1. 1 Quality of Routing Congestion Games in Wireless Sensor Networks Costas Busch Louisiana State University Rajgopal Kannan Louisiana State University Athanasios Vasilakos Univ. of Western Macedonia

2. 2 Introduction Price of Stability Price of Anarchy Outline of Talk

3. 3 Sensor Network Routing Each player corresponds to a pair of source-destination Objective is to select paths with small cost

4. 4 Main objective of each player is to minimize congestion: minimize maximum utilized edge

5. 5 A player may selfishly choose an alternative path that minimizes congestion Congestion Games:

6. 6 We consider Quality of Routing (QoR) congestion games where the paths are partitioned into routing classes: With service costs: Only paths in same routing class can cause congestion to each other

7. 7 An example: We can have routing classes Each routing class contains paths with length in range Service cost: Each routing class uses a different wireless frequency channel

8. 8 Player cost function for routing : Congestion of selected path Cost of respective routing class

9. 9 Social cost function for routing : Largest player cost

10. We are interested in Nash Equilibriums where every player is locally optimal Metrics of equilibrium quality: Price of StabilityPrice of Anarchy is optimal coordinated routing with smallest social cost

11. 11 Results: Price of Stability is 1 Price of Anarchy is

12. 12 Introduction Price of Stability Price of Anarchy Outline of Talk

13. 13 We show: QoR games have Nash Equilibriums (we define a potential function) The price of stability is 1

14. 14 number of players with cost Size of vector: Routing Vector

15. 15 Routing Vectors are ordered lexicographically = = == < <= =

16. If player performs a greedy move transforming routing to then: 16 Lemma: Proof Idea: Show that the greedy move gives a lower order routing vector

17. 17 Player Cost Before greedy move: After greedy move: Since player cost decreases:

18. 18 Before greedy move player was counted here After greedy move player is counted here

19. 19 > == No change Definite Decrease possible decrease possible increase or decrease Possible increase > END OF PROOF IDEA

20. 20 Existence of Nash Equilibriums Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium

21. 21 Price of Stability Lowest order routing : Is a Nash Equilibrium Achieves optimal social cost

22. 22 Introduction Price of Stability Price of Anarchy Outline of Talk

23. 23 We consider restricted QoR games For any path : Path lengthService Cost of path

24. 24 We show for any restricted QoR game: Price of Anarchy =

25. Path of player 25 Consider an arbitrary Nash Equilibrium edge maximum congestion in path

26. must have an edge with congestion Optimal path of player 26 In optimal routing : Since otherwise:

27. 27 In Nash Equilibrium :

28. 28 Edges in optimal paths of

29. 29

30. 30 Edges in optimal paths of

31. 31

32. 32 In a similar way we can define:

33. 33 We obtain sequences: There exist subsequence: Where: and

34. 34 Maximum edge utilization Minimum edge utilization Maximum path length Known relations

35. 35 We have: By considering class service costs, we obtain: