wireless communication technologies group 3/20/02ciss 2002, princeton 1 distributional properties of...

3/20/02 CISS 2002, Princeton 1

Wireless Communication Technologies Group

Distributional Properties of Inhibited Random Positions of Mobile Radio Terminals

Leonard E. Miller


National Institute of Standards and Technology

Gaithersburg, Maryland



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.



Wireless Network ModelingWhat is the difference between these two random networks?

R/D = 0.12, 0 = 0.00, N = 100: cavg = 0.441, navg = 4.41, havg = 5.08 R/D = 0.12, 0 = 0.05, N = 100: cavg = 1.00, navg = 3.34, havg = 8.42



• 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



Measures of Area Coverage

• A measure of the area coverage of a random placement of N nodes in a DD area can be based on the statistical variation of the number of nodes across regular subdivisions of the area, say "cells" of size D2/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.



Calculation of Area Coverage Measure

No inhibition dmin/D = 0.075

Treat each cell as a trial, calculate mean and variance



Results of calculation to test concept

0 0.0 0 0.05 0 0.075 Binomial

# nodes, n # cells Pn # cells Pn # cells Pn Pn

0 41 0.41 28 0.28 12 0.12 0.366

1 30 0.30 47 0.47 75 0.75 0.370

2 18 0.18 22 0.22 13 0.13 0.185

3 10 0.10 3 0.03 0 0.00 0.060

4 1 0.01 0 0.00 0 0.00 0.015

Sample mean 1.00 1.00 1.01 1.00

Sample variance 1.09 0.62 0.25 0.99



Probability of n nodes in a cell

inside

outside

# nodes placed Area remaining inside cell

Area remaining outside cell

0 1/N (N – 1)/N

1 1/N – A (N – 1)/N – A

2 1/N – 2A (N – 1)/N – 2A

3 1/N – 3A (N – 1)/N – 3A

… … …

k 1/N – kA (N – 1)/N – kAA: radius = minimum distance between nodes



Analytical Expression for Pn

max

1 1

max0

1 1Pr ,

n N n

i j N n

N Nn const iA jA n n

n N N

where A = area around a selected node that is “inhibited.” For A = 0,

1 1Pr

n N nN Nn

n N N



Comparison of Analysis, SimulationUsing A’ = E{A}

Estimates using (2) Result, 1000 trials

0 A nmaxMean Variance Mean Variance

0.00 0.00 N 1.000 0.990 1.000 0.989

0.01 0.00029 100 1.000 0.962 1.000 0.960

0.02 0.00105 9 0.998 0.883 1.000 0.893

0.03 0.00215 5 0.999 0.777 1.000 0.793

0.04 0.00345 3 1.000 0.653 1.000 0.682

0.05 0.00483 3 1.002 0.520 1.000 0.576

0.06 0.00620 2 1.062 0.461 1.000 0.475

0.07 0.00745 2 1.134 0.407 1.000 0.387

0.075 0.00800 2 1.169 0.376 1.000 0.347



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.



Conditional Mean and Variance

• Conditioned on the location p of a particular node, the number of neighbors for the node is the result of N1 binomial trials:

E{ | p; 0} = (N1) a(p; 0)

Var{ | p; 0} = (N1) a(p; 0) [1a(p; 0)]

where

a(p) min{1, (2 – 02)} p

Inhibited area

Communications area



Unconditional Mean and Variance

1 1 0

p 0 0 21 0

F FE p; Pr | F

1 F

d da

D D

where

2

21

8F , 0 1

2 3

2

2 2

2 2

p 2

E 1 F

Var 1 F 1 F

1 2 E p F

N

N

N N a



Example Simulation Results

Results diverge from theory for 0 > 0 because of sample size.



Scaling of Mean: For 400 nodes (four times the node density), halve the range and the inhibition distance to

get the same results for



Scaling of variance: inversely proportional to node density



Further Work

• Statistical relationship between #neighbors and connectivity, with and without inhibition– Means, variances– Correlation coefficients

• Methods for generating “random” networks with specified connectivity



Connectivity vs. #Neighbors--Relationship is statistical



Connectivity vs. #Neighbors--Correlation is positive for low connectivity



Connectivity vs. #Neighbors--Relation between averages

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