Fair Allocation and Network Ressources Pricing

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Fair Allocation and Network Ressources Pricing

• A simplified bi-level model

Moustapha BOUHTOU, Madiagne DIALLO, Laura
WYNTER France Telecom RD - IBM Reserach
Center University of Versailles, France
Madiagne.Diallo_at_prism.uvsq.fr
Work sponsored by France Télécom RD Under
contract 001B852
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Planning

• About Pricing Telecommunications
• Some Pricing schemes
• A Simplified Bi-Level Model
• Numerical Examples
• Madiagne.Diallo_at_prism.uvsq.fr

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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.
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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_at_prism.uvsq.fr

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A Simple Network
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OBJECTIVES

• Satisfy user demand and simultaneously obtain a
  fair flow, or a flow in user equilibrium.
• Avoid congestion
• Maximize operators profit
• 
  

Madiagne.Diallo_at_prism.uvsq.fr
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Simplified Bi-Level Model
Maximize user satisfaction AND
simultaneously Maximize operators profit


May take into account other objectives such that
maximizing profit on a set of links or routes.

Madiagne.Diallo_at_prism.uvsq.fr
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Mathematical method

• Consider a canonical problem

Min f(x) s.t. ?y d,
(1) ? y ? u, (2) d, y ?
0 x ?y (3)
? 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_at_prism.uvsq.fr
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Augmented Lagrangian
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 .
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!
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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

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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_at_prism.uvsq.fr
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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_at_prism.uvsq.fr

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Profit Maximization(Bouhtou, Diallo and Wynter
2003)

• When ? is not unique maximize profit with
• Max lt? , xgt
• s.t.
• ? ? (T ? P)
• Where P may be a set of bounds on feasible
  prices.
• 
  
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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
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Numerical example Bounded Prices Set
Initial Revenue (x)T? 46.3
Set of prices is bounded, we maximize profit
Max Revenue 79.54
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Numerical example Singleton Prices Set
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Numerical Example

M1 est LD
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Other Applications

• Transport
• Electricity

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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

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