Demand Based Rate Allocation

1. Demand Based Rate Allocation Arpita Ghosh and James Mammen {arpitag, jmammen}@stanford.edu EE 384Y Project 4th June, 2003 EE384Y

2. 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 EE384Y

3. 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 ipays an amount per unit time • Allocate rates “fairly”, based on • Decentralized solution EE384Y

4. 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? EE384Y

5. 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 EE384Y

6. B C A 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. EE384Y

7. 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 EE384Y

8. 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. EE384Y

9. 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. EE384Y

10. B C A 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 EE384Y

11. 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 EE384Y

12. 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) EE384Y

13. 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): EE384Y

14. 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 EE384Y

15. 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 EE384Y