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

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By: Gang ZhouComputer Science

DepartmentUniversity of Virginia

A Game-Theoretic Framework A Game-Theoretic Framework for Congestion Control in for Congestion Control in

General Topology NetworksGeneral Topology Networks

SYS793 Presentation

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Seminar on Coordinated SystemsGang ZhouOutlineOutline

Problem and MotivationProblem and Motivation The General Game-Theoretic FrameworkThe General Game-Theoretic Framework

The ModelThe Model Existence and Uniqueness of the Nash EquilibriumExistence and Uniqueness of the Nash Equilibrium

System Problem and Optimality of Nash System Problem and Optimality of Nash EquilibriumEquilibrium

A Congestion Control Scheme for Ad Hoc Wireless A Congestion Control Scheme for Ad Hoc Wireless NetworksNetworks

ConclusionsConclusions Discussion Discussion

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Seminar on Coordinated SystemsGang ZhouProblem and MotivationProblem and Motivation Congestion Control is an essential research issue in both Congestion Control is an essential research issue in both

wired network, such as Internet, and wireless networks, such wired network, such as Internet, and wireless networks, such as sensor networks. as sensor networks.

Users on the Internet are of noncooperative nature in terms Users on the Internet are of noncooperative nature in terms of their demand for network resourcesof their demand for network resources No specific information on other users’ flow rates.No specific information on other users’ flow rates. So cooperation among users is impossible.So cooperation among users is impossible.

Users on ad hoc wireless networks are also of noncooperative Users on ad hoc wireless networks are also of noncooperative nature as to their demand for network resourcesnature as to their demand for network resources No specific information on other users’ flow rates.No specific information on other users’ flow rates. Mobile users with no pre-existing fixed infrastructure Mobile users with no pre-existing fixed infrastructure Cooperation among users is also impossible.Cooperation among users is also impossible.

Game Theory is a perfect match for this noncooperative Game Theory is a perfect match for this noncooperative problemproblem

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Seminar on Coordinated SystemsGang Zhou






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Seminar on Coordinated SystemsGang ZhouThe ModelThe Model

Nodes set:Nodes set: Links set:Links set: User set:User set:

(M X 1) Flow rate vector:(M X 1) Flow rate vector: (L X 1) Link capacity (L X 1) Link capacity

vector:vector:

Routing matrix:Routing matrix:

Capacity constraints:Capacity constraints:

Flow rate upper-bound:Flow rate upper-bound:

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Utility functionUtility function Only depends on its flow rate!Only depends on its flow rate!

Price functionPrice function Indicates the current state of the networkIndicates the current state of the network

Cost functionCost function Supposed to model:Supposed to model:

User’s preferenceUser’s preference Current network statusCurrent network status

What should it be?What should it be?

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Seminar on Coordinated SystemsGang ZhouExistence and Uniqueness of the Nash Existence and Uniqueness of the Nash EquilibriumEquilibrium Nash Equilibrium definition in this contextNash Equilibrium definition in this context

NE here is defined as a set of flow rates and NE here is defined as a set of flow rates and corresponding set of costs, with the property that no corresponding set of costs, with the property that no user can benefit by modifying its flow while the other user can benefit by modifying its flow while the other players keep theirs fixed.players keep theirs fixed.

Mathematically speaking. is in NE, when of any Mathematically speaking. is in NE, when of any user is the solution to the following optimization problem user is the solution to the following optimization problem given all users on its path have equilibrium flow rates, given all users on its path have equilibrium flow rates, : :

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

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ConclusionsConclusions DiscussionDiscussion

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System goal:System goal: The sum of the utilities of users is maximized The sum of the utilities of users is maximized Aggregate cost at the links is minimizedAggregate cost at the links is minimized

or mathematically speaking:or mathematically speaking:

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Theorem 5.1: the unique NE of the game (Theorem Theorem 5.1: the unique NE of the game (Theorem 3.1) solves the following system problem:3.1) solves the following system problem:

where and satisfy assumptions A1-A4where and satisfy assumptions A1-A4

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ConclusionsConclusions DiscussionDiscussion

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Utility function:Utility function: is the user-specific preference parameter.is the user-specific preference parameter.

