Cell-based positioning method with single power-level of RNs
3.2 Positioning accuracy analysis in irregular net- net-work structurenet-work structure
3.2.2 Improving accuracy with power adjustment
If the transmitting power of the RN can be adjusted, then the coverage of the RN varies. Consider this problem: given a set of RN i, i = 1, . . . , n, with a fixed location and its transmission range ri, a ≤ ri ≤ b, we want to find a set of transmission ranges (r∗1, r∗2, . . . , r∗n) such that the positioning accuracy is optimized.
1 2
Figure 3.16: An example of region finding.
This problem is equivalent to finding a set of transmission ranges (r1∗, r2∗, . . . , rn∗) such that e(r1, r2, . . . , rn) = max1≤i≤{Ri} is minimized. That is,
z(r1∗, r∗2, . . . , r∗n) = min
(r1,r2,...,rn)e(r1, r2, . . . , rn) where a≤ ri ≤ b, i = 1, 2, . . . , n.
In this section, we applied probability analysis to find the optimization solution of power adjustment. However, finding this optimization solution is an NP-hard problem. Therefore, we applied Simulated Annealing (SA) to solve this optimiza-tion problem. Finally, we show the improving posioptimiza-tioning accuracy with power adjustment by simulation.
1) Simulated annealing (SA)
Recently, SA has become more popular for solving large-scale combinatorial opti-mization problems with approximate optiopti-mization solutions. The advantage of SA is that it provides a general purpose solution for a wide variety of combinatorial optimization problems. Thus, SA is used in many fields such as computer-aided
de-sign of integrated circuits, image processing, code-dede-signed, neural network theory and so on.
In general, the SA algorithm is similar to metal-cooling. During slow cooling, a metal rearranges the atoms into regular crystalline structures with high density and low energy. The SA algorithm starts with an initial solution s0 = (r10, r02, . . . , rn0), and finds the value of cost function e(s0) = e(r01, r20, . . . , rn0), also known as fit-ness function(see equation (1)). Let si be the current solution with cost function e(si). For each iteration j, generate a random neighbor sj of si and evaluate its cost function e(sj). If e(sj) ≤ e(si), then sj is accepted (i.e., set si = sj and e(si) = e(sj)). Otherwise, the sj will be accepted with the probability p = min{1, exp((e(si)− e(sj))/T )}. The parameter of T means the ”tempera-ture” which changed with parameter α for each iteration. This is known as the Metropolis criteria [27] and the pseudocode is shown below:
Step 1: Initialize the temperature T . Generate an initial solution s0 and set current solution si = s0.
Step 2: Generate a trial solution sj, a random neighbor of si. Step 3: Let e = e(sj)− e(si).
Step 4: If e ≤ 0, then the trial solution sj is accepted. Set current solution si = sj and e(si) = e(sj).
If e > 0, then the trial solution sj is accepted with the probability p = exp(−eT ) > d, where d is a random number in [0, 1]. Set current solution si = sj and e(si) = e(sj).
Otherwise, go to Step 2.
: blocked point : candidate point
Figure 3.17: Grid-based deployment.
Step 5: Repeat Steps 2-4 for It iterations.
Step 6: T = T × α.
Step 7: Repeat Steps 2-6, until T < Tstable.
In this simulation, we set It= 300, T = 1, Tstable = 0.05 and α = 0.85.
2) Simulation results
This simulation ran on networks of 25, 36, 49, . . . , 225 RNs in a square working area of 500 units × 500 units, respectively. We divided the service area into n grids (see Figure 3.17) where n is the number of RNs. As shown in Figure 3.17, white points, called candidate points, represent possible locations to place a RN in the grid. The black points represent block points where one cannot place a RN.
Assume that the signal of the RN must cover the entire grid.
Four types of adjustments are considered for improving the accuracy of the cell-based positioning method.
Type 1: Assume that the locations of RNs are given and the transmission ranges of the RNs are identical. We evaluate e(r, r, . . . , r) for r = a, a + δ, . . . , a + kδ where k = 0.6aδ and a is the minimal transmission range such that the entire service area is covered. Then find z(r∗, r∗, . . . , r∗) = min{e(r, r, . . . , r)|r = a, a + δ, . . . , a + 0.6aδ δ}. (In our simulation, we set δ = 1.)
