Routing in massively dense ad-hoc networks: Continum equilibrium

Transcription

1 Routing in massively dense ad-hoc networks: Continum equilibrium Pierre Bernhard, Eitan Altmann, and Alonso Silva I3S, University of Nice-Sophia Antipolis and CNRS INRIA Méditerranée POPEYE meeting, May

2 The problem Let Ω be a connected smooth open set in R 2, with Ω = R S T. Messages originate in S, with density σ(x), and have to be routed through Ω to R, with zero flux through T. At each point x Ω, messages flow in only one direction, but with a density ϕ of messages (intensity of the flow) that may vary from point to point. The intensity ϕ of messages flowing through a point x has a cost c(x, ϕ) (Delays, energy use,... ) Investigate both the collectively optimal routing and the Wardrop equilibria. 2

3 Data T S, σ(x) Ω c(x, ϕ) R T 3

4 The problem Let Ω be a connected smooth open set in R 2, with Ω = R S T. Messages originate in S, with density σ(x), and have to be routed through Ω to R, with zero flux through T. At each point x Ω, messages flow in only one direction, but with a density ϕ of messages (intensity of the flow) that may vary from point to point. The intensity ϕ of messages flowing through a point x has a cost per message c(x, ϕ) (delays, energy use,... ) hence a cumulated cost at x C(x, ϕ) = c(x, ϕ)ϕ. Investigate both the collectively optimal routing and the Wardrop equilibria. 4

5 The problem Let Ω be a connected smooth open set in R 2, with Ω = R S T. Messages originate in S, with density σ(x), and have to be routed through Ω to R, with zero flux through T. At each point x Ω, messages flow in only one direction, but with a density ϕ of messages (intensity of the flow) that may vary from point to point. The intensity ϕ of messages flowing through a point x has a cost per message c(x, ϕ) (delays, energy use,... ) hence a cumulated cost at x C(x, ϕ) = c(x, ϕ)ϕ. Investigate both the collectively optimal routing and the Wardrop equilibria. 5

6 The problem Let Ω be a connected smooth open set in R 2, with Ω = R S T. Messages originate in S, with density σ(x), and have to be routed through Ω to R, with zero flux through T. At each point x Ω, messages flow in only one direction, but with a density ϕ of messages (intensity of the flow) that may vary from point to point. The intensity ϕ of messages flowing through a point x has a cost per message c(x, ϕ) (delays, energy use,... ) hence a cumulated cost at x C(x, ϕ) = c(x, ϕ)ϕ. Investigate both the collectively optimal routing and the Wardrop equilibria. 6

7 The problem Let Ω be a connected smooth open set in R 2, with Ω = R S T. Messages originate in S, with density σ(x), and have to be routed through Ω to R, with zero flux through T. At each point x Ω, messages flow in only one direction, but with a density ϕ of messages (intensity of the flow) that may vary from point to point. The intensity ϕ of messages flowing through a point x has a cost per message c(x, ϕ) (delays, energy use,... ) hence a cumulated cost at x C(x, ϕ) = c(x, ϕ)ϕ. Investigate both the collectively optimal routing and the Wardrop equilibria. 7

8 The problem Let Ω be a connected smooth open set in R 2, with Ω = R S T. Messages originate in S, with density σ(x), and have to be routed through Ω to R, with zero flux through T. At each point x Ω, messages flow in only one direction, but with a density ϕ of messages (intensity of the flow) that may vary from point to point. The intensity ϕ of messages flowing through a point x has a cost per message c(x, ϕ) (delays, energy use,... ) hence a cumulated cost at x C(x, ϕ) = c(x, ϕ)ϕ. Investigate both the collectively optimal routing and the Wardrop equilibria. 8

9 Mathematical model : message flow Flow of messages represented as a vector field f(x) with f(x) = ϕ(x). Extend σ(x) = 0 for x T, and let Q = S T, and n(x) be the normal to Ω. The boundary conditions are thus x Q, n(x), f(x) = σ(x). No source nor sink of messages in Ω Will automatically imply S σ(x) ds + x Ω, divf(x) = 0. R n(x), f(x) ds = 0. 9

10 Mathematical model : message flow Flow of messages represented as a vector field f(x) with f(x) = ϕ(x). Extend σ(x) = 0 for x T, and let Q = S T, and n(x) be the normal to Ω. The boundary conditions are thus x Q, n(x), f(x) = σ(x). No source nor sink of messages in Ω Will automatically imply S σ(x) ds + x Ω, divf(x) = 0. R n(x), f(x) ds = 0. 10

