This cost can be obtained from the variational

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Unformatted text preview: op Equilibrium Existence, Uniqueness Application and Conclusion 2 3 4 Vijay Kamble (IIT Kharagpur, INDIA) Hierarchical Routing Games MURI meeting, Sept 9 2010 13 / 21 Existence, Uniqueness Properties of Multilevel Wardrop equilibria A surprising result for the cost at W.E. for the low priority flow: Proposition For a parallel network, the routing cost per packet for Low priority flow at Wardrop equilibrium β (p) is independent of the wardrop equilibrium high priority flow profile and only depends on the total flow at a particular p. This cost can be obtained from the variational inequalities which correspond to the Wardrop equilibrium of the total flow profile if the entire flow was routed as low priority: xl∗ (Tl (xl∗ ) − β (p)) = 0, l ∈ E Tl (xl∗ ) K ￿ l =1 (1) (2) (3) − β (p) ≥ 0, l ∈ E xl∗ = ¯(p) = r (γ p + (1 − p)) r ¯ where x ∗ is the total flow profile at Wardrop equilibrium. The total cost is then ¯ given by β (p) = β (p) + C L . Vijay Kamble (IIT Kharagpur, INDIA) Hierarchical Routing Games MURI meeting, Sept 9 2010 14 / 21 Existence, Uniqueness Properties of Multilevel Wardrop equilibria If γ = 1 then the Cost at W.E. for the low priority flow does not change with p! The result does not hold for general topologies. Further Proposition The total cost per unit High priority flow and Low priority flow at Wardrop ¯ equilibrium i.e. α(p) = α(p) + C L and β (p) = β (p) + C L respectively are ¯ convex, continuous and monotone decreasing in p, the proportion of the low priority flow. Further β (p) = α(p(1 − γ ))) for p ∈ [0, 1]. Vijay Kamble (IIT Kharagpur, INDIA) Hierarchical Routing Games MURI meeting, Sept 9 2010 15 / 21 Existence, Uniqueness Figure A tentative depiction: S.W.E. 0 Vijay Kamble (IIT...
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