A Game-Theoretic Framework for Congestion Control in General Topology Networks

1. A Game-Theoretic Framework for Congestion Control in General Topology Networks SYS793 Presentation

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. 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. 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. 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. 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. 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. Theorem 3.1: Under A1-A4, the network game admits a unique inner Nash equilibrium

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. System goal: • The sum of the utilities of users is maximized • Aggregate cost at the links is minimized or mathematically speaking:

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. 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. 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. 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. 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. 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. 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. 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.