Hyperfractals for wireless networks modelling

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HAL Id: tel-02868490

https://tel.archives-ouvertes.fr/tel-02868490

Submitted on 15 Jun 2020

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Hyperfractals for wireless networks modelling

Dalia-Georgiana Popescu

To cite this version:

Dalia-Georgiana Popescu. Hyperfractals for wireless networks modelling. Networking and Internet

Architecture [cs.NI]. Sorbonne Université, 2018. English. �NNT : 2018SORUS264�. �tel-02868490�

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

mobile nodes relays

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10⁰ 10¹ 10

10 10 10⁰

cumulated distance

density as function of cumulated distance cumulated distance F+1)

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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mH

mV x

x H

V

x

mV y x

mH

O O

15 20 25 30 35 40

0 2 4 6 8 10 12x 10⁹

nr hops

energy

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0 200 400 600 800 1000 0

0.5 1 1.5

2 x 10

⁵

index nodes

lo a d

0 500 1000

0 2 4 6 8 x 10

⁴

index nodes

lo a d

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0.5 1 1.5 2 2.5

x 10⁴ 0.6

0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4x 10⁴

Cartesian coordinate x

Cartesian coordinate y

Cologne network topology and coverage

Base stations locations Coverage

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0.7 0.8 0.9 1 1.1 1.2 1.3 1.4 x 10⁴ 1.45

1.5 1.55 1.6 1.65 1.7

x 10⁴

Cartesian coordiante x

Cartesian cordinate y

UE trajectory through the network

Base stations locations Coverage UE trajectory

1.4 1.45 1.5 1.55

x 10⁴ 1

1.1 1.2 1.3 1.4 1.5 1.6 1.7

x 10⁴

Cartesian coordinate x

Cartesian coordinate y

UE trajectory through the network

Base stations locations Coverage

UE trajectory

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

mobile nodes relays

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200 0 400 600 800 1000 1200 1400 1600 500

1000 1500

n

number of relays

computed, k

_max

=20

computed, k

_max

=40

computed, k

_max

=60

measured

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10

⁰

10

¹

10 10 10 10

⁰

cumulated distance

density as function of cumulated distance

cumulated distance

F+1)

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10 10 10 10^0.1 10⁰

cumulated distance

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10⁰ 10¹ 10

10 10⁰

cumulated distance

10

⁰

10 10 10

⁰

10

¹

cumulated distance

density as function of cumulated distance

cumulated distance

^F⁺¹⁾

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200 400 600 800 1000 1200 1400 1600 0.9

