Continuum Equilibria for Routing in Dense Ad-hoc Networks Eitan - - PowerPoint PPT Presentation

Problem Model Multi-Class Costs Conclusions

Continuum Equilibria for Routing in Dense Ad-hoc Networks

Eitan ALTMAN, Alonso SILVA*, Pierre BERNHARD, Merouane DEBBAH December 5, 2007

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Problem Model Multi-Class Costs Conclusions

Table of Contents

1

Statement Problem and Previous Works

2

The Network Model

3

Multi-Class Case

4

Congestion dependent/independent costs

5

Conclusions

Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 2/26

Problem Model Multi-Class Costs Conclusions

Table of Contents

1

Statement Problem and Previous Works

2

The Network Model

3

Multi-Class Case

4

Congestion dependent/independent costs

5

Conclusions

Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 3/26

Problem Model Multi-Class Costs Conclusions

Statement Problem

Study the global as well as the non-cooperative optimal solution for the routing problem among a large population of users. Find a general optimization framework for handling minimum cost paths in massively dense ad-hoc networks.

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Problem Model Multi-Class Costs Conclusions

Previous Works

Geometrical Optics

• P. Jacquet studies the routing problem as a parallel to an optics

problem. Drawback: He doesn’t consider interaction between each user’s decision. Electrostatics

• S. Toumpis studies the problem of the optimal deployment of

wireless sensor networks. Drawback: The local cost assumed is too particular (cost(T) = |T|2 where T is the flow).

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Problem Model Multi-Class Costs Conclusions

Previous Works

Road Traffic

• S. Dafermos studies the user-optimizing and the system-optimizing

pattern. Drawback: She doesn’t give a formal mathematical development and

• nly consider one class of traffic.

Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 5/26

Problem Model Multi-Class Costs Conclusions

Table of Contents

1

Statement Problem and Previous Works

2

The Network Model

3

Multi-Class Case

4

Congestion dependent/independent costs

5

Conclusions

Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 6/26

Problem Model Multi-Class Costs Conclusions

Important: The network is massively dense. Our objective: Preserve only the most relevant information to allow meaningful network optimization problems.

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Problem Model Multi-Class Costs Conclusions

50 100 20 40 60 80 100 120 140 160 180 200 50 100 20 40 60 80 100 120 140 160 180 200 50 100 20 40 60 80 100 120 140 160 180 200 50 100 20 40 60 80 100 120 140 160 180 200

Figure: Minimum cost routes (cost = distance2) where relay nodes are placed according to a spatial Poisson process of density λ(x, y) = a · (10−4x2 + 0.05) nodes/m2, for four increasing values of a.

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Problem Model Multi-Class Costs Conclusions

The Network Model

Let us consider in the plane X1 × X2: The node density function d(x1, x2) [nodes/m2], such that the total number of nodes on a region A, is then given by N(A) =

• A

d(x1, x2)dS.

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Problem Model Multi-Class Costs Conclusions

The information density function ρ(x1, x2) [bps/m2]: If ρ(x1, x2) > 0 then there is a distributed data source. If ρ(x1, x2) < 0 then there is a distributed data sink. We assume that the data created is equal to the data absorbed, i.e.

• X1×X2

ρ(x1, x2)dS = 0.

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Problem Model Multi-Class Costs Conclusions

Figure: The function T.

The traffic flow function T(x1, x2)[bps/m], such that: Its direction coincides with the direction of the flow of information at point (x1, x2). |T(x1, x2)| is the rate with which information crosses a linear segment perpendicular to T(x1, x2) centered on (x1, x2).

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Problem Model Multi-Class Costs Conclusions

The Conservation Equation

Over a surface Φ0 ⊆ X1 × X2 of arbitrary shape, we assume that

• Φ0

ρ(x1, x2)dS =

• ∂Φ0

[T · n(s)]dS where the vector n(s) is the unit normal vector perpendicular to ∂Φ0 at a boundary point ∂Φ0(s) and pointing outwards.

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Problem Model Multi-Class Costs Conclusions

Definition (Divergence) The divergence of a continuously differentiable vector field F = Fxˆ x + Fy ˆ y is defined to be the scalar-valued function: ∇ · F = ∂Fx ∂x + ∂Fy ∂y

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Problem Model Multi-Class Costs Conclusions

An equivalent definition: Given a sequence of areas Ak, with (x0, y0) ∈ intAk, s.t. the areas |Ak| → 0 with k, ⇒ ∇ · F(x0, y0) = lim

k→+∞

1 |Ak|

• ∂Ak

F(x, y) · ndS, where n(x, y) is the unitary external normal vector at (x, y).

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Problem Model Multi-Class Costs Conclusions

From the conservation equation holding for any smooth domain, then ∇ · T(x) := ∂T1(x) ∂x1 + ∂T2(x) ∂x2 = ρ(x).

