## 6 Simulation Results  
We consider a HetNet downlink system with 37 TPs, where only 19 TPs of these conduct actual communications to UEs, and the others wrap them around to produce virtual interference. The radius of each small cell is 250 m, since a dense deployment is considered. The system includes 100 RBs, each of which is under a bandwidth of 180 kHz. Therefore, the overall bandwidth of the system is 18 MHz. Additionally, \(2\times 2\) MIMO links are created using the space channel model (SCM) [17]. Each simulation lasts 20 TTIs, where a TTI is 1 ms. Important parameters used in the simulation are listed in Table 2.  
The simulation is carried out to prove the proposed algorithms effectiveness. A greedy algorithm, named max capacity, is also simulated as a benchmark. The max capacity algorithm tends to allocate resources to UEs with good channel conditions, in order to reach the optimal throughput of the network. In this way, UEs with worse channel conditions may no chance to communicate. Therefore, the fairness of the max capacity algorithm is unfavorable. Figures 4 and 5 compare the performances of the max capacity algorithm to that of the proposed one. Figure 4 demonstrates the average throughput per TP and fairness factor (as defined by Equation (5)) of the system under different resource allocation algorithms, when the transmit power of each TP is 20 Watts. It is obvious that max capacity algorithm achieves an outstanding throughput and a much worse fairness factor than the proposed one. The future mobile communication system targets to provide not only high throughput of the network, but also quality service to every UE. Therefore, the max capacity is no longer appropriate. The proposed algorithms have much better fairness factors. More importantly, as shown in Figure 5, the proposed centralized algorithm can also achieve an energy efficiency as good as that of max capacity. \begin{table}
\begin{tabular}{c|c} \hline
**Parameters** & **Value** \\ \hline Layout of cells & 37 hexagon cells; wrap-around used \\ Radius of cells & 250 m \\ Central frequency & 2 GHz \\ \(N_{\text{RB}}\), number of RBs & 100 \\ \(S_{m}\), limit of transmit power & 20 Watt \\ \(N_{\text{T}}\times N_{\text{R}}\) & 2 \(\times\) 2 \\ number of TTI / TTI & 20 /1 ms \\ \(\alpha\) & 0.1 \\ Channel model & SCM (path loss + shadowing + MIMO fading) \\ Minimal distance (TP and UE) & 35 m \\ Height of transmit/receive antenna & 35 m/1.5 m \\ Penetration loss & 20 dB \\ Traffic model & full buffer \\ Speed of UE & 10 m/s \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters in the simulation. Results in Figures 4 and 5 also compare the performance of the centralized algorithm proposed in Section 4 and that of the decentralized algorithm in Section 5. The results show that both energy efficiency and throughput are decreased when the decentralized algorithm is used. Figure 6 shows the system performances of the proposed algorithms when the transmit power of each TP is 40 Watts. Comparing the results to Figures 4 and 5, it can be seen that the energy efficiency of the centralized algorithm significantly decreases when the transmit power of each TP is up to 40 Watts, while the throughput does not increase. This proves that high-level transmit power is not appropriate in a dense network. Additionally, the results demonstrate that the decentralized algorithm is more robust, since both energy efficiency and throughput are changed a little when different transmit powers are used. Figure 4: System performances under different resource allocation algorithms (20 Watts, infinite \(C_{m}\), \(R_{\text{thres}}=180\) kbps). Figure 5: Energy efficiencies under different resource allocation algorithms (20 Watts, infinite \(C_{m}\), \(R_{\text{thres}}=180\) kbps). Simulations are also conducted under infinite, 100-Mbps and 80-Mbps backhaul limits, respectively, with a fixed \(R_{\text{thres}}\) of 360 kbps, to clearly illustrate the effect on system performance caused by backhaul constraints.