1 / 24

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 Engineering The Ohio State University { vurals, ekici }@ece.osu.edu. Introduction.

brent-leonard
Download Presentation

Analysis of Hop-Distance Relationship in Spatially Random Sensor Networks

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

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

  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. 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

  4. Analysis Topics • One dimensional networks: • Theoretical expressions for , and • Approximations of , and • Distribution approximation • Generalization to 2D networks

  5. Single-hop distance R Cover maximum distance in a hop! ri-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.

  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. Expected Value and Standard Deviation of dN • Computationally costly  Approximation required!

  8. 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!”

  9. Approximated E[ri] • Expected distance of hop i: • Expected value of vacant region rei: • Expected single-hop distance:

  10. Approximated σri • Variance of single-hop distance:

  11. Experimental, Theoretical, Approximated E[ri] and σri R=100 Approximated and theoretical results match the experimental ones almost perfectly

  12. Multi-hop distance dNApproximated E[dN] and σdN • Expected multi-hop distance, E[dN]: • Variance of multi-hop distance:

  13. Approximation of E[ri] • Theoretical expressions are computationally costly • Maximum number of hops • limited • Decaying oscillatory character around the approximation value

  14. Expected dN R=100 High density High density Low density Low density

  15. Standard Deviation of dN σdN Density increases

  16. 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

  17. 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

  18. Kurtosis of dN • Kurtosis of dN can be obtained by using determining moments of dN:

  19. Experimental vs. Approximated Kurtosis Values for Changing Node Density

  20. Experimental vs. Approximation Kurtosis Values for Changing Communication Range

  21. Mean Square Error between Multi-hop and Experimental Gaussian Distributions Highest Density Lowest Density

  22. Extensions to 2D Networks • Geometric complexity • Analysis is more complicated than 1D case regarding: • Definition • Modeling • Calculation of the expected value and standard deviation of distance • Definition of a region: • 1D : a line segment • 2D : an (irregular) area

  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. 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

More Related