1. A Scalable Network Resource Allocation Mechanism With Bounded Efficiency Loss IEEE Journal on Selected Areas in Communications, 2006 Johari, R., Tsitsiklis, J.N. Presented by Ma Man Lok and Wan Wing San

2. Agenda Introduction Single Link General Networks Simulation Conclusion Q & A

3. Introduction Network Resource Allocation –Network Traffic has grown exponentially User base increases Applications require increasing resource Applications require stricter Quality of Service –Introduce Usage-based Charges Resolve the allocation of resources to users Traffic management and congestion control

4. Introduction Congestion Pricing Mechanisms –Objective Users should pay for the additional congestion they create Encouraging the redistribution of the demand in space or in time

5. Introduction Congestion Pricing Mechanisms –Design Simple and Scalable end-to-end implementation Efficiency of resulting equilibria

6. Introduction Motivation –Recently proposed mechanisms Assume users are price taker –They do not anticipate the effect of their strategic decisions on the prices –Derive a alternative mechanisms by studying Cournot game

7. Introduction Cournot game –There is more than one firm –All firms produce a homogeneous product –Firms do not cooperate –Firms compete in quantities, and choose quantities simultaneously

8. Single Link Game –Multiple users compete for a single link –Strategies of the users represent their desired rates

9. Single Link Model –N users compete for a single link –Each user n has a utility function U n –Total data rate through the link incur a cost characterized by a cost function C

10. Single Link

11. It can be characterized as a optimization problem

12. Single Link Pricing Scheme –Assume users are price takers –Given a price μ > 0, user n choose x n to maximize –There exists a vector x and a scalar μ such that

13. Single Link Pricing Scheme –If users are not price takers –Alternative model Play a Cournot game to acquire a share of the link Notation x -n denote the vector of all rates chosen by users other than n Given x -n, user n choose x n to maximize

14. Single Link Pricing Scheme –Q n is similar to P n Except the user can anticipate –Nash Equilibrium (NE) exists for this game

15. Single Link Pricing Scheme –Assume p(q) = aq + b U n (0) ≥ 0 for all n –x s is any optimum solution of the problem –x is any NE of the game –The worst case efficiency loss is bounded by 1/3

16. General Networks Game Model Optimization Problem Payoff to User Bound of Efficiency Loss

17. Game Multiple users compete for network resources provided by multiple links Strategies of users represent their desired rate on paths which are combination of links

18. Model Assumption 1 & 2 still hold Network contains J, P and N as set of links, paths and users respectively Each path is a combination of some links –j  J, q  P and j  q Each user can own several paths –n  N and q  n Each path is owned by single user only –q  n, q'  n', n  n' and q  q'

19. Model (cont.) Rate allocated to path q: y q  0 Rate allocated to user n: d n =  q  n y q  0 Total rate on link j: f j =  q:j  q y q Utility of user n: U n (d n ) Cost of link j (overall users): C j (f j ) Price of link j of user n:  j (y) = p j (  q  n:j  q y q ) Total payment of user n:  q  n y q  j  q  j (y)

20. Model (cont.) Path-resource incidence matrix A –A jq = 1 if j  q –A jq = 0 if otherwise Path-user incidence matrix H –H nq = 1 if q  n –H nq = 0 if otherwise d = (d n, n  N), y = (y q, q  P) Ay = f, Hy = d

21. Optimization Problem

22. Payoff to User Price taker n Price anticipating user n where y -n = (y 1, …, y n-1, y n+1, …, y N )

23. Bound of Efficiency Loss Suppose p j (q j )=a j q j +b j for some a j >0, b j  0 Let y S be any solution to the optimization problem

24. Bound of Efficiency Loss Proof Sketch Establish relationship of N.E. of choosing rates on paths and N.E. of choosing rates on links Reduce analysis to individual games at each link, extend the bound for Single Link

25. Bound of Efficiency Loss Relationship Consider another game that each user n has to choose rate d jn at each link j User n can achieve max. rate by solving max-flow optimization problem

26. Bound of Efficiency Loss Relationship (cont.) Denote optimal objective value by z n (d n ) Price at each link j: p j (  n d jn ) Total payment of user n:  j d jn p j (  n d jn ) Payoff to price anticipating user n Suppose y is N.E. of game of (Q 1, …, Q N ) Define d jn =  q  n:j  q y q follows that

27. Bound of Efficiency Loss Relationship (cont.) U n (  q  n y q ) = U n (z n (d n )) –y n is feasible for the max-flow problem,  q  n y q  z n (d n )  U n (  q  n y q )  U n (z n (d n )) –For case that U n (  q  n y q ) < U n (z n (d n ))   q  n y q < z n (d n )  y n is not optimal and hence contradict with the assumption of N.E and so result follows Hence, the following hold at N.E.

28. Bound of Efficiency Loss Reducing analysis to individual link Let d n * be N.E. of the second game Replace U n (z n (d n )) by linear utility function  n T d n while keeping d n * as N.E. of the new game The second game can be decoupled into j Single Link game and hence the bound can be extended from the previous bound

29. Simulation Since the General Networks part is simply an extension of Single Link, only Single Link case is considered Objective: Test if the bound would be reached easily while assuming users are homogenous for simplicity Configuration –Both functions are non-linear Utility function U n (x) = 1 – e -kx Price function p(x) = e px –Both functions are linear Utility function U n (x) = kx Price function p(x) = px Result: achieved aggregate surplus is very close to the optimal value (within 3% loss)

30. Conclusion The scheme proposed by this paper is to –users choose the rate to send on paths –set the link price according to marginal cost of total rate allocated By using this scheme, the Efficiency Loss is bounded above by 1/3

31. Q & A