fair allocation and network ressources pricing fair allocation and network ressources pricing a...
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Fair Allocation and Network Ressources Fair Allocation and Network Ressources PricingPricing
A simplified bi-level model
Work sponsored by France Télécom R&D Under contract 001B852
Moustapha BOUHTOUBOUHTOU¤, Madiagne DIALLODIALLO*, Laura WYNTERWYNTER*
France Telecom R&DFrance Telecom R&D¤ - IBM Reserach CenterIBM Reserach Center**
University of Versailles, France [email protected]
Planning
• About Pricing Telecommunications
• Some Pricing schemes • A Simplified Bi-Level Model
• Numerical Examples
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 …
Some pricing schemes
Pricing taking into account users willingness• Priority pricing• Smart market pricing• Proportional Fairness pricing• …
Pricing independent of users willingness • Flat pricing• Paris metro pricing• …
OBJECTIVES
• Satisfy user demand and simultaneously obtain a fair flow, or a flow in user equilibrium.
• Avoid congestion
• Maximize operator´s profit
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.
Mathematical methodConsider a canonical problem
= od-route incident matrix, (od = origin-destination)d = demand , y = flow on route, = arc-route incident matrixu = 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)
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 .
Ifthe gradients y( y) of the active inequality constraints ( y u) and
the gradients y(y) of the equality contraints (y = d) are Linearly Independent
ThenThe Lagrange multipliers and for these constraints are unique
Uniqueness of Link Prices
Apply KKT Optimality Conditions at x*.
Application of KKTSolve 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
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
When is not unique maximize profit with:Max < , x*>
s.t. (T P)
Where P may be a set of bounds on feasible prices.
Profit Maximization(Bouhtou, Diallo and Wynter 2003)
Numerical example: Unbounded Prices Set
Link # 1 2 3 4 5 6 7 8 9 11 12 13
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 Capacity 100 1 100 8 2 100 100 1 2 100 4 100
Link Delay Data (4,4) (6,0) (7,5) (3,0) (6,0) (9,5) (8,6) (6,0) (6,0) (8,5) (7,0) (6,7)
Fair Link Flow x* 39 1 40 0 2 3 5 1 2 5 0 5
Initial Link Prices 0 74 0 0 4 0 0 74 4 0 0 0
5
3 7
456
2139
40
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
Numerical example: Bounded Prices Set
Link # 1 2 3 4 5 6 7 8 9
Link Definition (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2) (1,2)
Link Capacity 100 1.5 100 1.8 1 0.3 1.5 100 100
Link Delay 5x1+4x12 9x2 7x3+9x3
2 9x4 4x5 x6 6x7 4x8+5x82 6x9+2x9
2
Fair Link Flow Fair Link Flow xx** 11 1.51.5 1.21.2 1.81.8 11 0.30.3 1.51.5 33 1.51.5
Initial Link Prices 0 9 0 1.3 15 0 10.3 0 0
5 6
2
4
31
2.5
1
1.5
1
1.5 0.3
1.5
1.8
3
1.2 3
Initial Revenue = (x*)T = 46.3
Max Revenue = 79.54
Optimal Link Prices 0 13.3 0 1.3 41.8 0 10.3 0 0
Set of prices is bounded, we maximize profit
Numerical example: Singleton Prices Set
Link # 1 2 3
Link Definition (1,2) (1,3) (2,3)
Link Capacity 100 20 100
Link Delay 4x1+9x12 7x2 6x3+3x3
2
Fair Link Flow Fair Link Flow xx** 8080 2020 8080
Initial Link Prices 0 672 0
2
31
80
20
80
100
Active Constraints = 0 11 1
This matrix is cleary Linearly Independent
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 [email protected]
• 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.