Competitive Routing in Multi-User Communication Networks

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. Model and Formulation • 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:

7. Model and Formulation • 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. Model and Formulation • 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:

9. Model and Formulation • Assumptions of the cost function: • G1 It is a sum of user-link cost function: • G2 might be infinite • G3 is convex • G4 Whenever finite is 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 Links Users

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. User 2 Steps User 1

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

22. Part II – General Topology Network Users

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

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

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

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