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 Introduction Price of Stability Price of Anarchy Outline of Talk

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

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

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 Results: Price of Stability is 1 Price of Anarchy is

12 Introduction Price of Stability Price of Anarchy Outline of Talk

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

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

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

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 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 > == No change Definite Decrease possible decrease possible increase or 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 Introduction Price of Stability Price of Anarchy Outline of Talk

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

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

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

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

27 In Nash Equilibrium :

28 Edges in optimal paths of

29

30 Edges in optimal paths of

31

32 In a similar way we can define:

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

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

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