1. Competitive Routing in Multi-User Communication Networks Presentation By: Yuval Lifshitz In Seminar: Computational Issues in Game Theory (2002/3) By: Prof. Yishay Mansour Original Paper: A. Orda, R. Rom and N. Shimkin, “Competitive Routing in Multi-User Communication Networks”, pp. 964-971 in Proceedings of IEEE INFOCOM'93

2. Introduction Single Entity – Single Control Objective –Either centralized or distributed control –Optimization of average network delay –Passive Users Resource shared by a group of active users –Different measures of satisfaction –Optimizing subjective demands –Dynamic system

3. Introduction Questions: –Does an equilibrium point exists? –Is it unique? –Does the dynamic system converge to it?

4. Introduction What was done so far (1993): –Economic tools for flow control and resource allocation –Routing – two nodes connected with parallel identical links (M/M/c queues) –Rosen (1965) conditions for existence, uniqueness and stability

5. Introduction Goals of This Paper –The uniqueness problem of a convex game (convex but not common objective functions) –Use specificities of the problem (results cannot be derived directly from Rosen) –Two nodes connected by a set of parallel links, not necessarily queues –General networks

6. Set of m users: Set of n parallel communication links: User’s throughput demand – stochastic process with average: Fractional assignment Expected flow of user on link: Users flows fulfill the demand constraint: Total flow on link: Model and Formulation

7. Link flow vector: User flow configuration: System flow configuration: Feasible user flow – obey the demand constraint Set of all feasible user flows: Feasible system flow – all users flows are feasible Set of feasible system flows:

8. User cost as a function of the system’s flow configuration: Nash Equilibrium Point (NEP) –System flow configuration such that no user finds it beneficial to change its flow on any link –A configuration: that for each i holds: Model and Formulation

9. Assumptions of the cost function: –G1 It is a sum of user-link cost function: –G2 might be infinite –G3 is convex –G4 Whenever finiteis continuously differentiable –G5 At least one user with infinite flow (if exists) can change its flow configuration to make it finite

10. Model and Formulation Convex Game – Rosen guarantees the existence of NEP Kuhn-Tucker conditions for a feasible configuration to be a NEP We will investigate uniqueness and convergence of a system

11. Model and Formulation Type-A cost functions – is a function of the users flow on the link and the total flow on the link –The functions in increasing in both its arguments –The function’s partial derivatives are increasing in both arguments

12. Model and Formulation Type-B cost functions –Performance function of a link measures its cost per unit: –Multiplicative form: – cannot be zero, but might be infinite – is strictly increasing and convex – is continuously differentiable

13. Model and Formulation Type-C cost functions –Based on M/M/1 model of a link –They are Type-B functions –If then: else: – is the capacity of the link

14. Part I – Parallel links Users Links

15. Uniqueness Theorem: In a network of parallel links where the cost function of each user is of type-A the NEP is unique. Kuhn-Tucker conditions: for each user i there exists (Lagrange multiplier), such that for every link l, if : then:else: when:

16. Monotonicity Theorem: In a network of parallel links with identical type-A cost functions. For any pair of users i and j, if then for each link l. Lemma: Suppose that holds for some link l’ and users i and j. Then, for each link l:

17. Monotonicity If all users has the same demand then: If then Monotonic partition among users: User with higher demands uses more links, and more of each link

18. Monotonicity Theorem: In a network of parallel links with type-C cost functions. For any pair of links l and l’, if then for each user i. Lemma: Assume that for links l and l’ the following holds: Then:for each user j.

19. Convergence Two users sharing two links ESS – Elementary Stepwise System –Start at non-equilibrium point –Exact minimization is achieved at each stage –All operations are done instantly User’s i flow on link l at the end of step n :

20. Convergence Odd stage 2n-1: User 1 find its optimum when the other user’s 2n-2 step is known. Even stage 2n: User 2 find its optimum when the other’s user 2n-1 step is known. Steps User 1 User 2

21. Convergence Theorem: Let an ESS be initialized with a feasible configuration, Then the system configuration converges over time to the NEP, meaning: Lemma: Letbe two feasible flows for user 1. Andoptimal flows for user 2 against the above. If:then:

22. Part II – General Topology Users Network

23. Non-uniqueness NEP1 User 1 User 2 12 4 3 22,18 10,12 8,10 8,16 24,14 14,2 40

24. Non-uniqueness NEP2 User 1 User 2 12 4 3 20,23 18,5 2,12 8,16 22,18 4,13 40

25. Non-monotonous User 1 User 2 12 4 3 T(4,3)=5 7 4 T(3,1)=20 T(1,2)=1T(4,3)=4 T(3,1)=21

26. Diagonal Strict Convexity Weighted sum of a configuration: Pseudo-Gradient:

27. Diagonal Strict Convexity Theorem (Rosen): If there exists a vector for which the system is DSC. Then the NEP is unique Pseudo-Jacobian Corollary: If the Pseudo-Jacobian matrix is positive definite then the NEP is unique

28. Symmetrical Users All users has the same demand (same source and destination) Lemma: Theorem: A network with symmetrical users has a unique NEP

29. All-Positive Flows All users must have the same source and destination Type-B cost functions For a subclass of links, on which the flows are strictly positive, the NEP is unique.

30. Further Research General network uniqueness for type-B functions Stability (convergence) Restrictions on users (non non-cooperative games) Delay in measurements – “real” dynamic system