# 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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