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Fair Allocation and Network Ressources Pricing

Fair Allocation and Network Ressources Pricing. Moustapha BOUHTOU ¤ , Madiagne DIALLO * , Laura WYNTER * France Telecom R&D ¤ - IBM Reserach Center * University of Versailles, France Madiagne.Diallo@prism.uvsq.fr. A simplified bi-level model.

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Fair Allocation and Network Ressources Pricing

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  1. Fair Allocation and Network Ressources Pricing Moustapha BOUHTOU¤, Madiagne DIALLO*, Laura WYNTER* France Telecom R&D¤ - IBM Reserach Center* University of Versailles, France Madiagne.Diallo@prism.uvsq.fr A simplified bi-level model Work sponsored by France Télécom R&D Under contract 001B852

  2. Planning • About Pricing Telecommunications • Some Pricing schemes • A Simplified Bi-Level Model • Numerical Examples Madiagne.Diallo@prism.uvsq.fr

  3. About Pricing Telecoms Economic vs OR approaches? analytical methods when number of variables is small vs. numerical methods for (large-scale) networks Pricing? what? packets, transactions, bandwidth … how? flat rate, auctions, per volume … Objectives in pricing? max profit, min total delays,… Competition? How can pricing strategies take into account competion with other providers.

  4. Some pricing schemes • Pricing independent of users willingness • Flat pricing • Paris metro pricing • … Pricing taking into account users willingness • Priority pricing • Smart market pricing • Proportional Fairness pricing • …Madiagne.Diallo@prism.uvsq.fr

  5. OBJECTIVES • Satisfy user demand and simultaneously obtain a fair flow, or a flow in user equilibrium. • Avoid congestion • Maximize operator´s profit Madiagne.Diallo@prism.uvsq.fr

  6. Simplified Bi-Level Model Maximize user satisfaction AND simultaneously Maximize operator´s profit May take into account other objectives such that maximizing profit on a set of links or routes. Madiagne.Diallo@prism.uvsq.fr

  7. Mathematical method Min f(x) s.t. y = d, (1)  y  u, (2) d, y 0 x= y (3) Consider a canonical problem  = od-route incident matrix, (od = origin-destination) d = demand , y = flow on route,  = arc-route incident matrix u = capacity, x = total arc flow, x* = optimal arc flow Madiagne.Diallo@prism.uvsq.fr

  8. Augmented Lagrangian Solve a simple multi-flow problem: Associate to LinkPrices the Lagrange Multipliers for x =  y u. and the Lagrange Multiplyers  for constraintsy = d . At the optimum we get a unique link flow x* (for f strictly convex) and a price vector (x* ) for this optimal flow. However, the prices x*are not always unique!

  9. Uniqueness of Link Prices Apply KKT Optimality Conditions at x*. If the gradients y( y) of the active inequality constraints ( y  u) and the gradients y(y) of the equality contraints (y = d) are Linearly Independent Then The Lagrange multipliers  andfor these constraints are unique Madiagne.Diallo@prism.uvsq.fr

  10. Application of KKT Solve the relaxation minxf(x*)Tx + T (x - u) s.t. y = d, d, y, x 0 With f(x*) unknown we obtain the dual maxdT  s.t. [f(x*) +  ] -  0 Madiagne.Diallo@prism.uvsq.fr

  11. Link Price Polyhedron(Larsson and Patriksson 1998) T(,) = [f(x*) +  ] - T   0 (weak duality) [f(x*) +  ]T x* - dT = 0 (strong duality) T(x* - u)= 0  0 Madiagne.Diallo@prism.uvsq.fr

  12. Profit Maximization(Bouhtou, Diallo and Wynter 2003) When  is not unique maximize profit with: Max < , x*> s.t.  (T  P) Where P may be a set of bounds on feasible prices. Madiagne.Diallo@prism.uvsq.fr

  13. 40 1 2 5 39 40 5 1 5 6 5 4 1 2 5 5 2 3 7 Numerical example: Unbounded Prices Set Initial Revenue = (x*)T = 164 Set of Prices is unbounded thus we maximize profit over S={2, 4, 7, 9} Max Revenue over S = 904, * = {148, 8, 148, 8} Initial Revenue over S = 82

  14. 1.5 5 6 3 1 2.5 1.5 1 1.8 1 2 3 3 1.2 1.5 0.3 4 Numerical example: Bounded Prices Set Initial Revenue = (x*)T = 46.3 Set of prices is bounded, we maximize profit Max Revenue = 79.54

  15. 100 1 3 20 Active Constraints = 0 1 1 1 80 80 2 This matrix is cleary Linearly Independent Numerical example: Singleton Prices Set

  16. Other Applications • Transport • Electricity Madiagne.Diallo@prism.uvsq.fr

  17. Perspectives • Avoid T to be singleton or Correct it • by developing a characterization of the telecommunications networks that • exhibit sufficiently large Lagrange multiplier sets so as to permit • considerable revenue maximization. • Optimize over other objectives by studying more general bi-level programming model, freeing the prices of the complementary constraints that define them to be Lagrange multipliers. • Test whether this two-step procedure may come quit close to the true bi-level optimization problem Madiagne.Diallo@prism.uvsq.fr

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