1. Modeling End-to-end Distance for Given Number of Hops in Dense Planar Wireless Sensor Networks April. 2013 Chan-Myung Kim LINK@KoreaTech http://link.koreatech.ac.kr

2. ABSTRACT  We model the end-to-end distance for a given number of hops in dense planar Wireless Sensor Networks in this paper.  We derive that the closed-form formula for single hop distance and postulate Beta distribution for 2-hop distance.  When the number of hops increases beyond three, the multihop distance approaches Gaussian.  Our error analysis also shows the distance error is be minimized by using our model. LINK@KoreaTech 2

3. INTRODUCTION AND MOTIVATION  In Wireless Sensor Networks (WSN), knowledge of node location is often required in many applications.  Generally, the distances from a node with unknown location to several anchor nodes are estimated, and then a multilateration is applied to estimate the node location.  For those applications where the sensor nodes are overdensely deployed, the distance between the nodes are short and the variance of such distance is also small.  Therefore, it is quite promising to estimate the end-to-end distance based on the number of hops.  In this paper, we study the hopdistance relation in the planar WSN. LINK@KoreaTech 3

4. PRELIMINARIES  A. Skewness and Kurtosis  Skewness is a measure of symmetry, or more precisely, the lack of symmetry. A distribution, or sample set, is symmetric if it looks the same to the left and right of the center point. LINK@KoreaTech 4

5. PRELIMINARIES  A. Skewness and Kurtosis  Kurtosis is a measure of whether the data are peaked or flat relative to a normal distribution. LINK@KoreaTech 5

6. PRELIMINARIES  B. Chi-Square Test  Chi-square test is widely used to determine the goodness of fit of a distribution to a set of experimental data. LINK@KoreaTech 6

7. MODELING END-TO-END DISTANCE FOR GIVEN NUMBER OF HOPS  A. Problem Formulation  Firstly, our study on end-to-end distance for given number of hops is based on local coordinate system, which could be translated into a global coordinate system if enough nodes in the local coordinate system have known global coordinates.  Secondly, we assume the beacon packets are distributed in an ad hoc fashion. Under such circumstances, we have to assume the beacon packets are simply flooded throughout the sensor network LINK@KoreaTech 7

8. MODELING END-TO-END DISTANCE FOR GIVEN NUMBER OF HOPS  A. Problem Formulation  The problem of interest is to find the distance from a specific node to the anchor given this node is within i hops from the anchor. LINK@KoreaTech 8

9. MODELING END-TO-END DISTANCE FOR GIVEN NUMBER OF HOPS  B. Single-Hop Case  The problem of interest is to find the distance from a specific node to the anchor given this node is within i hops from the anchor.  And the conditional mean and variance are 2R/3 and R^2/18, respectively, LINK@KoreaTech 9

10. MODELING END-TO-END DISTANCE FOR GIVEN NUMBER OF HOPS  C. Two-Hop Case . LINK@KoreaTech 10

11. MODELING END-TO-END DISTANCE FOR GIVEN NUMBER OF HOPS  C. Two-Hop Case . LINK@KoreaTech 11

12. MODELING END-TO-END DISTANCE FOR GIVEN NUMBER OF HOPS  C. Two-Hop Case . LINK@KoreaTech 12

13. STATISTICAL ANALYSIS  All the simulation data are collected from such a scenario that N sensor nodes were uniformly distributed in a circular region of radius of 300 meters.The anchor node was placed at (0, 0).  We ran simulations for extensive settings of node density λ and transmission range R.  And for each setting of (N,R), we ran 300 simulations, in each of which all nodes are re-deployed from the beginning. LINK@KoreaTech 13

14. STATISTICAL ANALYSIS .. LINK@KoreaTech 14

15. STATISTICAL ANALYSIS .. LINK@KoreaTech 15

16. STATISTICAL ANALYSIS  Optimum Estimation and Error Analysis LINK@KoreaTech 16

17. CONCLUSIONS  In this paper, we study the modeling of the end-to-end distance for given number of hops in WSN.  The experiments showed that the distance does not increase linearly with the number of hops. Therefore, the distance should be analyzed for each number of hops.  We derived the distribution for single-hop distance and also showed that the complexity of derivation for multiple-hop distance is beyond practical interest.  Thus, we postulate Beta distribution for two-hop end-to-end distance and Gaussian distribution for three-and-more-hop end-to-end distance.  Computer simulations showed our postulated distributions agree well with the histograms.  We also show that the distance error can be minimized by exploiting the distribution knowledge. LINK@KoreaTech 17