A Uniform Continuum Model for the

Scaling of Ad-Hoc Networks

Ernst W GrundkeA Nur Zincir-Heywood

Faculty of Computer ScienceDalhousie University

Halifax NSwww.cs.dal.ca/~grundke/ants/continuumModel.html

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Acknowledgements

The Ant Colony:Nur Zincir-HeywoodAllan JostOwen YueDonald MorrisonNick Pilon

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In a mobile ad-hoc network, ...

•… nodes move•… nodes enter & leave•… communication is short-range wireless•… hosts = routers•… node power is scarce

What happens when such networks get large?

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A Continuum Model• Only the node density is known

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A Continuum Model• Only the node density is known• Think of nodes as “smeared out”

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A Uniform Model• Assume similar conditions everywhere:

no edge effects

XX

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Model

• Continuum of nodes• Uniform: no edge effects• Simple, optimistic assumptions• Mathematically tractable: no simulation• Dimensionless parameters • Analytical results• Approximate at best• J. B. S. Haldane, 1928, “On being the right size”

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Network Geometry

• Nearest neighbour distance d1

• Node density = 1/(r12)

r1 = d1/2

Voronoi cell

Node

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Network Geometry

• N nodes

• Max separation D

• D = d1(N-1)

D+d1

D

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Node Behaviour

• Generate user data randomly:

pT packets per unit time

• Finite transmission range R:d1 R D

• Forward user data packets

R

AB

C

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Nodes Behaviour (cont.)

• Motion:

pW link events per unit time

= pW/pT (the walk/talk ratio)

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Nodes Behaviour (cont.)

• Group of g nodes competing for a single channel:

g =(R+r1)2/r12

• Finite transmission bandwidth:pW<b, pT<b in isolation, pW<b/g, pT<b/g in network

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Forwarding User Data Packets

• Define forwarding overhead: = mean hops per packet

• Uniformity: data rate = pT (per node)

• If packets travel D/2 in hops of R, = (r1/R)(N-1)

• Example: <5 requires N<121.

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Ceiling (Forwarding)

pT < b/g

pT/b < R/(r1(N1)) r12 /(R+r1)2

= r1R/((R+r1)2(N1))

• Small range (R 2r1) is best; then

pT/b < 0.22/(N1)

• Example: If N=100 then pT/b < 2.5%

• Applies regardless of routing & mobility

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Ceiling (Forwarding)

pT/b < 0.22/(N1):

N Max pT/b

10 10.3%

100 2.47%

1,000 0.73%

10,000 0.22%

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Routing Traffic

• Define routing overhead: = packet transmissions caused by one link event

• Uniformity: data rate = pW (per node)

• Assume proactive routing; flat.• If each node broadcasts to (g1)/2,

N = (g1)/2.

• For R 2r1:

= (N/4)1

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Ceiling (Routing)

pW < b/g ...pW/b < 1/(2N) approx.

• Example: If N=100, pW/b < 0.5%.For b~1Mbps and 1000 bits/packet, pW < 5 link events/second

• Numerically not serious for modest N.

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Ceiling (Routing)

Routing: (1/N) Forwarding:

(1/N)

N Max pw/b Max pT/b

10 5% 10.3%

100 0.5% 2.47%

1,000 0.05% 0.73%

10,000 0.005% 0.22%

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Total Traffic: Data + Routing

pT + pW < b/g

( + ) pT < b/g ( = pW/pT)

• Forwarding traffic dominates: < /

• Routing traffic dominates: > /

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Power

• Average antenna power per node:

kR2(pT + pW) ~ (N)

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Other Dimensions

• See paper!

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Conclusions• For flat ad-hoc networks:

Keep N,R small; use hierarchies• Dimensionless parameters:

= pW/pT = the walk/talk ratio

= forwarding overhead = routing overhead

• In 2D: is (N/R), is (N/R2) / characterizes traffic type• Map simulations into a common space?

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?Comments?

Questions?

www.cs.dal.ca/~grundke/ants/continuumModel.html