Modeling End-to-end Distance for Given Number of Hops in Dense Planar Wireless Sensor Networks April Chan-Myung Kim

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. 2

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. 3

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. 4

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

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. 6

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 7

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. 8

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, 9

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

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

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

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. 13

STATISTICAL ANALYSIS .. 14

STATISTICAL ANALYSIS .. 15

STATISTICAL ANALYSIS  Optimum Estimation and Error Analysis 16

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. 17