Large wireless autonomic networks

Sensor networksPhilippe Jacquet

Future of internet• A galaxy of wireless mobile nodes

Toward massively dense networks

• Captors sensor networks

• Micro or nano-drones– Static or mobile– Several thousands nodes per hectar

Nano drones and droids

– Very small RF devices.

5. A simple wireless model• Physical model

– An infinite plan

– Emitters have same nominal power Q– Signal attenuation at distance r from emitter :

αr

Q

)2( >α

x

y

5. Physical model

– S is emitter set at time t

• Received signal at point z and time t

},,{ 21 KzzS=),( iii yxz =

)(SWz

€

Wz(S) =Q

zi − zα

zi ∈S

∑

z

iz1z 2z

A wireless model

• Emitters are distributed as a point Poisson process in the plan

– Signals sum

€

Wz(S) = Q z − zi

−α

i

∑€

density λ

€

z€

zi

€

S = S(λ )

A wireless model

• Signal distribution

€

E(e−θW (S(λ ))) = exp(−λπΓ(1− γ )Qγθ γ )

€

γ=2

α

Signal power Laplace transform

• Partition of the plan

€

Ai{ }i∈N

€

rαr

Q

€

W (S) = W (S ∩ Ai)i

∑

€

w(θ,λ ) = E(e−θW (S(λ ))) = E(e−θW (S(λ )∩ A i ))i

∏

€

E(e−θW (S(λ )∩ A i )) ≈λ Ai( )

k

k!k

∑ e−λ A i e−

Qθ

rαk

= exp λ Ai (e−

Qθ

rα−1)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

€

w(θ,λ ) = exp λ (e−

Qθ

rα−1)dA∫∫

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

Wireless Space capacity

• With signal over noise ratio K requirement

• Average area of correct reception

€

I(K) =sin(γπ )

γπK−γ

€

σ(K,λ ) = P(W (S(λ )) <1

Krα) =∫∫ I(K)

λ

€

I(10) ≈ 0.037066

€

I(K) = E P(| z − zi |−α

| z − z j |−α

j≠ i

∑> K)

i

∑ ⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

Wireless Space capacity

• Reception probability vs distance

• Optimal routing radius

€

rm = argmaxr>0

{rp(r,λ ,K)} =r1λ

K−

γ

2

€

p(r,λ ,K) = p(r λ Kγ

2 ,1,1)

€

p(r,1,1) = (−1)n sin(πnγ )

πn

∑ Γ(nγ)

n!r2n

€

p(r,1,1) =1− erf(r2

2) when α = 4

r

€

z

€

z

€

′ z

€

′ z

Wireless Space capacity

• Average number of hops

• Average per hop transmission number

• Net traffic density€

rm

€

z

€

′ z

€

z − ′ z

rm

€

1

p(rm )

A

€

ρ =λ rm p(rm )

E( z − ′ z )

Wireless space capacity

• Net traffic density

– Increases when λ increases.– Is there a limit on λ?€

ρ =λ rm p(rm )

E( z − ′ z )= λ

r1p(r1)

E( z − ′ z )K

−γ

2

Density limit

• Network must remain dense

• Gupta Kumar rule for non isolation

• Density limit

€

σ1 = πr12

σ A

€

πrm2 N

A=

σ 1

λK γ

N

A>>1

€

πrm2 N

A> log N

€

λ <K−γσ 1

N

A log N

Density limits

• Brut per node traffic limit :

• Net per node traffic limit :€

λ A

N<

σ 1K−γ

logN

€

ρA

N< p(r1) σ 1

A

E( z − ′ z )K

−γ

2 1

N log N

Space capacity result (Gupta-Kumar 2000)

• The capacity increases with the density

• Massively dense wireless networks€

Aρ = π r12 p1K

−γ

2A

E(| z − ′ z |)

N

log N= O

N

logN

⎛

⎝ ⎜

⎞

⎠ ⎟

N

capacity

Protocol on wireless network

• Every node sends hellos at frequency h– Hellos are not routed

• Traffic density due to hellos – No other traffic:

– Limit network size due to hellos€

λ =hN

A

€

logN <σ 1

hK−γ€

λ <K−γσ 1

N

A log N

Manageable neighborhood

• Average neighborhood size

• Maximum network size€

M =σ 1

λK−γ N

A

€

σ1K−γ ≈ 0.025 for α = 2.5 and K =10

€

Mmax = σ 1K−γ 1

h≈ 250 for h =

1

10,000

€

Nmax = eM max ≈ 3.7.10108

6. Interprétation

maxM

N

multi sauts

Nlog

voisinage moyen

M

voisinageunique

réseaudéconnecté

maxNwhen N<M:Single hop

Neighbor vontrol: remaining capacity

• Traffic density

€

λ =hN

A+ ρβ

N

M

€

σ1K−γ

M= h + μβ

N

M

€

μmax =σ 1K

−γ

β

1

N log N(1−

h

σ 1K−γ

logN)

positive since maxNN <

moralement NN log

1max ∝μ

libre

gestion duvoisinage

Time capacity paradox

• Mobility can create capacity in disconnected networks

• Delay Tolerant Networks

S

DX Xpath disruption!

S D

End-to-end path

S

DX

Xpath disruption!

nodelink

Information propagation speed

• Unit disk graph model• Random walk mobility

model

€

z €

′ z

Time capacity paradox

• Mobility creates capacitycapacity

time

capacity

timePermanently disconnected Permanently connected

Information propagation time

€

T( ′ z )