Price function:Price function:

is a network-wide constant which depends on factors is a network-wide constant which depends on factors like the type of the ad hoc network, number of users.like the type of the ad hoc network, number of users.

If an queue model is assumed, If an queue model is assumed, corresponds to the delay at the link . And hence the corresponds to the delay at the link . And hence the price is proportional to the aggregate delay on the price is proportional to the aggregate delay on the user’s path.user’s path.

Cost function:Cost function: What is it?What is it?

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The utility, price, and cost functions satisfy A1-A4, if The utility, price, and cost functions satisfy A1-A4, if parameters and are chosen appropriately.parameters and are chosen appropriately.

By Theorem 3.1, there exists unique inner NE.By Theorem 3.1, there exists unique inner NE.

By Theorem 5.1, this NE solves the following system By Theorem 5.1, this NE solves the following system problem:problem:

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Seminar on Coordinated SystemsGang ZhouConclusions Conclusions

Noncooperative game theoretic approach Noncooperative game theoretic approach provides an appropriate framework for developing provides an appropriate framework for developing congestion control schemes for communication congestion control schemes for communication networks.networks.

With suitable choice of cost functions, these With suitable choice of cost functions, these schemes are easily implementable.schemes are easily implementable.

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Seminar on Coordinated SystemsGang ZhouDiscussionDiscussion

How to decide the cost parameters and ?How to decide the cost parameters and ?

If the cost parameters and vary with network If the cost parameters and vary with network conditions, what will we do? Could we still use the conditions, what will we do? Could we still use the current framework or we need improvement?current framework or we need improvement?

What are your questions?What are your questions?

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Seminar on Coordinated SystemsGang ZhouReferencesReferences T. Alpcan and T. Basar. "A Game-Theoretic Framework for T. Alpcan and T. Basar. "A Game-Theoretic Framework for

Congestion Control in General Topology Networks“, Congestion Control in General Topology Networks“, in Proc. 41st in Proc. 41st IEEE Conference on Decision and ControlIEEE Conference on Decision and Control, Las Vegas, Nevada, , Las Vegas, Nevada, December 10-13, 2002.December 10-13, 2002.

E. Altman, T. Basar, T. Jimenez, and N. Shimkin, “Conpetitive E. Altman, T. Basar, T. Jimenez, and N. Shimkin, “Conpetitive routing in networks with polynomial costs”, routing in networks with polynomial costs”, in IEEE Transactions in IEEE Transactions on Automatic Controlon Automatic Control, vol. 47(1), pp. 92-96, January 2002., vol. 47(1), pp. 92-96, January 2002.

A. Orda, R. Rom, and N. Shimkin, “Competitive routing in A. Orda, R. Rom, and N. Shimkin, “Competitive routing in multiuser communication networks”, in multiuser communication networks”, in IEEE/ACM Transactions on IEEE/ACM Transactions on Networking,Networking, vol. 1, pp. 510-521, October 1993. vol. 1, pp. 510-521, October 1993.

E. Altman, T. Basar, and R. Srikant, “Nash equilibria for combined E. Altman, T. Basar, and R. Srikant, “Nash equilibria for combined flow control and routing in networks: asymptotic behavior for a flow control and routing in networks: asymptotic behavior for a large number of users”, in large number of users”, in IEEE Transactions on Automatic ControlIEEE Transactions on Automatic Control, , vol. 47(6), June 2002.vol. 47(6), June 2002.

T. Basar and R. Srikant, “Revenue-maximizing pricing and capacity T. Basar and R. Srikant, “Revenue-maximizing pricing and capacity expansion in a many-users regime”,expansion in a many-users regime”, in INFOCOM, New York, in INFOCOM, New York, June June 2002.2002.

Documents

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