Type 2: Assume that the locations of RNs are given and the transmission ranges ri, i = 1, 2, . . . , n, are in [a, 1.6a]. The SA method is applied to the network to find a set of transmission ranges (r∗1, r2∗, . . . , rn∗) such that z(r∗1, r2∗, . . . , rn∗)≈ min(r1,r2,...,rn)e(r1, r2, . . . , rn).
Type 3: Assume that the transmission ranges ri (i = 1, 2, . . . , n) of the RNs are given.
SA is applied to the network to allocate the locations (xi, yi) of RNs such that z(r1, r2, . . . , rn)≈ min(x1,y1),...,(xn,yn)e(r1, r2, . . . , rn) where (xi, yi) is the coordinate of RN i.
Type 4: This is a combination of Types 2 and 3. For a given (r1, r2, . . . , rn), SA is applied to allocate the locations of RNs. Then, use SA again to adjust the transmission ranges ri (i = 1, 2, . . . , n) of the RNs.
Because the optimal value of z(r1∗, r∗2, . . . , r∗n) is hard to find, we use a lower bound of z(r∗1, r2∗, . . . , rn∗), denoted as LB, for comparison. A lower bound of z(r∗1, r2∗, . . . , r∗n) is obtained as follows. By simulation, we can estimate the max-imum number of localization regions for each network. Then, the whole service area divided by the maximum number of localization regions can be used as a lower bound of z(r1∗, r∗2, . . . , r∗n). Table 3.3 summarizes the maximum number of localization regions and the lower bound of z(r∗1, r2∗, . . . , rn∗) for each network.
Table 3.3: The maximum numbers of localization regions and LBs of z(r∗1, r2∗, . . . , r∗n).
Number of RNs
in the network 25 36 49 64 81 100 121 144 169 196 255 Maximum
number of regions 168 246 325 416 499 601 673 739 798 865 893 LB of
z(r∗1, r2∗, . . . , r∗n) 1488 1016 769 601 501 416 371 338 313 289 280
In order to show the performance of the proposed methods, comparisons be-tween the accuracy both before and after the adjustments are made. Let Zi de-note the accuracy z(r1∗, r∗2, . . . , r∗n) found by Type i and Z0 denote the initial ac-curacy e(r01, r20, . . . , rn0) where (r10, r02, . . . , rn0) is an initial solution. Figure 3.18 shows the improvement rate γi of Type i, i = 1, 2, 4, where the improvement rate γi = |ZiZ−Z0|
0 × 100%. Besides, an upper bound of the improvement rate
¯
γ∗ = |LB−ZZ 0|
0 × 100% is also shown in Figure 3.18. From Figure 3.18, note that the accuracy can be improved up to 30% by the Type 4 method.
Figure 3.19 shows the relationships between the proportion of localization area to the entire service area and the number of RNs in the network. For example, the proportion of the lower bound is 250000338 × 100% = 0.14% for the network with 144 RNs. In Figure 3.20, it shows the average accuracy for various numbers of RNs. For Type 1 to 4 with 144 RNs, the average accuracy is within 0.15. In this figure, we learned that increasing the number of RNs from 25 to 144, the accuracy is significantly improved. However, when the number of RNs is more than 144, the improvement of accuracy becomes insignificant.
Finally, the improvement rates of the accuracy of Types 3 and 4 are compared
25 36 49 64 81 100 121 144 169 196 225 0
10 20 30 40 50 60 70 80 90 100
Number of RNs
Improvement rate
(%)
Type 1 ( γ1 ) Type 2 ( γ2 ) Type 4 ( γ4 ) Upper bound
Figure 3.18: The improvement rates of Types 1, 2, and 4, and the upper bound.
in Figure 3.21. The results of the two types are almost the same. This means that for accuracy, the factor of allocating the RNs’ locations is more significant than the factor of adjusting transmission ranges. If we can place the RNs in appropriate places, the accuracy of the cell-based positioning method will be better.