Distributional Properties of Inhibited Random

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Nov 21, 2013 (3 years and 9 months ago)

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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