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Problem Model Multi-Class Costs Conclusions

Table of Contents

1

Statement Problem and Previous Works

2

The Network Model

3

Multi-Class Case

4

Congestion dependent/independent costs

5

Conclusions

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Problem Model Multi-Class Costs Conclusions

Multi-Class Case

For each class j ∈ J: ∇ · Tj(x) = ρj(x), ∀ x ∈ Φ, where Tj is the traffic flow function of class j. ρj is the information density function of class j.

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Problem Model Multi-Class Costs Conclusions

Considering g(x, T(x)) as a generic local cost function at point x, a multi-class optimization problem would then be: minimize Z over the flow distributions {T j

i }

Z =

• Φ

g(x, T(x))dx1dx2 subject to ∇ · Tj(x) = ρj(x), j = 1, ..., ν ∀x ∈ Φ.

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Problem Model Multi-Class Costs Conclusions

Kuhn-Tucker conditions implies for i = 1, 2: ∂g(x, T) ∂T j

i

+ ∂ζj(x) ∂xi = 0 if T j

i (x) > 0

∂g(x, T) ∂T j

i

+ ∂ζj(x) ∂xi ≥ 0 if T j

i (x) = 0.

where the ζj(x) are Lagrange multipliers.

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Problem Model Multi-Class Costs Conclusions

It follows also that necessarily ζj(x) = 0 ∀x ∈ ∂Φ where T(x) > 0. This will provide in some cases the boundary condition to recover ζj.

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Problem Model Multi-Class Costs Conclusions

Example

Let the general cost function be g(x, T(x)) =

• i=1,2

gi(x, T(x))Ti(x). Affine cost per packet: gi(x, T(x)) = 1 2ki(x)Ti(x) + hi(x). Then the Kuhn-Tucker conditions simplify to ki(x)Ti(x) + hi(x) + ∂ζ(x) ∂xi = 0 if Ti(x) > 0 ki(x)Ti(x) + hi(x) + ∂ζ(x) ∂xi ≥ 0 if Ti(x) = 0.

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Problem Model Multi-Class Costs Conclusions

Assume ki(·) > 0. Let ai := 1/ki, and b s.t. bi := hi/ki. Assume that there exists a solution where T(x) > 0 for all x. Then Ti(x) = −

• ai(x)∂ζ(x)

∂xi + bi(x)

• .

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Problem Model Multi-Class Costs Conclusions

The function ζ(·) can be found as the solution in H1

0(Φ) of the elliptic

equation (an equality in H−1(Φ))

• i

∂ ∂xi

• ai(x) ∂ζ

∂xi

• + ∇·b(x) + ρ(x) = 0 .

Well behaved Dirichlet problem, known to have a unique solution in H1

0(Φ) (J. L. Lions).

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Problem Model Multi-Class Costs Conclusions

Table of Contents

1

Statement Problem and Previous Works

2

The Network Model

3

Multi-Class Case

4

Congestion dependent/independent costs

5

Conclusions

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Problem Model Multi-Class Costs Conclusions

Congestion independent cost

(0, 0) (a, 0) (a, b) (0, b) Γ+

1

Γ−

2

Γ−

3

Γ+

4

Let Φ be a rectangle and ∂Φ = Γ1 ∪ Γ2 ∪ Γ3 ∪ Γ4. The local cost depends on the direction but is independent of the flow i.e. c(x) = (c1(x), c2(x)).

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Problem Model Multi-Class Costs Conclusions

The cost incurred by a user traveling along a path p is given by the line integral cp =

• p

c · dx. Let V j(x) be the minimum cost to go from a point x to a set Bj ⊆ Γ2 ∪ Γ3, j = 1, ..., ν V j(x) = min

• c1(x)dx1 + V j(x1 + dx1, x2), c2(x)dx2 + V j(x1, x2 + dx2)
• Thus

0 = min

• c1(x) + ∂V j(x)

∂x1 , c2(x) + ∂V j(x) ∂x2

• ∀x ∈ Bj, V j(x) = 0.

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Problem Model Multi-Class Costs Conclusions

Congestion dependent cost

Last equations still holds but ci = ci(x, Ti). We introduce a potential function ψ defined by ψ(x, T) =

• i=1,2

Ti ci(x, s)ds so that for both i = 1, 2: ci(x, Ti) = ∂ψ(x, T) ∂Ti . Then, the user equilibrium flow is the one obtained from the global

• ptimization problem where we use ψ(x, T) as local cost.

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Problem Model Multi-Class Costs Conclusions

Table of Contents

1

Statement Problem and Previous Works

2

The Network Model

3

Multi-Class Case

4

Congestion dependent/independent costs

5

Conclusions

Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 25/2

Problem Model Multi-Class Costs Conclusions

We study a setting to describe the network in terms of macroscopic parameters rather than in terms of microscopic parameters. These macroscopic quantities retain just enough information to allow meaningful and tractable problems for the routing optimization

• f the network.

We solve the routing problem for the affine cost per packet in a multi-class case. In directional antennas we found cases where user-optimizing and system-optimizing solution coincides.

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Problem Model Multi-Class Costs Conclusions

Thank you !

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