0.91 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99

Fraction of total nr of points in giant component

200 400 600 800 1000 1200 1400 1600

30 40 50 60 70 80 90 100 110

Number of points N nr points outside of the giant component N^1/3

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mH

mV x

x H

V

x

mV y x

mH

O O

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15 20 25 30 35 40 0

2 4 6 8 10 12x 10⁹

nr hops

energy

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15 20 25 30 35 0.6

0.8 1 1.2 1.4 x 10

⁵

k hops

m in im u m e n e rg y

10

^1.3

10

^1.4

10

⁴

10

⁵

10

⁶

k hops

m in im u m e n e rg y

22 24 26 28 30 32

200 250 300 350 400

k hops

m in im u m e n e rg y

10

^1.37

10

^1.43

10

^1.49

10

^2.4

10

^2.5

k hops

m in im u m e n e rg y

0 20 40 60 80

0.5 1 1.5 2 2.5

3 x 10

⁵

k hops

m in im u m e n e rg y

10

^1.3

10

^1.8

10

⁴

10

⁵

10

⁶

k hops

m in im u m e n e rg y

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20 30 40 50 60 70 1

2 3 4 5 x 10

⁵

k hops

m in im u m e n e rg y

10

^1.5

10

^1.6

10

^1.7

10

^5.1

10

^5.3

10

^5.5

k hops

m in im u m e n e rg y

10 15 20 25 30 35 40

1 2 3 4 5 6x 10⁷

k hops

minimum energy

10^1.1 10^1.3 10^1.5

10^7.1 10^7.3 10^7.5 10^7.7

k hops

minimum energy

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10² 10⁶

10⁸ 10¹⁰

path length

path maximum energy

n=500 n=800

10¹ 10²

10⁶ 10⁸ 10¹⁰

path length

path maximum energy n=500

n=800

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0 200 400 600 800 1000 0

0.5 1 1.5

2 x 10

5

index nodes

lo a d

0 500 1000

0 2 4 6 8 x 10

⁴

index nodes

lo a d

400 5 600 800 1000 1200

10 15 20

nr points d

_m

=3,d

_r

=3 d

_m

=3, d

_r

=4.3

400 600 800 1000 1200

14 15 16 17 18 19

nr points

d el ay

d

_m

=3,d

_r

=3

d

_m

=3, d

_r

=4.3

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400 600 800 1000 1200 10

12 14 16 18

nr points d

_m

=3,d

_r

=3 d

_m

=3, d

_r

=4.3

400 6 600 800 1000 1200

6.5 7 7.5 8 8.5

nr points

d el ay

d

_m

=3,d

_r

=3

d

_m

=3, d

_r

=4.3

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0 2000 4000 6000 8000 0

1 2 3 4 5 6 7 8

n

average broadcast time

X inflexion point O(nh) O(n log n)

1.2 1.25 1.3 1.35 1.4

2.5 3 3.5

x 10

⁴

time

nu m be r of n od es c on ta m in at ed

with teleportation

no teleportation

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200 300 400 500 600 700 800 0

5 10 15 20 25 30 35 40 45 50

n delay (ms) direct route

upper bound in direct route lower bound in direct route

2000 300 400 500 600 700 800

10 20 30 40 50

n

delay (ms)

direct route

upper bound in direct route lower bound in direct

200 300 400 500 600 700 800

0 5 10 15 20 25 30 35 40 45 50

n

delay (ms)

direct route

2000 300 400 500 600 700 800

5 10 15 20 25 30 35 40 45 50

n

delay (ms)

diverted route

upper bound in diverted route lower bound in diverted route

200 300 400 500 600 700 800

0 10 20 30 40 50

n

delay (ms)

diverted route upper bound in diverted lower bound in diverted

200 300 400 500 600 700 800

0 5 10 15 20 25 30 35 40 45 50

n

delay (ms)

diverted route

upper bound in diverted route lower bound in diverted route

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2000 300 400 500 600 700 800 10

20 30 40 50

n

delay (ms)

direct route

2000 300 400 500 600 700 800

10 20 30 40 50

n

delay (ms)

2000 300 400 500 600 700 800

5 10 15 20 25 30 35 40 45 50

n

delay (ms)

direct route

200 300 400 500 600 700 800

0 5 10 15 20 25 30 35 40 45 50

n

delay (ms)

direct route

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200 300 400 500 600 700 800 0

10 20 30 40 50

n

delay

direct route

200 300 400 500 600 700 800

0 10 20 30 40 50

n

delay

200 300 400 500 600 700 800

0 5 10 15 20 25 30 35 40 45 50

n

delay

direct route

200 300 400 500 600 700 800

0 5 10 15 20 25 30 35 40 45 50

n

delay

direct route

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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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0 1000 2000 3000 4000 5000 0

10 20 30 40 50

n

br oa dc as t t im e

upper bound simulations lower bound

0 1000 2000 3000 4000 5000

0 5 10 15 20 25 30 35

n

br oa dc as t t im e

upper bound simulations lower bound

0 1000 2000 3000 4000 5000

0 5 10 15 20 25 30

n

br oa dc as t t im e

upper bound simulations lower bound

0 1000 2000 3000 4000 5000

0 5 10 15 20

n

br oa dc as t t im e

upper bound

simulations

lower bound

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0 1000 2000 3000 4000 5000 0

10 20 30 40

n

br oa dc as t t im e

upper bound simulations lower bound

0 1000 2000 3000 4000 5000

0 5 10 15 20

n

br oa dc as t t im e

upper bound simulations lower bound

0 1000 2000 3000 4000 5000

0 5 10 15 20

n

br oa dc as t t im e

upper bound simulations lower bound

0 1000 2000 3000 4000 5000

0 2 4 6 8 10 12

n

br oa dc as t t im e

upper bound

simulations

lower bound

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