Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks

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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks

Analysis of Hop-Distance Relationship in Spatially Random

Sensor Networks

Serdar Vural and Eylem Ekici

Department of Electrical and Computer EngineeringThe Ohio State University

{ vurals, ekici }@ece.osu.edu

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Introduction

• Random deployment of sensor networks is widely assumed for various applications

• Performance metrics that depend on sensor positions:– Coverage– Delay– Energy Consumption– Throughput …

• If sensor locations are unknown, modeling sensor locations becomes important for:

– Pre-deployment: Estimate metrics probabilistically

– Post deployment: Use simple metrics (e.g. hop count) for fine-granularity location/distance estimations

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Aim

• Find the relationship between hop count and Euclidean distance– Distribution of maximally covered distance dN

in N hops• Important for distance estimations through

broadcasting

– Need to know spatial distribution of sensors• Spatially uniform with density λ

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Analysis Topics

• One dimensional networks:

– Theoretical expressions for , and– Approximations of , and– Distribution approximation

• Generalization to 2D networks

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Single-hop distance

Rri-1 rei-1

R

ri rei• The pdf of a single-hop-distance in a one dimensional network [1] is:

[1] Y.C. Cheng, and T.G. Robertazzi, “Critical Connectivity Phenomena in Multi-hop Radio Models,“ IEEE Transactions on Communications, vol. 37, pp. 770-777,July 1989.

Cover maximum distance in a hop!

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Multi-hop distance

• Consider sensors at the maximum distance to a transmitting node• The pdf of a multi-hop-distance in a one dimensional network:

regionvacantr

Rrr

i

i

e

ie

:

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Expected Value and Standard Deviation of dN

• Computationally costly Approximation required!

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Expected Value and Standard Deviation of dN

Approximations for:

• Expected value and standard deviation of ri

• Expected value and standard deviation of Dn

ASSUMPTION:

“Single-hop distances are identically distributed … but not independent!”

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Approximated E[ri]

• Expected value of vacant region rei:

• Expected distance of hop i:

• Expected single-hop distance:

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Approximated σri

• Variance of single-hop distance:

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Experimental, Theoretical, Approximated E[ri] and σri

Approximated and theoretical results match the experimental ones almost perfectly

R=100

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Multi-hop distance dN

Approximated E[dN] and σdN

• Expected multi-hop distance, E[dN]:

• Variance of multi-hop distance:

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Approximation of E[ri]

• Theoretical expressions are computationally costly

Maximum number of hops limited

• Decaying oscillatory character around the approximation value

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Expected dN

R=100

High density

Low density

High density

Low density

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Standard Deviation of dN

σdN

Density increases

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Distribution of dN

Observation: • Closed form solutions very costly to obtain• Multi-hop distance distribution resembles Gaussian distribution with mean E[dN] and std. dev. σdN

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Distribution of dN

• A statistical measure to test Gaussianity is required Kurtosis[2]:

• Kurtosis expression is complicated for multi-hop

• Can we approximate?

[2] A. Hyvarinen, J. Karhunen, and E. Oja (2001), “Independent Component Analysis,“ John Wiley & Sons

3])[(

])[()(

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4

xxE

xxExkurt

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Kurtosis of dN

Kurtosis of dN can be obtained by using determining moments of dN:

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Experimental vs. Approximated Kurtosis Values for Changing Node Density

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Experimental vs. Approximation Kurtosis Values for Changing Communication Range

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Mean Square Error between Multi-hop and Experimental Gaussian Distributions

Highest Density

Lowest Density

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Extensions to 2D Networks

Geometric complexity Analysis is more complicated than

1D case regarding:1. Definition2. Modeling 3. Calculation of the expected value

and standard deviation of distance

Definition of a region:1D : a line segment2D : an (irregular) area

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Directional Propagation Model

2

2 ii rA

• Angular slice S(α,R)

• Find the farthest sensor withinS(α,R) at each hop

• A chain of such hops forms a multi-hop distance

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2

21 ie rRAi

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Conclusions

• The distribution of the maximum Euclidean distance for a given number of hops is studied

• Theoretical expressions are computationally costly• Presented efficient approximation methods• Multi-hop-distance distribution resembles Gaussian

distribution Possible to model by Gaussian pdf

• Need only the mean and the variance values• Highly accurate results that match experimental and

theoretical results obtained• A model is also proposed for 2D Sensor Networks

Documents

Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks