large wireless autonomic networks sensor networks philippe jacquet
TRANSCRIPT
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 )