1. Charge-Sensitive TCP and Rate Control Richard J. La Department of EECS UC Berkeley November 22, 1999

2. Motivation Network users have a great deal of freedom as to how they can share the available bandwidth in the network The increasing complexity and size of the Internet renders centralized rate allocation impractical –distributed algorithm is desired Two classes of flow/congestion control mechanisms –rate-based : directly controls the transmission rate based on feedback –window-based : controls the congestion window size to adjust the transmission rate and backlog

3. Motivation Transmission Control Protocol (TCP) does not necessarily results in a fair or efficient allocation of the available bandwidth Many algorithms have been proposed to achieve fairness among the connections Fairness alone may not be a suitable objective –most algorithms do not reflect the user utilities or preferences –good rate allocation should not only be fair, but should also maximize the overall utility of the users

4. Model Network with a set J of links and a set I of users –

5. Model (Kelly) –system is not likely to know –impractical for a centralized system to compute and allocate the user rates

6. Model (Kelly)

7. Background (Kelly’s work) One can always find vectors and such that 1) solves for all 2) solves 3) 4) is the unique solution to

8. Fairness Max-min fairness : –a user’s rate cannot be increased without decreasing the rate of another user who is already receiving a smaller rate –gives an absolute priority to the users with smaller rates (weighted) proportional fairness : – is weighted proportionally fair with weight vector if is feasible and for any other feasible vector

9. Fluid Model (Mo & Walrand) where

10. Fluid Model (Mo & Walrand) Theorem 1 (Mo & Walrand) : For all w there exists a unique x that satisfies the constraints (1)-(4) –this theorem tells us that the rate vector is a well defined function of the window sizes w. –denote the function by x(w) –x(w) is continuous and differentiable at an interior point –q(w) may not be unique, but the sum of the queuing delay along any route is well defined

11. Mo & Walrand’s Algorithm (p, 1)-proportionally fair algorithm : where

12. Mo & Walrand’s Algorithm Theorem 2 (Mo & Walrand) : The window sizes converge to a unique point w * such that for all Further, the resulting rate at the unique stable point w * is weighted proportionally fair that solves NETWOKR(A, C ; p).

13. Pricing Scheme Price per unit flow at a switch is the queuing delay at the switch, i.e., –the total price per unit flow of user i is given by where is connection i’s queue size at resource j

14. User Optimization & Assumption User optimization problem : where is the price per unit flow, which is the queuing delay Assumption 1 : The optimal price is a decreasing function of.

15. Examples of Utility Functions

16. Price Updating Rule At time t, each user i updates its price according to

17. Price Updating Rule Define a mapping to be Fixed point of the mapping T is a vector p such that T(p) = p. Theorem : There exists a unique fixed point p * of the mapping T, and the resulting rate allocation from p * is the optimal rate allocation x * that solves SYSTEM(U,A,C).

18. Algorithm I Suppose that users update their prices according to Assumption 2 : There exists M > 0 such that (a) for all p such that (b) for all p such that

19. Convergence in Single Bottleneck Case Theorem : Under the assumptions 1 and 2, the user prices p(n) converges to the unique fixed point of the mapping T under both Jacobi and the totally asynchronous update schemes as.

20. Algorithm II Suppose that users update their window sizes according to where

21. Assumption & Convergence Assumption 3: The utility functions satisfy where Theorem : Under assumption 3, the window sizes converge to a unique stable point of the algorithm II, where the resulting rates solve SYSTEM(U,A,C).