11 Mathematical model : message flow Flow of messages represented as a vector field f(x) with f(x) = ϕ(x). Extend σ(x) = 0 for x T, and let Q = S T, and n(x) be the normal to Ω. The boundary conditions are thus x Q, n(x), f(x) = σ(x). No source nor sink of messages in Ω Will automatically imply S σ(x) ds + x Ω, divf(x) = 0. R n(x), f(x) ds = 0. 11

12 Mathematical model : message flow Flow of messages represented as a vector field f(x) with f(x) = ϕ(x). Extend σ(x) = 0 for x T, and let Q = S T, and n(x) be the normal to Ω. The boundary conditions are thus x Q, n(x), f(x) = σ(x). No source nor sink of messages in Ω Will automatically imply S σ(x) ds + x Ω, divf(x) = 0. R n(x), f(x) ds = 0. 12

13 Mathematical model : costs Let e θ = (cos θ, sin θ) be the direction of travel of a message. Total cost incurred in a path from x(t 0 ) = x 0 R to x(t 1 ) = x 1 S is J(e θ ( )) = x1 x 0 c ( x, f(x) ) ds = t1 t 0 c ( x(t), f(x(t)) ) dt. Let C(x, ϕ) := c(x, ϕ)ϕ Total (collective) cost of congestion is G(f( )) = Ω c( x, f(x) ) f(x) dx. 13

14 Mathematical model : costs Let e θ = (cos θ, sin θ) be the direction of travel of a message. Total cost incurred in a path from x(t 0 ) = x 0 R to x(t 1 ) = x 1 S is J(e θ ( )) = x1 x 0 c ( x, f(x) ) ds = t1 t 0 c ( x(t), f(x(t)) ) dt. Let C(x, ϕ) := c(x, ϕ)ϕ Total (collective) cost of congestion is G(f( )) = Ω c( x, f(x) ) f(x) dx = Ω C( x, ϕ(x) ) dx. 14

15 Global optimum : necessary conditions Dualize the constraint divf = 0 with a dual variable p(x), use Green s formula to rewrite lagrangian L. Fréchet derivative D f L(f, p) = 0 yields x : f (x) 0, D 2 C(x, f 1 (x) ) f (x) f (x) = p(x). If D 2 C(x, ϕ)/ϕ un bounded as ϕ 0, then at ϕ = 0, 0 ϕ L yields x : f (x) = 0, D 2 C(x, 0) p(x). 15

16 Equivalent formulation Solving the necessary conditions can be stated as : find two scalar functions p( ) and ϕ( ) (in carefully chosen function spaces), such that x Ω, p(x) D 2 C(x, ϕ(x)), x : ϕ(x) 0, p(x) = D 2 C(x, ϕ(x)), x R, p(x) = 0, and with f (x) := ϕ(x) D 2 C(x, ϕ(x)) p(x), x Ω, divf (x) = 0, x Q, n(x), f (x) = σ(x). 16

17 Wardrop equilibrium The Hamilton-Jacobi-Carathéodory-Bellman equation of the problem is x Ω, min θ e θ, V (x) + c(x, f (x) ) = 0, x R, V (x) = 0. and the optial direction of motion is opposite to V. Wherever f (x) = 0, this coincides with previous equations upon replacing p by V and D 2 C by c. Hence one can look for the Wardrop equilibrium by solving the global optimization problem with C(x, ϕ) = ϕ 0 c(x, ψ)dψ. Generalizes Beckman, 1952, Beckman et al Theorem If c(x, ϕ) = c(x)ϕ α, α > 0, any global equilibrium where f (x) = 0 over Ω is a Wardrop equilibrium. 17

18 Wardrop equilibrium The Hamilton-Jacobi-Carathéodory-Bellman equation of the problem is x Ω, V (x) + c(x, f (x) ) = 0, x R, V (x) = 0. and the optial direction of motion is opposite to V. Wherever f (x) = 0, this coincides with previous equations upon replacing p by V and D 2 C by c. Hence one can look for the Wardrop equilibrium by solving the global optimization problem with C(x, ϕ) = ϕ 0 c(x, ψ)dψ. Generalizes Beckman, 1952, Beckman et al Theorem If c(x, ϕ) = c(x)ϕ α, α > 0, any global equilibrium where f (x) = 0 over Ω is a Wardrop equilibrium. 18

