Charge-Sensitive TCP and Rate Control

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 uniquex 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 allpsuch 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).