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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.


Elliptical areas associated Gaussian mobile coordinates.
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f6-j62mil: Elliptical areas associated Gaussian mobile coordinates.

Mentions: Instead of assuming that the mobiles are randomly located in a rectangular area, we now assume that the x and y coordinates of the mobile locations have Gaussian distributions. That is, we assume that the pdfs of the x and y coordinates are independent and have the following pdfs:px(α)=1σ12πe−α2/2σ12,−∞<α<∞(3.1a)andpy(β)=1σ22πe−β2/2σ22,−∞<β<∞(3.1b)where σ1 and σ2 are, respectively, the standard deviations of the x and y coordinates. Without loss of generality, we assume that σ1 = λσ2, where λ is an area shape parameter, with λ ≤ 1. The joint pdf of the coordinates is given bypx,y(α,β)=12πσ1σ2exp{−12[(ασ1)2+(βσ2)2]}.(3.2a)Note that the joint pdf in Eq. (3.2a) is the special case of the bivariate Gaussian pdf with uncorrelated random variables (RVs); the more general case of correlated Gaussian coordinates can be treated by using a simple transformation of the coordinate system. As illustrated in Fig. 6, the elliptical area defined by the equation(ασ1)2+(βσ2)2=k2(3.2b)contains percent of the mobile positions, or about 39 % of the mobile positions when k = 1, 86 % when k = 2, and 99 % when k = 3. The elliptical area containing nearly all the positions corresponds to the rectangular area shown in Fig. 1, so that the Gaussian-coordinate model can easily be related to the uniformly distributed mobile model when it is convenient. For example, an ellipse just fitting inside the rectangle of Fig. 1 has the area and contains of the mobile positions for the rectangular, uniform distribution. This same percentage for the Gaussian-coordinate model is contained in the elliptical area given by Eq. (3.2b) with k = 1.754, so that the two models are roughly equivalent when and or D1 ≈ 3.5 σ1 and D2 ≈ 3.5 σ2.


Distribution of Link Distances in a Wireless Network
Elliptical areas associated Gaussian mobile coordinates.
© Copyright Policy - open-access
Related In: Results  -  Collection

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

f6-j62mil: Elliptical areas associated Gaussian mobile coordinates.
Mentions: Instead of assuming that the mobiles are randomly located in a rectangular area, we now assume that the x and y coordinates of the mobile locations have Gaussian distributions. That is, we assume that the pdfs of the x and y coordinates are independent and have the following pdfs:px(α)=1σ12πe−α2/2σ12,−∞<α<∞(3.1a)andpy(β)=1σ22πe−β2/2σ22,−∞<β<∞(3.1b)where σ1 and σ2 are, respectively, the standard deviations of the x and y coordinates. Without loss of generality, we assume that σ1 = λσ2, where λ is an area shape parameter, with λ ≤ 1. The joint pdf of the coordinates is given bypx,y(α,β)=12πσ1σ2exp{−12[(ασ1)2+(βσ2)2]}.(3.2a)Note that the joint pdf in Eq. (3.2a) is the special case of the bivariate Gaussian pdf with uncorrelated random variables (RVs); the more general case of correlated Gaussian coordinates can be treated by using a simple transformation of the coordinate system. As illustrated in Fig. 6, the elliptical area defined by the equation(ασ1)2+(βσ2)2=k2(3.2b)contains percent of the mobile positions, or about 39 % of the mobile positions when k = 1, 86 % when k = 2, and 99 % when k = 3. The elliptical area containing nearly all the positions corresponds to the rectangular area shown in Fig. 1, so that the Gaussian-coordinate model can easily be related to the uniformly distributed mobile model when it is convenient. For example, an ellipse just fitting inside the rectangle of Fig. 1 has the area and contains of the mobile positions for the rectangular, uniform distribution. This same percentage for the Gaussian-coordinate model is contained in the elliptical area given by Eq. (3.2b) with k = 1.754, so that the two models are roughly equivalent when and or D1 ≈ 3.5 σ1 and D2 ≈ 3.5 σ2.

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.