## 2 System Model  
We consider a two-tier LTE-A system with a BS supported by L3 relays as shown in Figure 1. The relays are assumed to be In-band type 1b [3] and full duplex, placed in the center of BS to the most distant user. A total of \(K\) users and \(N\) RBs are considered with users placed at a uniform distance from BS. The total powers of BS and RN are denoted by \(P_{total}^{BS}\) and \(P_{total}^{RN}\), respectively. The LTE-A system uses OFDMA transmission in the downlink. Let the system bandwidth is \(B\) with \(N\) number of RB, then, \(W=\frac{B}{N}\) is the bandwidth of one RB. We express the channel gains \(g_{k,n}^{Direct\_link}\) and \(g_{k,n}^{Relay\_link}\) for user \(k\) where \(k\in\mathcal{K}=\{1...,K\}\) on RB \(n\) where \(n\in\mathcal{N}=\{1...,N\}\) for BS and RN respectively. Practically, the channel gain depends upon various factors, including thermal noise at receiver, receiver noise figure, antenna gains, distance between transmitter and receiver, path loss exponent, log normal shadowing and fading. Therefore, for all the links, we can write  
\[g_{k,n}=-\varrho-\phi 10\log_{10}d_{k}-\zeta_{k,n}+10\log_{10}h_{k,n} \tag{1}\]  
In the above equation, \(\varrho\) (83.46 dB) is a constant depending upon thermal noise at receiver, receiver noise figure, and antenna gains, \(\phi\) is path loss exponent, \(d_{k}\) is the distance in Km from UE \(k\) to the BS/relay, \(\zeta_{k,n}\) (10.5 dB) is shadowing parameter modeled by a normally distributed random variable with standard deviation 8 dB, and \(h_{k,n}\) corresponds to the Rayleigh fading channel coefficient of user \(k\) in subchannel \(n\)[21]. \begin{table}
\begin{tabular}{l l} \hline \hline
**Notation** & **Definition** \\ \hline \(\mu_{k,n}\) & the RB assignment indicator \\ \(R_{k}\) & capacity of user \(k\) is given by the Shannon Capacity \\ \(SNR_{k,n}\) & the signal-to-noise ratio for user \(k\) on RB \(n\) \\ \(B\) & System bandwidth \\ \(N\) & Number of RB \\ \(W=\frac{B}{N}\) & the RB bandwidth \\ \(P_{total}^{RN}\), \(P_{total}^{BS}\), \(P_{total}^{BS}\) & the total power at which RN and BS transmit \\ \(g_{k,n}^{Relay\_link}\) & \(g_{k,n}^{Direct\_link}\) \\ \(h_{k,n}^{Relay\_link}\), \(h_{k,n}^{Direct\_link}\) & random channel coefficients for user \(k\) on RB \(n\) for RN and BS \\ \(SNR_{k,n}^{Direct\_link}\) & signal-to-noise ratio for user \(k\) via Direct Link \\ \(SNR_{k,n}^{Relay\_link}\) & signal-to-noise ratio for user \(k\) via Relay Link \\ \(SINR_{k,n}^{Direct\_link}\) & signal-to-Interference-and-noise ratio for user \(k\) via Direct Link \\ \(\alpha_{k}\) & proportional rate constraint for user \(k\) \\ \(\lambda_{k}\) & rate parameter for user \(k\) \\ \(D_{k}\) & allocated RB set for user \(k\) \\ \(R_{k}\) & rate matrix for user \(k\) \\ \(\hat{p}_{k,n}\) & optimal power allocation \\ \(P_{T}\) & total transmit power \\ \(\theta\) & Lagrangian multiplier \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of notations. The throughput of user \(k\) is given by,  
\[R_{k}= \begin{cases}\frac{1}{2}\frac{B}{N}\sum\limits_{n=1}^{N}\mu_{k,n}\text{log} _{2}(1+SNR_{k,n}),&\text{Access link users}\\ \frac{B}{N}\sum\limits_{n=1}^{N}\mu_{k,n}\text{log}_{2}(1+SNR_{k,n}),&\text{ Direct link users}\end{cases}  \tag{2}\]  
where the factor \(1/2\) in access link shows the two time-slots transmission from BS-RN and RN-UE, and \(\mu_{k,n}\) is the binary variable such that \(\mu_{k,n}=1\) when RB \(n\) is allocated to the user \(k\), \(SNR_{k,n}\) is the maximum average signal-to-noise ratio for user \(k\) between direct and relay links.