Demand Based Rate Allocation

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Demand Based Rate Allocation

• Arpita Ghosh and James Mammen
• arpitag, jmammen_at_stanford.edu

EE 384Y Project 4th June, 2003
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Outline

• Motivation
• Previous work
• Insight into proportional fairness
• Demand-based max-min fairness
• Decentralized algorithm
• Global stability using a Lyapunov function
• Fairness of the fixed point
• Conclusion

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Motivation

• Current scenario
• Rate allocation in the current internet is
  determined by congestion control algorithms
• Achieved rate is not a function of demand
• Our problem
• Network with N users, L links, with capacities
• Fixed route for each user, specified by routing
  matrix A
• User i pays an amount per unit time
• Allocate rates fairly, based on
• Decentralized solution

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Fairness for a single link

• users, single link with capacity
• User i pays to the link
• Weighted fair allocation of rates
• Decentralized solution
• Price of the resource,
• Each users rate payment/price
• What is fair for a network?

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Previous Work

• Proportional fairness Kelly, Maulloo Tan, 98
• A feasible rate vector x is proportionally fair
  if for every other feasible rate vector y
• Proposed decentralized algorithm, proved
  properties
• Generalized notions of fairness Mo Walrand,
  2000
• -proportional fairness A feasible rate
  vector x is
• fair if for any other feasible rate vector y
• Special cases proportional
  fairness
• 
  max-min fairness

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Two Ways to Allocate Fairly

• Method 1 User i splits its payment over
  the links it uses, so as to maximize the minimum
  proportional allocation on each link.

• Method 2 Each link allocates proportionally
  fair rates to users based on their total payment
  to the network the rate of user i is the minimum
  of these rates.

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What proportional fairness means

• We show that allocating rates according to Method
  1 leads to a proportionally fair solution for the
  case of two users and any network
• We conjecture it to be true for N users based on
  observation from several examples
• This gives insight into proportional fairness
• Total payment split across links so as to
  maximize rate
• Number of links used matter

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Payment-based max-min fairness

• Max-min fairness
• A feasible rate vector x is max-min fair if no
  rate can be increased without decreasing some
  s.t.
• This definition of fairness does not take into
  account the payments made by users
• We introduce a new notion of fairness
• Weighted max-min fairness
• A feasible rate vector x is weighted max-min
  fair if no rate can be increased without
  decreasing some rate s.t.

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Weighted max-min fairness Interpretations

• Rate allocation x is weighted max-min fair if
  rate for a user cannot be increased without
  decreasing the rate for some other user who is
  already paying as much or more per unit rate
• Weighted max-min fairness is max-min fairness
  with rates replaced by rate per unit payment
• Assuming to be integers, weighted max-min
  fairness can be thought of as max-min fairness
  with flows for user i.

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Decentralized Approach

• How decentralized algorithms work
• Each link sets its price based on total traffic
  through it
• User i adjusts based on the prices through
  its links
• Price is an increasing function of traffic
  through link, to maximize utilization while
  preventing loss or congestion
• Consider the following example
• Rate of user i depends on minimum allocated rate,
  equivalently, on the highest priced link on its
  path

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A Decentralized Algorithm

• Consider the following decentralized
  algorithm(A)
• User i adjusts based on the highest price on
  its path
• Link j sets price based on total traffic
• We want to show that this algorithm converges to
  the weighted max-min fair solution

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Continuous approximation to (A)

• Outline of proof
• Series of continuous approximations to discrete
  (A)
• Construct a Lyapunov function to show global
  stability
• Show that unique fixed point is weighted max-min
  fair
• Differential equation corresponding to (A)
• 
  
• Approximation to max function as
  gives
• 
  (C)

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Lyapunov function

• We show that L(x) is a Lyapunov function for
  (C)
• This means all trajectories of diff. eqn (C) will
  converge to the unique maximum of L(x)
• By appropriately choosing prices , the
  maximizing x for L(x) is the solution to
• (P)

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Fairness of Decentralized Algorithm

• Finally we show that solution of (P) approaches
  the weighted max-min fair solution as
• Thus the decentralized algorithm converges to the
  weighted max-min fair solution
• Simulation results with a network of buffers also
  show that discrete time algorithm (A) converges
  to weighted max-min fair rate allocation

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Conclusions

• We provided insight into proportional fairness
• We introduced the notion of weighted max-min
  fairness
• We proposed a decentralized algorithm for
  weighted max-min fairness, and proved its global
  stability and convergence to the desired solution