Omega}|=N_{\text{MBS}}+N_{\text{SBS}}\) is the total number of BSs. UEs are randomly distributed following a PPP with density of \(\lambda_{\text{U}}\). The set of UEs is denoted by \(M^{\Omega}=\{m|m=1,2,\cdots,|M^{\Omega}|\}\). UEs located at the edges of cells usually suffer from severe intercell interference and low capacity. To reduce the interference and enhance peak data rates of cell edge users, joint transmission Coordinated Multi-Point (termed as JT CoMP) is considered in the network architecture, which allows multiple BSs in the neighborhood to cooperatively serve a specific UE simultaneously. The association relationship between UEs, \(m\), and BS, \(b\), is denoted by a bit number \(x_{m,b}(m\in M^{\Omega},b\in B^{\Omega})\) defined as:  
\[x_{m,b}=\left\{ \begin{array}{ll}1&\text{UE $m$ is associated with BS $b$}\\ 0&\text{otherwise}\end{array} \right.. \tag{1}\]  
Thus, the entire association result between BSs and UEs in the considering network can be presented by:  
\[\mathbf{X}=\left[ \begin{array}{cccc}x_{1,1}&x_{1,2}&\cdots&x_{1,|B^{\Omega}| }\\ x_{2,1}&x_{2,2}&\cdots&x_{2,|B^{\Omega}|}\\ \vdots&\vdots&\ddots&\vdots\\ x_{|M^{\Omega}|,1}&x_{|M^{\Omega}|,2}&\cdots&x_{|M^{\Omega}|,|B^{\Omega}|} \end{array} \right]. \tag{2}\]  
The SINR of a specific UE in a downlink transmission can be presented in terms of \(x_{m,b}\) as follows,  
\[\Gamma_{m}=\frac{\sum\limits_{b\in B^{\Omega}}x_{m,b}P_{b}g_{m,b}}{\sum\limits _{b^{\prime}\in B^{\Omega}}(1-x_{m,b^{\prime}})P_{b^{\prime}}g_{m,b}+\sigma^{ 2}}, \tag{3}\]  
where \(P_{b}\) is the transmit power of BS \(b\), \(g_{m,b}\) is the channel gain between BS \(b\) and UE \(m\), and \(\sigma^{2}\) is the variance of additive white Gaussian noise (AWGN). Equation (3) suggests that the more BSs are associated with UE \(m\), the better service it can obtain. However, the necessary overhead to accomplish a JT CoMP transmission involving too much BSs is unacceptable. Thus, it is better to narrow the associating BSs of a UE into \(N\) BSs in close proximity. As in 4G LTE and 5G NR (New Radio) wireless standards, resource block (RB) is considered the unit of time and spectrum resource for allocation in this paper. Assume that the bandwidth of each RB is \(W\) and the total number of RBs is \(N_{\text{RB}}\). Each UE in the network can occupy a part of resource for transmission. Let \(\beta_{m}\) denote the proportion that the resource assigned to UE \(m\) out of all. Then the data rate of a downlink transmission to UE \(m\) can be given by:  
\[R_{m}=W\left|\beta_{m}N_{\text{RB}}\right|\log_{2}\left(1+\Gamma_{m}\right), \tag{4}\]where \(|x|\) is the minimum integer smaller than or equal to \(x\); \(\Gamma_{m}\) is defined by Equation (3). Let \(B_{m}^{\Omega}\subseteq B^{\Omega}\) denote the set consisting of BSs associated with UE \(m\), i.e., \(B_{m}^{\Omega}=\{b|b\in B^{\Omega}\), and \(x_{m,b}=1\}\). Similarly, let \(M_{b}^{\Omega}\subseteq M^{\Omega}\) denote the set consisting of UEs associated with BS \(b\), i.e., \(M_{b}^{\Omega}=\{m|m\in M^{\Omega},\text{ and }x_{m,b}=1\}\). The resource that BS \(b\) can assign to UE \(m\) should be no more than \(\frac{1}{|M_{b}^{\Omega}|}\), where \(|M_{b}^{\Omega}|\) represents the total number of UEs associated with BS \(b\). In the case where JT CoMP is employed, the resource that UE \(m\) can be obtained is restricted by the most heavy-loading BS among those associated with UE \(m\).