## 2 System Models  
In this paper, the LTE communication systems-based resources allocation issues are investigated. The architecture of the systems is shown in Figure 1. In this hybrid system, D2D UEs and traditional cellular users' equipment (CeUEs) share all resources. In Figure 1, three relay stations are deployed in the cellular system, where UEs are uniformly and randomly distributed into relay stations. In this study, the base station is located in the center, while the three relay stations form a triangle around the base station. Each relay station signal has a coverage with a radius of 200 m. The CeUEs are uniformly and randomly distributed in each relay station within range. Each D2D pair includes a receiver UE and a transmitter UE distributed at a distance of 80 m from the relay station. The channel models include Raleigh fading, shadowing fading and path loss [17]. In Figure 1, each uplink data transfer includes two hops. In the first hop, the \(u_{l}\)-th UE transmits the signal to the \(l\)th relay station by the channel gain \(h_{u,l}^{(n)}\). The \(l\)th relay station relays the transmission to the base station using the \(n\)th resource block (RB). However, when the \(u_{j}\)-th D2D UE transmits signal to the \(l\)th relay station, the interference link gain \(g_{u_{j},l}^{(n)}\) will be transmitted to the \(l\)th relay station. In this study, the channel models include path loss and shadowing fading. Thus, the fading channel model from UE to relay station (UE-relay) can be expressed by:  
\[PL_{u_{l},\ell}(\ell)\ =\ 103.8+20.9\log(\ell)+L_{su}+10\log(\zeta), \tag{1}\]  
where 103.8 is the antenna gain; \(L_{su}\) is shadowing fading with a log-normal distributed random variable where the standard deviation \(\sigma\) = 10; 10 log(\(\zeta\)) is the Rayleigh fading effect; and \(\ell\) is the distance between UE to relay station. Similarly, the fading channel model from relay station to BS (relay-BS) can be expressed by [12]:  
\[PL_{\ell,\epsilon NB}(\ell)\ =\ 100.7+23.5\log(\ell)+L_{su}+10\log(\zeta). \tag{2}\]  
It is assumed that Base Station (BS) knows the Channel State Information (CSI) of all channels. Thus, the unit Power Signal-to-Interference-Plus-Noise Ratio (Unit Power SINR) of the first hop can be expressed by:  
\[\gamma_{u_{l},l,1}^{(n)}\ =\ \frac{P_{u_{l},l}^{(n)}h_{u_{l},l}^{(n)}}{ \sum\limits_{ \begin{array}{c}\forall u_{j}\in U_{j}\\ j\
eq l,j\in L\end{array} }x_{u_{j}}P_{u_{j},l}^{(n)}g_{u_{j},l}^{(n)}+\sigma^{ 2}}, \tag{3}\]  
where \(h_{u_{l},l}^{(n)}\) is the channel gain from the \(u_{l}\)-th UE to the \(l\)th relay with the \(n\)th RB; and \(x_{u_{j}}\ =\ 1\) or 0. Each UE can only use one RB, with \(x_{u_{j}}\ =\ 1\) indicating one RB and \(x_{u_{j}}\ =\ 0\) indicating no RB. Furthermore, \(U_{j}\) is the set of D2D UEs in the \(j\)th relay area; \(P_{u_{l},j}^{(n)}\) and \(P_{u_{j},l}^{(n)}\) are the transmission power of the \(u_{l}\)-th UE and the \(u_{j}\)-th CeUE, respectively; \(P_{u_{l},j}^{(n)}g_{u_{j},l}^{(n)}\) is the interference link gain from the \(u_{j}\)-th CeUE to the \(l\)th relay;  
Figure 1: Architecture of systems with deployment of three relay stations. \(\sigma^{2}~{}=~{}N_{0}B_{RB}\). \(N_{0}\) is power spectral density of the added white Gaussian noise (AWGN); and \(B_{RB}\) is bandwidth of a RB. Similarly, the unit power SINR of the second hop can be expressed by:  
\[\gamma_{l,ENB,2}^{(n)}~{}=~{}\frac{P_{l,enB}^{(n)}h_{l,enB}^{(n)}}{\sum\limits_ { \begin{array}{c}\forall u_{j}\in U_{j}\\ j\
eq l,j\in L\end{array} }x_{u_{j}}P_{j,enB}^{(n)}g_{j,enB}^{(n)}+\sigma^{2}}, \tag{4}\]  
where \(h_{l,enB}^{(n)}\) is the channel gain from the \(l\)th relay to the Base Station (eNB) with the \(n\)th RB; \(P_{l,enB}^{(n)}\) and \(P_{j,enB}^{(n)}\) are the transmission power of the \(l\)-th relay and the \(j\)-th relay, respectively; and \(g_{j,enB}^{(n)}\) is the interference link gain from the \(j\)th relay to the eNB with the \(n\)th RB. In this study, there are two links in the second hop. Equation (4) expresses the SINR of the \(l\)th relay station to BS.