# Energy-Efficient Resource and Power Allocation for Underlay Multicast Device-to-Device Transmission
## 3 System Model and Problem Formulation
Therefore, we can calculate the normalized transmission rate of the \(m\)-th D2D cluster as  
\[r_{m}^{D2D}=\sum\limits_{n=1}^{N}x_{m,n}\log_{2}\Bigl{(}1+P_{m}^{D2D_{m}}\gamma_ {m,n}^{D2D}\Bigr{)} \tag{5}\]  
As a result, the transmission rate of \(m\)-th D2D cluster is expressed as  
\[R_{m}^{D2D}=|\mathcal{K}_{m}|r_{m}^{D2D} \tag{6}\]  
Similarly, we can formulate the normalized transmission rate for a CU \(n\) which use RB \(n\) as  
\[R_{n}^{CU}=\log_{2}\Bigl{(}1+P_{n}^{CU}\gamma_{n}^{CU}\Bigr{)} \tag{7}\]  
In order to guarantee the quality of service (QoS) requirement of each UE, a threshold is set. Specifically, for the \(m\)-th D2D multicast cluster, the above requirement is expressed as:  
\[P_{m}^{D2D_{m}}\gamma_{m}^{D2D}\geq\Gamma_{\min}^{D2D} \tag{8}\]  
The above expression is explained as in order to ensure reliable transmission, the signal-to-interference-and-noise ratio (SINR) of the user who has the worst channel condition among a D2D multicast cluster should be above certain threshold. Again, for a CU, this requirement is set as  
\[P_{n}^{CU}\gamma_{n}^{CU}\geq\Gamma_{\min}^{CU} \tag{9}\]  
where \(\Gamma_{\min}^{D2D}\) and \(\Gamma_{\min}^{CU}\) represent the minimum SINR threshold to ensure reliable D2D multicast and cellular transmission specified by the system. Combined with the maximum transmit power constraints for CUs and D2D clusters and by substituting expressions (2) and (3) into (8) and (9), the transmit power range of a cluster head and a CU is expressed as  
\[\left\{ \begin{array}{l}\frac{\Gamma_{\min}^{D2D}}{\min-\frac{ \Gamma_{\max}^{D2D}}{\sum\limits_{n=1}^{N}x_{m,n}\Gamma_{\min}^{CU}\Gamma_{ \max}^{D2D}+\sigma_{n}^{2}}}\\ \frac{\Gamma_{\min}^{CU}}{\frac{\Gamma_{\min}^{CU}}{\frac{\Gamma_{\min}^{CU}}{ \sum\limits_{n=1}^{N}x_{m,n}\Gamma_{\min}^{D2D}\sigma_{n}^{D2D}+\sigma_{n}^{2}} }}\leq P_{n}^{CU}\leq P_{\max}^{CU},\forall m\in\mathcal{M},n\in\mathcal{N} \\ \frac{\Gamma_{\max}^{CU}}{\frac{\Gamma_{\min}^{CU}}{\sum\limits_{n=1}^{N}x_{m, n}\Gamma_{\min}^{D2D}\sigma_{n}^{D2D}\sigma_{n}^{D2D}+\sigma_{n}^{2}}}\leq P_{n}^{CU} \leq P_{\max}^{CU},\forall m\in\mathcal{M},n\in\mathcal{N}\\ \frac{\Gamma_{\max}^{CU}}{\sum\limits_{n=1}^{N}x_{m,n}\Gamma_{\max}^{D2D}\sigma _{n}^{D2D}\sigma_{n}^{D2D}+\sigma_{n}^{2}}\end{array} \right. \tag{10}\]  
where \(P_{\max}^{CU}\) and \(P_{\max}^{D2D}\) represent the maximum allowed transmit power of a CU and a D2D user, respectively. From expression (10), we can deduce that besides the channel condition factor, the transmit power of the cluster head as well as the CU who shares the RB resource with a D2D multicast cluster are intertwined. In order to guarantee reliable transmission, the transmit power of different kinds of UEs should be considered. Given the limited energy capacity of each device, the objective of this paper is to maximize the overall energy efficiency of all D2D multicast clusters, which is also a hottest research aspect. According to [13], the energy-efficiency (EE) of a single D2D multicast cluster \(m\) can be expressed as  
\[\eta_{m}=\frac{R_{m}^{D2D}}{P_{m}^{D2D_{m}}} \tag{11}\]  
Combing Equations (1)-(11) and since there are \(M\) D2D multicast clusters, the energy efficiency of all the D2D multicast clusters is defined as the ratio of total D2D multicast transmission data rates to the overall consumed power of all clusters. Consequently, the energy efficiency optimization problem can be expressed as \[\max_{\forall m\in\mathcal{M}}\eta_{all}=\frac{\sum\limits_{m=1}^{M}R_{m}^{D2D}}{ \sum\limits_{m=1}^{M}P_{m}^{D2D_{m}}} \tag{12}\]  
s.t.