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 Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 1/26
Problem Model Multi-Class Costs Conclusions Table of Contents Statement Problem and Previous Works 1 The Network Model 2 Multi-Class Case 3 Congestion dependent/independent costs 4 Conclusions 5 Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 2/26
Problem Model Multi-Class Costs Conclusions Table of Contents Statement Problem and Previous Works 1 The Network Model 2 Multi-Class Case 3 Congestion dependent/independent costs 4 Conclusions 5 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. Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 3/26
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). Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 4/26
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 only 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 Statement Problem and Previous Works 1 The Network Model 2 Multi-Class Case 3 Congestion dependent/independent costs 4 Conclusions 5 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. Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 6/26
Problem Model Multi-Class Costs Conclusions 200 200 200 200 180 180 180 180 160 160 160 160 140 140 140 140 120 120 120 120 100 100 100 100 80 80 80 80 60 60 60 60 40 40 40 40 20 20 20 20 0 0 0 0 0 50 100 0 50 100 0 50 100 0 50 100 Figure: Minimum cost routes ( cost = distance 2 ) where relay nodes are placed according to a spatial Poisson process of density λ ( x , y ) = a · (10 − 4 x 2 + 0 . 05) nodes / m 2 , for four increasing values of a . Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 7/26
Problem Model Multi-Class Costs Conclusions The Network Model Let us consider in the plane X 1 × X 2 : The node density function d ( x 1 , x 2 ) [ nodes / m 2 ], such that the total number of nodes on a region A , is then given by � N ( A ) = d ( x 1 , x 2 ) d S . A Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 8/26
Problem Model Multi-Class Costs Conclusions The information density function ρ ( x 1 , x 2 ) [ bps / m 2 ]: If ρ ( x 1 , x 2 ) > 0 then there is a distributed data source . If ρ ( x 1 , x 2 ) < 0 then there is a distributed data sink . We assume that the data created is equal to the data absorbed, i.e. � ρ ( x 1 , x 2 ) d S = 0 . X 1 × X 2 Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 9/26
Problem Model Multi-Class Costs Conclusions Figure: The function T . The traffic flow function T ( x 1 , x 2 )[ bps / m ], such that: Its direction coincides with the direction of the flow of information at point ( x 1 , x 2 ). | T ( x 1 , x 2 ) | is the rate with which information crosses a linear segment perpendicular to T ( x 1 , x 2 ) centered on ( x 1 , x 2 ). Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 10/2
Problem Model Multi-Class Costs Conclusions The Conservation Equation Over a surface Φ 0 ⊆ X 1 × X 2 of arbitrary shape, we assume that � � ρ ( x 1 , x 2 ) dS = [ T · n ( s )] dS Φ 0 ∂ Φ 0 where the vector n ( s ) is the unit normal vector perpendicular to ∂ Φ 0 at a boundary point ∂ Φ 0 ( s ) and pointing outwards. Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 11/2
Problem Model Multi-Class Costs Conclusions Definition (Divergence) The divergence of a continuously differentiable vector field F = F x ˆ x + F y ˆ y is defined to be the scalar-valued function: ∇ · F = ∂ F x ∂ x + ∂ F y ∂ y Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 12/2
Problem Model Multi-Class Costs Conclusions An equivalent definition: Given a sequence of areas A k , with ( x 0 , y 0 ) ∈ i ntA k , s.t. the areas | A k | → 0 with k , 1 � ⇒ ∇ · F ( x 0 , y 0 ) = lim F ( x , y ) · n dS , | A k | k → + ∞ ∂ A k where n ( x , y ) is the unitary external normal vector at ( x , y ). Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 13/2
Problem Model Multi-Class Costs Conclusions From the conservation equation holding for any smooth domain, then ∇ · T ( x ) := ∂ T 1 ( x ) + ∂ T 2 ( x ) = ρ ( x ) . ∂ x 1 ∂ x 2 Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 14/2
Problem Model Multi-Class Costs Conclusions Table of Contents Statement Problem and Previous Works 1 The Network Model 2 Multi-Class Case 3 Congestion dependent/independent costs 4 Conclusions 5 Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 15/2
Problem Model Multi-Class Costs Conclusions Multi-Class Case For each class j ∈ J : ∇ · T j ( x ) = ρ j ( x ) , ∀ x ∈ Φ , where T j is the traffic flow function of class j . ρ j is the information density function of class j . Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 15/2
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 = g ( x , T ( x )) d x 1 d x 2 subject to Φ ∇ · T j ( x ) = ρ j ( x ) , j = 1 , ..., ν ∀ x ∈ Φ . Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 16/2
Problem Model Multi-Class Costs Conclusions Kuhn-Tucker conditions implies for i = 1 , 2: + ∂ζ j ( x ) ∂ g ( x , T ) T j = 0 if i ( x ) > 0 ∂ T j ∂ x i i + ∂ζ j ( x ) ∂ g ( x , T ) T j ≥ 0 if i ( x ) = 0 . ∂ T j ∂ x i i where the ζ j ( x ) are Lagrange multipliers. Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 17/2
Problem Model Multi-Class Costs Conclusions It follows also that necessarily ζ j ( x ) = 0 ∀ x ∈ ∂ Φ where T ( x ) > 0 . This will provide in some cases the boundary condition to recover ζ j . Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 18/2
Problem Model Multi-Class Costs Conclusions Example Let the general cost function be � g ( x , T ( x )) = g i ( x , T ( x )) T i ( x ) . i =1 , 2 Affine cost per packet: g i ( x , T ( x )) = 1 2 k i ( x ) T i ( x ) + h i ( x ) . Then the Kuhn-Tucker conditions simplify to k i ( x ) T i ( x ) + h i ( x ) + ∂ζ ( x ) = 0 if T i ( x ) > 0 ∂ x i k i ( x ) T i ( x ) + h i ( x ) + ∂ζ ( x ) ≥ 0 if T i ( x ) = 0 . ∂ x i Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 19/2
Problem Model Multi-Class Costs Conclusions Assume k i ( · ) > 0. Let a i := 1 / k i , and b s.t. b i := h i / k i . Assume that there exists a solution where T ( x ) > 0 for all x . Then � � a i ( x ) ∂ζ ( x ) T i ( x ) = − + b i ( x ) . ∂ x i Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 20/2
Problem Model Multi-Class Costs Conclusions The function ζ ( · ) can be found as the solution in H 1 0 (Φ) of the elliptic equation (an equality in H − 1 (Φ)) ∂ � a i ( x ) ∂ζ � � + ∇· b ( x ) + ρ ( x ) = 0 . ∂ x i ∂ x i i Well behaved Dirichlet problem, known to have a unique solution in H 1 0 (Φ) (J. L. Lions). Altman, Silva, Bernhard, Debbah Continuum Equilibria for Routing in Dense Ad-hoc Networks 21/2
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