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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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 of Hop-Distance Relationship in Spatially Random Sensor Networks
Analysis Topics
• One dimensional networks:
– Theoretical expressions for , and– Approximations of , and– Distribution approximation
• Generalization to 2D networks
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Expected Value and Standard Deviation of dN
• Computationally costly Approximation required!
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Approximated E[ri]
• Expected value of vacant region rei:
• Expected distance of hop i:
• Expected single-hop distance:
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Approximated σri
• Variance of single-hop distance:
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Experimental, Theoretical, Approximated E[ri] and σri
Approximated and theoretical results match the experimental ones almost perfectly
R=100
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Multi-hop distance dN
Approximated E[dN] and σdN
• Expected multi-hop distance, E[dN]:
• Variance of multi-hop distance:
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Expected dN
R=100
High density
Low density
High density
Low density
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Standard Deviation of dN
σdN
Density increases
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Kurtosis of dN
Kurtosis of dN can be obtained by using determining moments of dN:
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Experimental vs. Approximated Kurtosis Values for Changing Node Density
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Experimental vs. Approximation Kurtosis Values for Changing Communication Range
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
Mean Square Error between Multi-hop and Experimental Gaussian Distributions
Highest Density
Lowest Density
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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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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Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks
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