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Distribution of Link Distances in a Wireless Network

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ABSTRACT

The probability distribution is found for the link distance between two randomly positioned mobile radios in a wireless network for two representative deployment scenarios: (1) the mobile locations are uniformly distributed over a rectangular area and (2) the x and y coordinates of the mobile locations have Gaussian distributions. It is shown that the shapes of the link distance distributions for these scenarios are very similar when the width of the rectangular area in the first scenario is taken to be about three times the standard deviation of the location distribution in the second scenario. Thus the choice of mobile location distribution is not critical, but can be selected for the convenience of other aspects of the analysis or simulation of the mobile system.

No MeSH data available.


Plot of the link distance cdf for a rectangular deployment area (D1 = ζD2 ≤ D2).
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f4-j62mil: Plot of the link distance cdf for a rectangular deployment area (D1 = ζD2 ≤ D2).

Mentions: In Appendix A, it is shown that the cdf for the link distance between two mobiles that are randomly positioned in a rectangular area is given by Eq. (2.8). For a square area with D1 = D2 = D, or ζ = 1, the cdf reduces to Eq. (2.9).Fd(γ=ξD1)={0,ξ<0ζξ2[12ζξ2−43ξ(1+ζ)+π,]0≤ξ<123ζξ2−1(2ξ2+1)−16ζ(8ξ3+6ζξ2−ζ) +2ζξ2sin−1(1/ξ),1≤ξ<ζ−123ζξ2−1(2ξ2+1)−12ζ2(ξ4+2ξ2−13) +23ξ2−ζ−2(2ζ2ξ2+1)+16ζ−2−ξ2 +2ζξ2{sin−1(1/ξ)−cos−1(1/ζξ)},ζ−1≤ξ<1+ζ−21,1+ζ−2≤ξ.(2.8)Fd(γ=ξD)={0,ξ<0ξ2(12ξ2−83ξ+π),0≤ξ<143ξ2−1(2ξ2+1)−(12ξ4+2ξ2−13) +2ξ2[sin−1(1/ξ)−cos−1(1/ξ)],1≤ξ<21,2≤ξ.(2.9)Example plots of Eqs. (2.8) and (2.9) are shown in Fig. 4. For example, note from Fig. 4 that the median link distance (the value γ of for which the cdf equals 0.5) is approximately for the case of ζ = 1. In fact, solving Fd(ξ) = 0.5 numerically for ζ = 1 yields ξmed = 0.5120. Additional median values for this distribution are given in Table 1 for different values of ζ.


Distribution of Link Distances in a Wireless Network
Plot of the link distance cdf for a rectangular deployment area (D1 = ζD2 ≤ D2).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4862812&req=5

f4-j62mil: Plot of the link distance cdf for a rectangular deployment area (D1 = ζD2 ≤ D2).
Mentions: In Appendix A, it is shown that the cdf for the link distance between two mobiles that are randomly positioned in a rectangular area is given by Eq. (2.8). For a square area with D1 = D2 = D, or ζ = 1, the cdf reduces to Eq. (2.9).Fd(γ=ξD1)={0,ξ<0ζξ2[12ζξ2−43ξ(1+ζ)+π,]0≤ξ<123ζξ2−1(2ξ2+1)−16ζ(8ξ3+6ζξ2−ζ) +2ζξ2sin−1(1/ξ),1≤ξ<ζ−123ζξ2−1(2ξ2+1)−12ζ2(ξ4+2ξ2−13) +23ξ2−ζ−2(2ζ2ξ2+1)+16ζ−2−ξ2 +2ζξ2{sin−1(1/ξ)−cos−1(1/ζξ)},ζ−1≤ξ<1+ζ−21,1+ζ−2≤ξ.(2.8)Fd(γ=ξD)={0,ξ<0ξ2(12ξ2−83ξ+π),0≤ξ<143ξ2−1(2ξ2+1)−(12ξ4+2ξ2−13) +2ξ2[sin−1(1/ξ)−cos−1(1/ξ)],1≤ξ<21,2≤ξ.(2.9)Example plots of Eqs. (2.8) and (2.9) are shown in Fig. 4. For example, note from Fig. 4 that the median link distance (the value γ of for which the cdf equals 0.5) is approximately for the case of ζ = 1. In fact, solving Fd(ξ) = 0.5 numerically for ζ = 1 yields ξmed = 0.5120. Additional median values for this distribution are given in Table 1 for different values of ζ.

View Article: PubMed Central - PubMed

ABSTRACT

The probability distribution is found for the link distance between two randomly positioned mobile radios in a wireless network for two representative deployment scenarios: (1) the mobile locations are uniformly distributed over a rectangular area and (2) the x and y coordinates of the mobile locations have Gaussian distributions. It is shown that the shapes of the link distance distributions for these scenarios are very similar when the width of the rectangular area in the first scenario is taken to be about three times the standard deviation of the location distribution in the second scenario. Thus the choice of mobile location distribution is not critical, but can be selected for the convenience of other aspects of the analysis or simulation of the mobile system.

No MeSH data available.