# Game Theory-Based Load-Balancing Algorithms for Small Cells Wireless Backhaul Connections +
## 3 Results
### Simulation Results and Discussions
#### 3.3.1 Evaluation of the Proposed LB Algorithm over Parallel Virtual Tunnels
The results presented above are valid in the situation when the total transfer rate of the users' flows is less than the total capacity of the communication tunnels, but the difference between the requested rate and the offered capacity is relatively small when it is important to fill each tunnel as close to capacity as possible. The selected flow mixture represents such a situation. If this difference is large, it does not matter which LB algorithm is used, the tunnels being capable of accommodating the source flows. The analysis presented above shows that if the tunnels are congested and real-time applications are used by many users the Combinatorial Auction algorithm is the best solution, this algorithm being capable to fulfill with high probability the latency requirements of real-time applications. If the level of congestion on the tunnels is reduced the simple Auction-based algorithm is the best choice having better performance than the Multiple Knapsack algorithm but lower implementation complexity. Due to the lack of control, the Selfish Routing algorithm is not a practical option, the Multiple Knapsack algorithm being the best solution if the auction-based GT algorithms cannot be used. Figure 11: Statistical characterization of the packet delays inserted by the bests of the considered LB algorithms: (**a**) The pdfs of the packet delays; (**b**) The cdfs of the packet delays. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**LB Algorithm** & \multicolumn{3}{c}{**cdf Function/Packet Delay (ms)**} \\ \hline Combinatorial Auction & 0.23/20 ms & 0.58/40 ms & 0.8/60 ms & 0.95/80 ms \\ Auction & 0.08/20 ms & 0.22/40 ms & 0.36/60 ms & 0.54/80 ms \\ Selfish routing no signaling penalty & 0.01/20 ms & 0.09/40 ms & 0.21/60 ms & 0.28/80 ms \\ Multiple Knapsack & 0.025/20 ms & 0.1/40 ms & 0.21/60 ms & 0.38/80 ms \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Values of the packet delay cdfs for the LB algorithms considered in Figure 11.  
Figure 12 presents the price of anarchy, \(max_{NE}\) (20), for the four tunnels considered in the performed experiment. Figure 12a presents the evolution of the price of the anarchy with the absolute value of the total rate \(R_{max}<R_{av-tunnel}\), for each tunnel, while in Figure 12b the variation of the price of anarchy with the relative value of the total rate \(R_{max}/R_{av-tunnel}\) is presented. The price of anarchy increases abruptly when the total rate \(R_{max}\) approaches the average transfer rate of the tunnel, but for values of \(R_{max}\) smaller than \(R_{av-tunnel}\) the variation of the price of anarchy is relatively slow with the increase of \(R_{max}\) (see Figure 12a). Furthermore, we can see that as the total rate \(R_{max}\) approaches the \(R_{av-tunnel}\) value of the tunnel transfer rate significant gains in system transfer rate can be obtained by using more efficient LB algorithms than the relatively simple selfish routing. If we compare the variation of the price of anarchy for the considered tunnels with the values of the \(R_{max}/R_{av-tunnel}\) ratio (the relative total rate) results that the variation of the price of anarchy has (approximately) the same behavior no matter the average rate/std rate ratio, \(R_{av-tunnel}/R_{std-tunnel}\), of the tunnel (see Figure 12b).