5.3 PDF and CDF of Connectivity
5.3.3 With Toroidal Metrics But Without Considering R 2
In the above simulations, we give bounded transmission radius R2 in each models and force communication probability to zero while two nodes’ distance exceeds R2. How-ever, in the fast fading channel model and slow fading channel model, the radio signal strength decays with distance, therefore. There may be weakly radio signal while two nodes are far away more than R2. Next we simulate model without consider R2. Fig.
5.21 shows the fast fading model, with R1 = 0, R2 = 38.1 and f (r) = exp
³
−100r2
´ , for
0.010 0.02 0.03 0.04 0.05 0.06 0.2
0.4 0.6 0.8 1
l=200
density
probability
l=500 l=800
Not isolated Connected Theoretical
Figure 5.16: The cdfs of Diso, Dcon and Dth in the squares of l = 200, l = 500 and l = 800 over Nakagami fading channels with R1 = 0, R2 = 23.8 and f (r) =
³r2 50+ 1
´ e−r250.
Table 5.1: The density differences between Diso and Dcon for p = 0.5.
l = 200 l = 500 l = 800 Disk model 0.0022 0.00170 0.001100 Bernoulli model 0.0013 0.00038 0.000338 Slow fading model 0.0006 0.00011 0.000016 Fast fading model 0.0002 0.00013 0.000038
l = 200, l = 500and l = 800 without consider R2. Compared with the above simulation results, we can observe that removing R2 constraint doesn’t cause drastic change on the number of deploy nodes. Although it still has weakly signal strength while two nodes
’ distance exceeds R2, the signal strength is not strong enough to support two nodes’
communication.
0.010 0.02 0.03 0.04 0.05 0.06
0.010 0.02 0.03 0.04 0.05 0.06
Chapter 6
Conclusions
In this work, we assume that the distribution of wireless nodes is modeled by a Poisson point process with mean density over a deployment region instead of a uniform deployment. Also, a generalized link model is proposed. By deploying nodes with ade-quate mean density, we can get the expected number of isolated nodes and the distribution of the number of isolated nodes. Theoretical derivation and simulations are elaborately demonstrated. In conclusion, the expected number of isolated nodes is e−ω with certain constant ω and the distribution of the number of isolated nodes is asymptotically Poisson with mean e−ω. This information can help developer to design a wireless ad hoc and sensor network or to make decisions based on the demand of performance. Hopefully, the future research hopefully can loosen the restrictions on the deployment method and the deployment region.
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