1. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 1 Distributional Properties of Inhibited Random Positions of Mobile Radio Terminals Leonard E. Miller Wireless Communication Technologies Group National Institute of Standards and Technology Gaithersburg, Maryland

2. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 2 Abstract/Outline Subject: Spatial distribution properties of randomly generated points representing the deployment of radio terminals (nodes) in an area. Focus: Measures of area coverage, connectivity. Focus: Influence of “inhibition” process that controls the minimum distance between nodes. –Cheng & Robertazzi, "A New Spatial Point Process for Multihop Radio Network Modeling," Proc. 1990 IEEE Internat'l Conf. on Comm., pp. 1241-1245. Sampling of results relating measures of connectivity.

3. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 3 Wireless Network Modeling What is the difference between these two random networks?

4. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 4 Both networks are generated using uniform distributions for x and y positions, but the second network adds the requirement or “inhibition” that nodes cannot be closer than R/D =  0 = 0.05. The average number of neighbors per node is lower for the inhibition process in this example (4.41 vs. 3.34), but the average node-pair connectivity is higher (1.00 vs. 0.44) because the nodes are placed more evenly in the space. Intuitively, the network with the minimum distance requirement also provides better “area coverage.” In this paper, a measure of area coverage is developed that shows the effect of inhibition quantitatively. Also, expressions are given for the mean and variance of the average number of neighbors per node. Node positions are “inhibited” for one network

5. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 5 Measures of Area Coverage A measure of the area coverage of a random placement of N nodes in a D  D area can be based on the statistical variation of the number of nodes across regular subdivisions of the area, say "cells" of size D 2 / N. On the average, for a random distribution of node locations, one would expect one node per cell. The variance of the number of nodes per cell then would reflect the uniformity of the distribution of the node locations among the cells and hence the degree to which the node location process produces an even pattern of coverage for the area.

6. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 6 Calculation of Area Coverage Measure No inhibitiond min /D = 0.075 Treat each cell as a trial, calculate mean and variance

7. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 7 Results of calculation to test concept Binomial # nodes, n# cellsPnPn PnPn PnPn PnPn 0410.41280.28120.120.366 1300.30470.47750.750.370 2180.18220.22130.130.185 3100.1030.0300.000.060 410.0100.000 0.015 Sample mean1.00 1.011.00 Sample variance1.090.620.250.99

8. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 8 Probability of n nodes in a cell inside outside # nodes placedArea remaining inside cell Area remaining outside cell 01/N(N – 1)/N 11/N – A(N – 1)/N – A 21/N – 2A(N – 1)/N – 2A 31/N – 3A(N – 1)/N – 3A ……… k1/N – kA(N – 1)/N – kA A: radius = minimum distance between nodes

9. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 9 Analytical Expression for P n where A = area around a selected node that is “inhibited.” For A = 0,

10. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 10 Comparison of Analysis, Simulation Using A’ = E{A} Estimates using (2)Result, 1000 trials 00 A n max Mean VarianceMeanVariance 0.00 N 1.0000.9901.0000.989 0.010.000291001.0000.9621.0000.960 0.020.0010590.9980.8831.0000.893 0.030.0021550.9990.7771.0000.793 0.040.0034531.0000.6531.0000.682 0.050.0048331.0020.5201.0000.576 0.060.0062021.0620.4611.0000.475 0.070.0074521.1340.4071.0000.387 0.0750.0080021.1690.3761.0000.347

11. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 11 Mean, Variance of # Neighbors The simplest measure of connectivity is the average number of neighbors per node,. = # connections (links) / # nodes The analysis in this paper gives the mean value of with and without inhibition in the selection of node locations. The analytical values are compared to simulated values, plus empirical values of the variance of the number of neighbors are obtained.

12. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 12 Conditional Mean and Variance Conditioned on the location p of a particular node, the number of neighbors for the node is the result of N  1 binomial trials: E{ | p;  0 } = ( N  1 ) a ( p;  0 ) Var{ | p;  0 } = ( N  1 ) a ( p;  0 ) [ 1  a ( p;  0 )] where a ( p )  min{1,  (  2 –  0 2 )} p Inhibited area Communications area

13. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 13 Unconditional Mean and Variance where

14. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 14 Example Simulation Results Results diverge from theory for  0 > 0 because of sample size.

15. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 15 Scaling of Mean: For 400 nodes (four times the node density), halve the range and the inhibition distance to get the same results for

16. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 16 Scaling of variance: inversely proportional to node density

17. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 17 Further Work Statistical relationship between #neighbors and connectivity, with and without inhibition –Means, variances –Correlation coefficients Methods for generating “random” networks with specified connectivity

18. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 18 Connectivity vs. #Neighbors --Relationship is statistical

19. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 19 Connectivity vs. #Neighbors --Correlation is positive for low connectivity

20. Wireless Communication Technologies Group 3/20/02CISS 2002, Princeton 20 Connectivity vs. #Neighbors --Relation between averages