Results – Path Loss-based Model

First Experiment: Effect of AP Parameters on Their Number

The experiments we carry out here aim to check the coherence of the scheme we developed to find the minimal number of APs. We run the scheme for various values of the parameters of the APs and observe the effect on their numbers. The results of our experiments are shown in Table 2.1. Results are reported in more detail in Appendix B. In this table, dmax represents the coverage distance, while t1is the processing time for the scheme to complete.

Table 2.1: Path loss-based model – effect of AP parameters on their number

Pt[dBm] Rth[dBm] dmax[m] No. of APs t1[h]

20 -85 1770 1 AP 0:0:0.6 20 -76 628 2 APs 0:0:0.8 20 -73 444.6 3 APs 0:0:1.8 20 -70 314.7 6 APs 0:0:6.5 20 -55 55.97 50 APs 1:18:53. 15 -76 353.18 4 APs 0:0:3.3 15 -73 250 6 APs 0:0:11.1 15 -70 177 13 APs 0:0:42.0 15 -55 31.477 72 APs 4:43:3.39

As shown in Table 2.1, the AP parameters (Pt and Rth) affect the number of APs, as expected. For large coverage distances (large values of Pt(20 dBm) and low values of Rth (-85 dBm)), one AP can cover all the users, as shown in Figure 2.2.

As the coverage area shrinks, the number of APs increases. By increasing the value of Rth to -73 dBm and keeping Pt at 20 dBm, the required number of APs is increased, as expected. Figure 2.3 shows the number, and positions of the APs.

Figure 2.2: Path loss-based model – number and placement of AP when Pt = 20 dBm, Rth= -85 dBm, and ψ = 0

Figure 2.3: Path loss-based model – number and placement of APs when Pt= 20 dBm and Rth= -73 dBm

Table 2.1 shows that six APs are required to cover the users when Pt decreases to 15 dBm while Rth remains at -73 dBm. This is due to the coverage distance decreasing from 444.6 m to 250 m. The position of APs is shown in Figure 2.4. As shown in this figure, the size of each demand area is larger than the coverage distance of the AP. There- fore, one AP is not sufficient to cover one demand area. As the coverage distance decreases further to 31.47 m, seventy two APs are required to cover the area. The reason for so many APs is that most of the users are 100 m away from their closest neighbour, and only a few APs can cover more than one user.

Evaluation of a Multi-objective Functions Model 2.3. Numerical Experiments

Figure 2.4: Path loss-based model – number and placement of APs when Pt= 15 dBm and Rth= -73 dBm

We observed that when the coverage distance halves, the design area requires more than twice the number of APs. For example, in the case where the coverage distance changes from 628 m to 314.77 m, the number of APs increases from two to six, and when it changes from 353 m to 177 m, the number of APs increases from four to thirteen. This is due to the position of potential users in each demand area. Some of them are further away from the others at the centre of the area, as shown in Figure 2.1.

In general our numerical experiments show satisfactory results.

Second Experiment: Evaluation of Each Objective Function

Figure 2.2 shown in the first part of the experiment, is for the case when ψ = 0. In this case, the first objective function in problem (2.10) is ignored and only the objective function (2.4) is effective. The aim of this function is to ensure that the users that are far away from an AP enjoy an acceptable level of signal quality. Figure 2.2 clearly shows that this goal is obtained.

When ψ = 1, the second objective function in problem (2.10) is ignored and only the objective function (2.3) becomes effective. The aim of this function is to improve the average signal quality in the entire area. An example of this case is shown in Figure 2.5. As can be seen from this figure, the AP is placed roughly at the centre of the design area close

Figure 2.5: Path loss-based model – optimal number and placement of AP when Pt= 20 dBm, Rth= -85 dBm, and ψ = 1

to the densely populated potential users. Users on the top that are far away from the AP can be disconnected, as they are close to the border of AP’s coverage. Therefore, the objective is not achieved.

Third experiment: Effect of ψ Parameter on the Position of the AP

The balancing parameter ψ has an important role in the model. The aim of this experiment is to evaluate how this parameter should be selected. We run the experiment for a simple case where only one access point is required (Pt= 20 dBm and Rth= -85 dBm) to cover the potential users. The parameter ψ is varied between 0 and 1 and its effect on the placement of the AP, as well as the processing time tt is recorded. The results are reported in Table 2.2.

Table 2.2: Path loss-based model – effect of ψ parameter on the position of the AP ψ Position of APs tt[h] 0 [600.002,749.999] 0:0:0.616 0.03 [593.628,753.398] 0:0:0.632 0.05 [589.279,755.718] 0:0:0.628 0.07 [584.853,758.079] 0:0:0.752 0.075 [587.763,756.527] 0:0:0.380 0.0756 [584.633,758.196] 0:0:0.430 0.0757 [584.326,758.36] 0:0:0.430 0.0758 [200.502,0.865142] 0:0:0.131 0.076 [200.501,0.865487] 0:0:0.117

Evaluation of a Multi-objective Functions Model 2.3. Numerical Experiments

Table 2.2 – continued from previous page

ψ Position of APs t1[h] 0.2 [200.548,0.836348] 0:0:0.110 0.3 [200.556,0.831281] 0:0:0.139 0.4 [200.561,0.827888] 0:0:0.112 0.5 [200.573,0.819882] 0:0:0.123 0.6 [200.582,0.813493] 0:0:0.102 0.7 [200.59,0.80709] 0:0:0.100 0.8 [200.597,0.802179] 0:0:0.102 0.9 [200.603,0.798081] 0:0:0.104 1 [200.608,0.794256] 0:0:0.102

The results show clearly two trends: when ψ is small enough (here, ψ ≤ 0.0757), the solution is closer to the minimiser of the second objective function (2.4). In the interval ψ ∈ [0.0757, 0.0758] one can observe a sudden change. For ψ ≥ 0.0758, the solution is close to the minimiser of the second objective function (2.3).

As can been from the results, the choice of the best possible value of the parameter ψ can be quite difficult. Another approach is described in Chapters 3 and 4.