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

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 Objectives in pricing? max profit, min total delays,… Competition? How can pricing strategies take into account competion with other providers. Pricing? what? packets, transactions, bandwidth … how? flat rate, auctions, per volume …

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

5. A Simple NetworkO 1 23 D d u1u1 u6u6 u3u3 u5u5 u2u2 u4u4  = 1 1 1 od-route incident matrix  = 1 0 0 0 1 1 0 1 0 1 1 0 0 0 1 arc-route incident matrix d: demand value u i: capacity of arc i fictive arc

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

7. Simplified Bi-Level Model Madiagne.Diallo@prism.uvsq.fr 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.

8. Mathematical method 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 Min f(x) s.t.  y = d, (1)  y  u, (2) d, y  0 x =  y (3) Madiagne.Diallo@prism.uvsq.fr

9. Augmented Lagrangian 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! Solve a simple multi-flow problem: Associate to Link Prices the Lagrange Multipliers for x =  y  u. and the Lagrange Multiplyers  for constraints  y = d.

10. 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 Uniqueness of Link Prices Madiagne.Diallo@prism.uvsq.fr Apply KKT Optimality Conditions at x*.

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

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

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

14. Numerical example: Unbounded Prices Set Link #123456789111213 Link Definition(1.6)(1.5)(2.1)(3.2)(3.5)(3.7)(4.5)(5.6)(5.7)(5.2)(6.3)(7.4) Link Capacity1001 82 12 4 Link Delay Data (4,4)(6,0)(7,5)(3,0)(6,0)(9,5)(8,6)(6,0) (8,5)(7,0)(6,7) Fair Link Flow x * 39140023512505 Initial Link Prices 07400400 4000 5 37 456 21 39 40 1 1 5 5 5 5 2 2 Initial Revenue = (x * ) T = 164 Max Revenue over S = 904, * = {148, 8, 148, 8} Set of Prices is unbounded thus we maximize profit over S ={2, 4, 7, 9} Initial Revenue over S = 82

15. Numerical example: Bounded Prices Set Link #123456789 Link Definition(1,2) Link Capacity1001.51001.810.31.5100 Link Delay5x 1 +4x 1 2 9x 2 7x 3 +9x 3 2 9x 4 4x 5 x6x6 6x 7 4x 8 +5x 8 2 6x 9 +2x 9 2 Fair Link Flow x * 11.51.21.810.31.531.5 Initial Link Prices 0901.315010.300 56 2 4 31 2.5 1 1.5 1 0.3 1.5 1.8 3 1.2 3 Initial Revenue = (x * ) T = 46.3 Max Revenue = 79.54 Optimal Link Prices 013.301.341.8010.300 Set of prices is bounded, we maximize profit

16. Numerical example: Singleton Prices Set Link #123 Link Definition(1,2)(1,3)(2,3) Link Capacity10020100 Link Delay4x 1 +9x 1 2 7x 2 6x 3 +3x 3 2 Fair Link Flow x * 802080 Initial Link Prices 06720 2 31 80 20 80 100 Active Constraints = 0 1 1 This matrix is cleary Linearly Independent

17. Numerical Example 1  = od-route incident matrix t 2 2 d = 4 ar 1 ar 2 ar 3 s 2 1 0 0 1 1 0 1 M 1 = M 1 est LD arc-route incident matrix r 1 r 2 ar 1 ar 2 ar 3 1 0 0 1 1 0  = y = (y 1, y 2 )

18. Other Applications Transport Electricity Madiagne.Diallo@prism.uvsq.fr

19. Perspectives 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 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.