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Demand Based Rate Allocation. Arpita Ghosh and James Mammen {arpitag, [email protected] EE 384Y Project 4 th June, 2003. Outline. Motivation Previous work Insight into proportional fairness Demand-based max-min fairness Decentralized algorithm

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Demand based rate allocation

Demand Based Rate Allocation

Arpita Ghosh and James Mammen

{arpitag, [email protected]

EE 384Y Project

4th June, 2003



  • 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



  • 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 ipays an amount per unit time

    • Allocate rates “fairly”, based on

    • Decentralized solution


Fairness for a single link
Fairness for a single link

  • users, single link with capacity

  • User ipays to the link

  • Weighted fair allocation of rates:

  • Decentralized solution:

    • Price of the resource,

    • Each user’s rate = payment/price

  • What is fair for a network?


Previous work
Previous Work

Proportional fairness[Kelly, Maulloo & Tan, ’98]

  • A feasible rate vector xis 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


Two ways to allocate fairly




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.


What proportional fairness means
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


Payment based max min fairness
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 xis weighted max-min fair if no rate can be increased without decreasing some rate s.t.


Weighted max min fairness interpretations
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.


Decentralized approach




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


A decentralized algorithm
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


Continuous approximation to a
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



Lyapunov function
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



Fairness of decentralized algorithm
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



  • 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