sensors are required to maintain the desired coverage ratio. Interestingly, when the border effects are ignored, the sensor density does not increase with the perimeter since the area is constant. On the other hand, a FoI with a larger perimeter has a larger boundary region, and for all the sensors near the boundary, a part of their sensing region falls outside the FoI. This result shows that the border effects are too important to be neglected in a FoI with a large perimeter, when the desired coverage ratio is high.
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8 10 12 14 16 18
Sensordensity(x10−6)
Perimeter of the convex field (in 103m) Border effects,E[η]=0.8
No border effects,E[η]=0.8 Border effects,E[η]=0.9 No border effects,E[η]=0.9
Figure 3.9 Relationship between the sensor density and the perimeter of the FoI for a desired coverage ratio under border effects.
• Impact of sensing range: We study the impact of the sensing range on the coverage ratio for two different densities of deployment in a FoI. The sensors are uniformly deployed in a FoI with an area of 4×106m2 and a perimeter of 12×103m.
Fig. 3.10 shows an increase in the expected coverage ratio with the sensing range for two different densities of deployment. Obviously, as the sensing range of the sensors increases, the coverage ratio increases but rapidly for a larger density. An interesting observation from this result is that for a lower density of sensors, the border effects are prominent. Similar to the results in the previous section, we observe that the border effects are important for a higher coverage ratio with a low density of sensors. Thus, some work in the literature avoids border effects by considering dense deployment of
they tend to have a smaller effective sensing area. The results in Figs. 3.9 and 3.10 show that the border effects are negligible in the case of a smaller perimeter and a lower ratio of the sensing range to the area of the FoI.
0 0.2 0.4 0.6 0.8 1
10 50 100 150 200 250 300 350 400 450
Expectedcoverageratio
Sensing range (in m)
Border effects,ρ=8.75×10−6 Border effects,ρ=6.25×10−4 No border effects,ρ=8.75×10−6 No border effects,ρ=6.25×10−4
Figure 3.10 Relationship between the expected coverage ratio and the sensing range of sensors.
• Comparison with sensor density computed in [1]: Finally, we compare the CSD estimated in this work with that obtained in [1]. As mentioned earlier, the work in [1] does not consider the exact geometry of the FoI but considers only the area and the perimeter. Due to this, the number of sensors computed with their analysis is not necessarily minimal. For the scenario described in Example 1, we compute the sensor density for a given coverage ratio, using both Eq. 3.24 and the results in [1].
The sensing range, perimeter, and the area of the FoI are 100m, 6×103m, and 106m2, respectively. Fig. 3.11 compares the sensor densities for different coverage ratios. It can be seen that the sensor density estimated with our analysis is always lower than that obtained in [1]. The difference in the estimated sensor density is significant for higher coverage ratio. Thus, we conclude that our approach estimates the sensor density better and hence lowers the cost of stochastic deployment of WSNs. The results in Figs. 3.9 and 3.11 show that considering the geometry of the FoI and not just the area estimates the CSD better.
• Impact of a rectangular boundary over the FoI: We study the impact of a rectangular boundary over the FoI. We compare the required number of sensors
0 50 100 150 200 250
0.5 0.6 0.7 0.8 0.9 0.99
Sensordensity(x10−6 )
Expected coverage ratio Based on the proposed CSD
Based on the existing work
Figure 3.11 Comparison of the sensor density obtained with Eq. 3.24 and that in [1] for different coverage ratios.
estimated in this work with that obtained using a rectangular boundary over the FoI.
We compute the required number of sensors for a given coverage ratio, using Eq. 3.24.
We create a rectangular boundary over the FoI and compute the required number of sensors for the given coverage ratio as shown in Fig. 3.12. The sensing range of the sensors is 100m. Fig. 3.13 compares the required number of sensors for different coverage ratios. It can be seen that the required sensors estimated with our analysis is always lower than that obtained using a rectangular boundary over the FoI. The difference in the estimated sensors is significant for higher coverage ratio. Thus, we conclude that our approach estimates the number of sensors better and hence lowers the cost of stochastic deployment of WSNs.
y
(3000,0) x (2300,1800) (2000,2000)
(4800,0) (2000,0)
(3700,2000)
(4800,1000) (4800,2000)
Figure 3.12 A rectangular boundary over a convex polygon shaped FoI.
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0.5 0.6 0.7 0.8 0.9
NumberofsensorsforE[η]
Expected coverage ratio Based on the proposed result Rectangular boundary over the FoI
Figure 3.13 Comparison of the required number of sensors obtained with Eq. 3.24 and a rectangular boundary over the FoI for different cov- erage ratios.
very realistic. We discuss how the analysis proposed in our work can be extended when the assumptions are not valid.
When the FoI is not in the shape of a polygon but is in an irregular shape, it is hard to estimate the effective sensing area exactly. An irregular-shaped region can be approximated with a polygonal shape and then our analysis can be applied.
Circumscribing an irregular shaped FoI with a polygon can be done efficiently by using the algorithms proposed in [43]. With this approach, our analysis for the CSD can be used for even an irregular-shaped FoI.
Similarly, our analysis is based on the assumption that the sensing region of a sensor is circular. In practice, the sensing region may not be a perfect disc. A few other sensing models are described in [4]. The work in [44] estimated the inner radius of a non-disc region by inscribing a circle inside the region. Once the inner radius is estimated it can be used in our analysis for other sensing models.