1. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 1 Serdar Vural and Eylem Ekici Department of Electrical and Computer Engineering The Ohio State University { vurals, ekici }@ece.osu.edu

2. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 2 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

3. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 3 Aim Find the relationship between hop count and Euclidean distance –Distribution of maximally covered distance d N in N hops Important for distance estimations through broadcasting –Need to know spatial distribution of sensors Spatially uniform with density λ

4. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 4 Analysis Topics One dimensional networks: –Theoretical expressions for, and –Approximations of, and –Distribution approximation Generalization to 2D networks

5. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 5 Single-hop distance R r i-1 r ei-1 R riri r ei 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!

6. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 6 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:

7. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 7 Expected Value and Standard Deviation of d N Computationally costly  Approximation required!

8. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 8 Expected Value and Standard Deviation of d N Approximations for: Expected value and standard deviation of r i Expected value and standard deviation of Dn ASSUMPTION: “Single-hop distances are identically distributed … but not independent! ”

9. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 9 Approximated E[r i ] Expected value of vacant region r ei : Expected distance of hop i: Expected single-hop distance:

10. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 10 Approximated σ ri Variance of single-hop distance:

11. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 11 Experimental, Theoretical, Approximated E[r i ] and σ ri Approximated and theoretical results match the experimental ones almost perfectly R=100

12. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 12 Multi-hop distance d N Approximated E[d N ] and σ dN Expected multi-hop distance, E[d N ]: Variance of multi-hop distance:

13. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 13 Approximation of E[r i ] Theoretical expressions are computationally costly  Maximum number of hops limited Decaying oscillatory character around the approximation value

14. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 14 Expected d N R=100 High density Low density High density Low density

15. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 15 Standard Deviation of d N σ dN Density increases

16. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 16 Distribution of d N Observation: Closed form solutions very costly to obtain Multi-hop distance distribution resembles Gaussian distribution with mean E[d N ] and std. dev. σ dN

17. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 17 Distribution of d N 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

18. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 18 Kurtosis of d N Kurtosis of d N can be obtained by using determining moments of d N :

19. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 19 Experimental vs. Approximated Kurtosis Values for Changing Node Density

20. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 20 Experimental vs. Approximation Kurtosis Values for Changing Communication Range

21. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 21 Mean Square Error between Multi-hop and Experimental Gaussian Distributions Highest Density Lowest Density

22. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 22 Extensions to 2D Networks Geometric complexity  Analysis is more complicated than 1D case regarding: 1.Definition 2.Modeling 3.Calculation of the expected value and standard deviation of distance Definition of a region: 1D : a line segment 2D : an (irregular) area

23. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 23 Directional Propagation Model Angular slice S(α,R) Find the farthest sensor within S(α,R) at each hop A chain of such hops forms a multi-hop distance

24. Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks 24 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