1. By: Gang Zhou Computer Science Department University of Virginia 1 A Game-Theoretic Framework for Congestion Control in General Topology Networks SYS793 Presentation

2. Seminar on Coordinated Systems Gang Zhou 2 Outline  Problem and Motivation  The General Game-Theoretic Framework  The Model  Existence and Uniqueness of the Nash Equilibrium  System Problem and Optimality of Nash Equilibrium  A Congestion Control Scheme for Ad Hoc Wireless Networks  Conclusions  Discussion

3. Seminar on Coordinated Systems Gang Zhou 3 Problem and Motivation  Congestion Control is an essential research issue in both wired network, such as Internet, and wireless networks, such as sensor networks.  Users on the Internet are of noncooperative nature in terms of their demand for network resources  No specific information on other users’ flow rates.  So cooperation among users is impossible.  Users on ad hoc wireless networks are also of noncooperative nature as to their demand for network resources  No specific information on other users’ flow rates.  Mobile users with no pre-existing fixed infrastructure  Cooperation among users is also impossible.  Game Theory is a perfect match for this noncooperative problem

4. Seminar on Coordinated Systems Gang Zhou 4  Problem and Motivation  The General Game-Theoretic Framework  The Model  Existence and Uniqueness of the Nash Equilibrium  System Problem and Optimality of Nash Equilibrium  A Congestion Control Scheme for Ad Hoc Wireless Networks  Conclusions  Discussion

5. Seminar on Coordinated Systems Gang Zhou 5 The Model  Nodes set:  Links set:  User set:  (M X 1) Flow rate vector:  (L X 1) Link capacity vector:  Routing matrix:  Capacity constraints:  Flow rate upper-bound:

6. Seminar on Coordinated Systems Gang Zhou 6  Utility function  Only depends on its flow rate!  Price function  Indicates the current state of the network  Cost function  Supposed to model:  User’s preference  Current network status  What should it be?

7. Seminar on Coordinated Systems Gang Zhou 7 Existence and Uniqueness of the Nash Equilibrium  Nash Equilibrium definition in this context  NE here is defined as a set of flow rates and corresponding set of costs, with the property that no user can benefit by modifying its flow while the other players keep theirs fixed.  Mathematically speaking. is in NE, when of any user is the solution to the following optimization problem given all users on its path have equilibrium flow rates, :

8. Seminar on Coordinated Systems Gang Zhou 8  Theorem 3.1: Under A1-A4, the network game admits a unique inner Nash equilibrium

9. Seminar on Coordinated Systems Gang Zhou 9  Problem and Motivation  The General Game-Theoretic Framework  The Model  Existence and Uniqueness of the Nash Equilibrium  System Problem and Optimality of Nash Equilibrium  A Congestion Control Scheme for Ad Hoc Wireless Networks  Conclusions  Discussion

10. Seminar on Coordinated Systems Gang Zhou 10  System goal:  The sum of the utilities of users is maximized  Aggregate cost at the links is minimized or mathematically speaking:

11. Seminar on Coordinated Systems Gang Zhou 11  Theorem 5.1: the unique NE of the game (Theorem 3.1) solves the following system problem: where and satisfy assumptions A1-A4

12. Seminar on Coordinated Systems Gang Zhou 12  Problem and Motivation  The General Game-Theoretic Framework  The Model  Existence and Uniqueness of the Nash Equilibrium  System Problem and Optimality of Nash Equilibrium  A Congestion Control Scheme for Ad Hoc Wireless Networks  Conclusions  Discussion

13. Seminar on Coordinated Systems Gang Zhou 13  Utility function:  is the user-specific preference parameter.  Price function:  is a network-wide constant which depends on factors like the type of the ad hoc network, number of users.  If an queue model is assumed, corresponds to the delay at the link. And hence the price is proportional to the aggregate delay on the user’s path.  Cost function:  What is it?

14. Seminar on Coordinated Systems Gang Zhou 14  The utility, price, and cost functions satisfy A1-A4, if parameters and are chosen appropriately.  By Theorem 3.1, there exists unique inner NE.  By Theorem 5.1, this NE solves the following system problem:

15. Seminar on Coordinated Systems Gang Zhou 15  Problem and Motivation  The General Game-Theoretic Framework  The Model  Existence and Uniqueness of the Nash Equilibrium  System Problem and Optimality of Nash Equilibrium  A Congestion Control Scheme for Ad Hoc Wireless Networks  Conclusions  Discussion

16. Seminar on Coordinated Systems Gang Zhou 16 Conclusions  Noncooperative game theoretic approach provides an appropriate framework for developing congestion control schemes for communication networks.  With suitable choice of cost functions, these schemes are easily implementable.

17. Seminar on Coordinated Systems Gang Zhou 17 Discussion  How to decide the cost parameters and ?  If the cost parameters and vary with network conditions, what will we do? Could we still use the current framework or we need improvement?  What are your questions?

18. Seminar on Coordinated Systems Gang Zhou 18 References  T. Alpcan and T. Basar. "A Game-Theoretic Framework for Congestion Control in General Topology Networks“, in Proc. 41st IEEE Conference on Decision and Control, Las Vegas, Nevada, December 10-13, 2002.  E. Altman, T. Basar, T. Jimenez, and N. Shimkin, “Conpetitive routing in networks with polynomial costs”, in IEEE Transactions on Automatic Control, vol. 47(1), pp. 92-96, January 2002.  A. Orda, R. Rom, and N. Shimkin, “Competitive routing in multiuser communication networks”, in IEEE/ACM Transactions on Networking, vol. 1, pp. 510-521, October 1993.  E. Altman, T. Basar, and R. Srikant, “Nash equilibria for combined flow control and routing in networks: asymptotic behavior for a large number of users”, in IEEE Transactions on Automatic Control, vol. 47(6), June 2002.  T. Basar and R. Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users regime”, in INFOCOM, New York, June 2002.