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

Transcription

1 A Game-Theoretic Framework for Congestion Control in General Topology SYS793 Presentation! By:! Computer Science Department! University of Virginia 1 Outline 2 1

2 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 existing fixed infrastructure! Cooperation among users is also impossible.! Game Theory is a perfect match for this noncooperative problem 3 4 2

3 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: 5! 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? 6 3

4 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 r is the solution to the following optimization problem given all users on its path have equilibrium flow rates, : 7! Theorem 3.1: Under A1-A4, A4, the network game admits a unique inner Nash equilibrium 8 4

5 9! System goal:! The sum of the utilities of users is maximized! Aggregate cost at the links is minimized or mathematically speaking: 10 5

6 ! Theorem 5.1: the unique NE of the game (Theorem 3.1) solves the following system problem: where and satisfy assumptions A1-A4 A

7 ! Utility function:! is the user-specific specific preference parameter.! Price function:! is a network-wide 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? 13! The utility, price, and cost functions satisfy A1-A4, 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: 14 7

8 15 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. 16 8

9 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? 17 References! T. Alpcan and T. Basar.. "A Game-Theoretic Framework for Congestion Control in General Topology, in Proc. 41st IEEE Conference on Decision and Control,, Las Vegas, Nevada, December 10-13, 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 , 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 , 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