1. Capacity Allocation in Networks Under Noncooperative Elastic Users Instructor: Ishai Menache Eliron Amir Winter 2006

2. Project Goals Investigate uniqueness of Nash equilibrium point under non- cooperative routing. Find optimal capacity allocation under non-cooperative routing.

3. Definitions Set of users: Set of links: Flow configuration of user i: Flow configuration of link l: System flow configuration: Cost function:

4. Definitions Type-A cost function:

5. Nash Equilibrium A reasonable model for a working point in a multiple user game. Occurs when no player has anything to gain by changing his own strategy unilaterally. NEP – Nash Equilibrium Point.

6. Uniqueness of NEP Proved for parallel networks in: Ariel Orda, Raphael Rom and Nahum Shimkin, "Competitive routing in multi- user communication networks,1993.

7. Uniqueness of NEP Multiple NEPs for general topologies under type-A assumption. 2005, topology-dependant proof for uniqueness, Richman and Shimkin. Its unknown whether under type-B cost functions, a uniqueness exists for general topologies. Type-B:

8. Uniqueness of NEP Directions investigated during the project : Proving uniqueness through convex potential functions, Ramesh Johari and John N. Tsitsiklis, "Efficiency loss in a network resource allocation games". Finding monotone properties of parallel components, in order to create reduction to the original parallel-link proof.

9. Elastic Users We concentrate on parallel link networks Throughput demand is not constant, and depends on network congestion. Additional term to the cost function:

10. Uniqueness of NEP for Elastic Users - Proof Idea: Reduction to the plastic problem. Add another link,, to the network. Assign it with a cost function,. Where, is a dummy parameter, set higher than the typical flow of the network.

11. Capacity Allocation Type-C functions: Yannis A. Korilis, Aurel A. Lazar and Ariel Orda, "Capacity allocation under noncooperative routing. Transferring capacity from any link, to a link with initially higher capacity reduces the cost of all users. Best capacity allocation in term of overall cost, is achieved when we put all the capacity in one link.

12. Capacity Allocation Network provider goal is to maximize its profit,, with the right capacity allocation. Prices are static and must not be modified. Users are elastic, with M/M/1 latencies. An added term to the users cost function: The proof we uniqueness allows injective mapping between capacity configuration and flow configuration.

13. Capacity allocation New cost function: We shell consider two cases : – Symmetrical users,. – Non symmetrical users (type-A cost function).

14. Simulations Matlab based script. Best response method – Each user in its turn minimizes his own cost function until convergence is achieved. We check if the flow configuration was indeed a NEP by checking the KKT conditions. Although no theoretical proof for convergence of (synchronized) dynamics exists, in practice all experiments converge to the (unique) NEP.

15. Experimental Results Symmetrical users:

16. Experimental Results Symmetrical users:

17. Experimental Results Best capacity allocation for network provider is achieved when one link only is active. Users throughput is higher when all the capacity is allocated to only one link. Larger throughput larger profit. A discontinuity in derivative of flow/cost occurs when the users switch from one link to the other

18. Experimental Results Non symmetrical users:

19. Experimental Results Non symmetrical users:

20. Experimental Results Non symmetrical users:

21. Experimental Results Different sensitivity of users to delay costs results in different behavior of users flow functions. Non symmetry allows us to find an example where the peak is in the center. Sometimes it is worthy to split the capacity between several links.

22. Experimental Results Three links – Three users

23. Experimental Results Multiple 3 links/3 users simulations to check if splitting capacity is beneficial. Low variance in utility parameters (i.e., users close to symmetrical), only in 5% of cases it is worthy to split capacity. High variance in utility parameters (i.e. users close to non symmetrical), in 45% of cases it is worthy to split the capacity. Larger networks might increase probability.

24. Conclusions and future work Uniqueness of the NEP in parallel networks with elastic users. We have seen that in some cases, splitting the capacity is beneficial for the network administrator. Find an analytical solution (e.g., optimization based) for the capacity allocation problem.