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Demand Based Rate Allocation

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Demand Based Rate Allocation

Arpita Ghosh and James Mammen

{arpitag, jmammen}@stanford.edu

EE 384Y Project

4th June, 2003

EE384Y

- 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

EE384Y

- 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

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- 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?

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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

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B

C

A

- 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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- 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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- 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.

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- 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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B

C

A

- 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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- 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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- 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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- 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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- 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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- 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

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