Quality of Routing Congestion Games in Wireless Sensor Networks

1. Quality of Routing Congestion Games in Wireless Sensor Networks Costas Busch Louisiana State University RajgopalKannan Louisiana State University AthanasiosVasilakos Univ. of Western Macedonia

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

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

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

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

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

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 Stability Price of Anarchy is optimal coordinated routing with smallest social cost

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

27. In Nash Equilibrium :

28. Edges in optimal paths of

30. Edges in optimal paths of

32. In a similar way we can define:

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

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

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