19 Wardrop equilibrium The Hamilton-Jacobi-Carathéodory-Bellman equation of the problem is x Ω, V (x) + c(x, f (x) ) = 0, x R, V (x) = 0. and the optial direction of motion is opposite to V. Wherever f (x) = 0, this coincides with previous equations upon replacing p by V and D 2 C by c. Hence one can look for the Wardrop equilibrium by solving the global optimization problem with C(x, ϕ) = ϕ 0 c(x, ψ)dψ. Generalizes Beckman, 1952, Beckman et al, Theorem If c(x, ϕ) = c(x)ϕ α, α > 0, any global equilibrium where f (x) = 0 over Ω is a Wardrop equilibrium. 19

20 Wardrop equilibrium The Hamilton-Jacobi-Carathéodory-Bellman equation of the problem is x Ω, V (x) + c(x, f (x) ) = 0, x R, V (x) = 0. and the optial direction of motion is opposite to V. Wherever f (x) = 0, this coincides with previous equations upon replacing p by V and D 2 C by c. Hence one can look for the Wardrop equilibrium by solving the global optimization problem with C(x, ϕ) = ϕ 0 c(x, ψ)dψ. Generalizes Beckman, 1952, Beckman et al, 1956 Theorem If c(x, ϕ) = c(x)ϕ α, α > 0, any global equilibrium where f (x) = 0 over Ω is a Wardrop equilibrium. 20

21 Linear congestion costs If c(x, ϕ) = c(x)ϕ, the equations for p uncouples from ϕ and becomes a standard mixed Neuman-Dirichlet elliptic PDE: ( ) 1 x Ω, div c(x) p(x) = 0, p x Q, (x) = c(x)σ(x), n x R, p(x) = 0. (Existence and) uniqueness. Solution e.g. via standard finite element method. f (x) = 1 c(x) p(x). 21

22 No congestion, minimize delays We investigate the case where the cost incurred are only linked to the location, but the network is uncongested. Then c(x, ϕ) = c(x). Now D 2 C(x, ϕ) = c(x) and D 2 C/ϕ as ϕ 0. Equivalent formulation: Find two scalar functions p( ) and ψ( ) such that x Ω, ψ(x) 0, p(x) c(x), ψ(x)[ p(x) c(x)] = 0, x Ω, ψ(x) p(x) + ψ(x), p(x) = 0, x Q, ψ(x) n(x), p(x) = σ(x). Then, f (x) = ψ(x) p(x). We propose an algorithm à la Uzawa, proved convergent if f (x) 0 over Ω. Proves uniqueness in that case. 22

23 No congestion, minimize delays We investigate the case where the cost incurred are only linked to the location, but the network is uncongested. Then c(x, ϕ) = c(x). Now D 2 C(x, ϕ) = c(x) and D 2 C/ϕ as ϕ 0. Equivalent formulation: Find two scalar functions p( ) and ψ( ) such that x Ω, ψ(x) 0, p(x) c(x), ψ(x)[ p(x) c(x)] = 0, x Ω, ψ(x) p(x) + ψ(x), p(x) = 0, x Q, ψ(x) n(x), p(x) = σ(x). Then, f (x) = ψ(x) p(x). We propose an algorithm proved convergent if f (x) 0 x Ω. Proves uniqueness in that case. No satisfactory result for the general case. 23

24 No congestion, minimize delays We investigate the case where the cost incurred are only linked to the location, but the network is uncongested. Then c(x, ϕ) = c(x). Now D 2 C(x, ϕ) = c(x) and D 2 C/ϕ as ϕ 0. Equivalent formulation: Find two scalar functions p( ) and ψ( ) such that x Ω, ψ(x) 0, p(x) c(x), ψ(x)[ p(x) c(x)] = 0, x Ω, ψ(x) p(x) + ψ(x), p(x) = 0, x Q, ψ(x) n(x), p(x) = σ(x). Then, f (x) = ψ(x) p(x). We propose an algorithm proved convergent if f (x) 0 x Ω. Proves uniqueness in that case. No satisfactory result for the general case. 24

25 Bibliography Martin Beckman : A continuous model for transportation, Econometrica, 20, pp , M. Beckman, C.B. McGuire and C.B. Winsten : Studies in the Economics of Transportation, Yale University Press,