Wireless Communication Technologies Group
3/20/02
CISS 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
Wireless Communication Technologies Group
3/20/02
CISS 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.
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
3
Wireless Network Modeling
What is the difference between these two random networks?
R
/
D
= 0.12,
x
0
= 0.00,
N
= 100: c
avg
= 0.441, n
avg
= 4.41, h
avg
= 5.08
R
/
D
= 0.12,
x
0
= 0.05,
N
= 100: c
avg
= 1.00, n
avg
= 3.34, h
avg
= 8.42
Wireless Communication Technologies Group
3/20/02
CISS 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
=
x
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
Wireless Communication Technologies Group
3/20/02
CISS 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.
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
6
Calculation of Area Coverage Measure
No inhibition
d
min
/
D
= 0.075
Treat each cell as a trial, calculate mean and variance
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
7
Results of calculation to test concept
0
0.0
x
0
0.05
x
0
0.075
x
Binomial
# nodes,
n
# cells
P
n
# cells
P
n
# cells
P
n
P
n
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
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
8
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
–
2
A
(
N
–
1)/
N
–
2
A
3
1/
N
–
3
A
(
N
–
1)/
N
–
3
A
…
…
…
k
1/
N
–
kA
(
N
–
1)/
N
–
kA
A
: radius = minimum
distance between nodes
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
9
Analytical Expression for
P
n
max
1 1
max
0
1 1
Pr,
n N n
i j N n
N
N
n const iA jA n n
n
N N
where
A
= area around a selected node that is “inhibited.”
For
A
= 0,
1 1
Pr
n N n
N
N
n
n
N N
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
10
Comparison of Analysis, Simulation
Using
A
’ = E{
A
}
Estimates using (2)
Result, 1000 trials
x
0
A
n
max
Mean
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
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
11
Mean, Variance of # Neighbors
•
The simplest measure of connectivity is the average
number of neighbors per node,
n
.
•
n
= # connections (links) / # nodes
•
The analysis in this paper gives the mean value of
n
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.
Wireless Communication Technologies Group
3/20/02
CISS 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{
n

p;
x
0
} = (
N
1
)
a
(
p;
x
0
)
Var{
n

p;
x
0
} = (
N
1
)
a
(
p;
x
0
) [
1
a
(
p;
x
0
)]
where
a
(
p
)
min{1,
p
(
x
2
–
x
0
2
)}
p
Inhibited area
Communications
area
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
13
Unconditional Mean and Variance
1 1 0
p 0 0 2
1 0
F F
E p; Pr  F
1 F
d d
a
D D
x x
x x x x
x
where
2
2
1
8
F,0 1
2 3
x x
x x p x
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
n x
n x x
x
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
14
Example Simulation Results
Results diverge from theory for
x
0
> 0
because of sample size.
Wireless Communication Technologies Group
3/20/02
CISS 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
n
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
16
Scaling of variance: inversely proportional to node density
Wireless Communication Technologies Group
3/20/02
CISS 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
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
18
Connectivity vs. #Neighbors

Relationship is
statistical
Wireless Communication Technologies Group
3/20/02
CISS 2002, Princeton
19
Connectivity vs. #Neighbors

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

Relation between averages
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