# Game Theory-Based Load-Balancing Algorithms for Small Cells Wireless Backhaul Connections +
## 3 Results
### Simulation Results and Discussions
#### 3.3.3 Evaluation of the Proposed User Clustering Algorithms
In Figure 15c the curves converge around 500 ms while in Figure 15b the curves converge around 1500 ms. Still, should be noticed that the differences in the performance depicted in Figure 15b,c, i.e., the distribution of the packet's delays, are small. Figure 16 presents the pdf of the packet delays in the system (the sojourn time) in the scenario with three APs and the following cases:  
* the three APs work separately, and no user clustering algorithm is used. The clustering of the users is performed only at the beginning of the simulation by attaching each user to the AP from which it receives the largest power-see Figure 16a. * a cooperative user clustering algorithm is used, and the APs power adjustment step is set to 1 dB-see Figure 16b. * a cooperative user clustering algorithm is used, and the APs power adjustment step is set to 2 dB-see Figure 16c. A cooperative user clustering algorithm with separate a power adjustment step for the central AP (1 dB) and for the side APs (0.5 dB) was also tested in the same conditions, but the results are very similar to those presented in Figure 16b, i.e., 1 dB power adjustment step. In Figure 16 are presented the pdfs of the packet delays on each tunnel and the combined tunnel used by the simulated system. In Figure 16a, when no clustering algorithm is used, there is a significant difference between the load of tunnel 2 (red curve) and tunnels 1 and 3 (blue and magenta curves), these two last tunnels being more loaded, especially tunnel 1 (blue curve). If the clustering algorithms are used, the load of the APs and the outgoing tunnels attached to the APs will change, and the load of tunnel 1 will decrease significantly, its load being distributed between tunnel 2 and 3 which will become more loaded, especially tunnel 3. Should be noticed that the performance of the clustering algorithm depends significantly on the position of the users and the characteristics of the service flows generated by the users  
Figure 15: The Cumulative Density Functions of the packet delays on the aggregated virtual tunnels when 2 APs and different user clustering algorithms are used: (**a**) The cdfs of the packet delays when no user clustering algorithm is used; (**b**) The cdfs of the packet delays when the non-cooperative user clustering algorithm is used; (**c**) The cdfs of the packet delays when the cooperative user clustering algorithm is used. and an optimal (more exactly close to optimal) distribution of the traffic between APs is not possible without using more complex clustering algorithms with more complex interaction between the APs and the users. The difference between the load of the APs in the 3 situations mentioned above is more visible if we analyze the cdfs of the packet delays depicted in Figure 17. In Figure 17a all curves converge around 3500 ms, while in Figure 17b,c all curves converge around 1500 ms. The performance of LB among the APs for power adjustment step 1 dB and 2 dB are very close. It should be noticed that the clustering algorithms, like the LB algorithms acting on the backhaul communication tunnels, require access only to the queue of the IP stack and the driver of the NIC that provides the AP-user connectivity. Identification of the data flows and the measurement of the parameters of these flows can be achieved by analyzing the content of the IP stack queue while the load of the AP-users connection can be assessed by analyzing the evolution in time of the length of the NIC's driver queue. Figure 16: Statistical characterization of the packet delays on the aggregated virtual tunnels when 3 APs and different user clustering algorithms are used: (**a**) The pdfs of the packet delays when no user clustering algorithm is used; (**b**) The pdfs of the packet delays when the cooperative user clustering algorithm with 1 dB power adjustment step is used; (**c**) The pdfs of the packet delays when the cooperative user clustering algorithm with 2 dB power adjustment step is used.