## 2 System Model
### System Throughput  
D2D underlaid cellular network is considered to be located in a region with radius R, where BS is the center of this region as illustrated in Figure 3 (each D2D pair is described as a point due to the short distance between transmitter and receiver). D2D pairs and CUs are independent identically distributed (i.i.d.) around the BS within the region, and the positions of D2D receivers follow a uniform distribution inside the region that is close enough to satisfy the maximum distance constraints of D2D communication. Furthermore, the channel is modeled as the Rayleigh fading channel, and its response follows the i.i.d. complex Gaussian distribution. Figure 2: Session setup of D2D communication. As the system model is established independently in a single cell, the interference from other cells is ignored. D2D pairs and CUs interfere mutually due to the resource sharing in the same sub-channel. Hence, the total number of available RBs and D2D pairs in the communication region are assumed to be \(M\) and \(N\), where \(X=\{1,2,3,\ldots,M\}\) and \(Y=\{1,2,3,\ldots,N\}\), respectively. The set of RBs is denoted as \(R=\left\{RB_{1},RB_{2},\ldots,RB_{j},\ldots,RB_{M}\right\}\) and let \(DP=\left\{D2D_{1},\ D2D_{2},\ldots,\ D2D_{i},\ldots,\ D2D_{N},\right\}\) be the set of D2D pairs. In the uplink, the \(j\)-th CU transmits signal \(s_{c}\) to the BS, and the \(i\)-th D2D pair reuses the same RB to transmit signal \(s_{d}\). Here, a matching matrix is used to reveal the state of resource allocation. **Definition 1**. : _The resource sharing between D2D pairs and RBs is denoted as a matching matrix \(\beta^{\text{CD}}=\left[\beta^{\text{CD}}_{i,j}\right]_{N\times M}\), where \(\beta^{\text{CD}}_{i,j}\in\{0\,\ 1\}\) is the binary decision variable to show the result of allocation as_  
\[\beta^{\text{CD}}_{i,j}=\left\{ \begin{array}{c}1,\text{ if D2D\_i}\text{ reuse RB\_j}\\ 0,\text{ otherwise}\end{array} \right. \tag{1}\]  
Thus, the received signals at D2D receiver and BS are given respectively as  
\[y_{D\_i}=d^{-\frac{q}{i}}_{i,i}h_{i}s_{d}+\beta^{\text{CD}}_{i,i}d^{-\frac{q}{ i}}_{i,j}h_{i,j}s_{c}+N_{d} \tag{2}\]  
\[y_{\text{C\_j}}=d^{-\frac{q}{i}}_{j,i}h_{j,\text{B}}s_{c}+\beta^{\text{CD}}_{ i,i}d^{-\frac{q}{i}}_{i,\text{B}}h_{i,\text{B}}s_{d}+N_{c} \tag{3}\]  
where \(h_{x,y}\) represents the channel response of the \(x\)-\(y\) link that is from device \(x\) to \(y\), and distributes independently as complex Gaussian distribution \(\mathcal{CN}(0,1)\). Hence, the path-loss model is assumed to be distance dependent and \(\alpha\) is the path-loss exponent. \(N_{d}\) and \(N_{c}\) denote the additive noise at D2D receivers and the BS, which is distributed as \(\mathcal{CN}\big{(}0,\sigma^{2}\big{)}\). Finally, the distance from device x to y is denoted as \(d_{x,y}\). The signal-to-interference-plus-noise ratio (SINR) at D2D receivers are given by [28]  
\[\gamma_{D\_i}=\frac{p_{D\_i}d^{-\alpha}_{i,i}\big{|}h_{i,i}\big{|}^{2}}{I^{i} _{cd}+I^{i}_{dd}+N_{d}} \tag{4}\]  
Figure 3: Illustration of a communication area where users are normally distributed around the base station. where \(I_{cd}^{i}=\sum_{j=1}^{M}\beta_{i,j}^{CD}p_{C,j}d_{j,j}^{-\alpha}\abs{h_{j,j}}^{2}\) is the interference from CUs; \(I_{dd}^{i}=\sum\limits_{k\in\mathcal{D}U_{j},\ k\
eq i}p_{C,k}d_{k,j}^{-\alpha} \abs{h_{k,j}}^{2}\) denotes the interference from neighboring D2D pairs in the same sub-channel and \(DU_{j}\) is D2D pairs in j-th sub-channel; \(D\_i\) and \(C\_j\) represent i-th D2D pair and j-th CU; \(p_{D\_j}\) and \(p_{C\_j}\) are the transmit power of i-th D2D pair and j-th CU, respectively. The CUs have no interference to each other since CUs communicate with orthogonal spectrum resources in the cellular system. Therefore, the received SINR of cellular uplink at BS is given by  
\[\gamma_{C\_j}=\frac{p_{C\_j}d_{j,B}^{-\alpha}\abs{h_{j,B}}^{2}}{I_{dc}^{j}+N_{ C}} \tag{5}\]  
where \(I_{dc}^{j}=\sum_{i=1}^{N}\beta_{i,j}^{CD}p_{D\_j}d_{i,B}^{-\alpha}\abs{h_{i,B} }^{2}\).