June 4, 2003EE384Y1 Demand Based Rate Allocation Arpita Ghosh and James Mammen {arpitag, EE 384Y Project 4 th June, 2003

June 4, 2003EE384Y2 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

June 4, 2003EE384Y3 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

June 4, 2003EE384Y4 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 user’s rate = payment/price What is fair for a network?

June 4, 2003EE384Y5 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

June 4, 2003EE384Y6 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. A BC

June 4, 2003EE384Y7 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

June 4, 2003EE384Y8 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.

June 4, 2003EE384Y9 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.

June 4, 2003EE384Y10 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 BC

June 4, 2003EE384Y11 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

June 4, 2003EE384Y12 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)

June 4, 2003EE384Y13 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):

June 4, 2003EE384Y14 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

June 4, 2003EE384Y15 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