Quality of Routing Congestion Games in Wireless Sensor Networks

Transcription

1 Qualty of Routng ongeston Gaes n Wreless Sensor Networs ostas Busch Lousana State Unversty Rajgoal Kannan Lousana State Unversty Athanasos Vaslaos Unv. of Western Macedona 1

2 Outlne of Tal Introducton Prce of Stablty Prce of Anarchy 2

3 Sensor Networ Routng Each layer corresonds to a ar of source-destnaton Objectve s to select aths wth sall cost 3

4 Man objectve of each layer s to nze congeston: nze axu utlzed edge layer congeston 3 4

5 ongeston Gaes: A layer ay selfshly choose an alternatve ath that nzes congeston congeston 1 3 5

6 We consder Qualty of Routng (QoR congeston gaes where the aths are arttoned nto routng classes: Q Q 1 2 Q Wth servce costs: S( Q S( Q S( Q 1 2 Only aths n sae routng class can cause congeston to each other 6

7 An exale: We can have O(log n routng classes Each routng class Q j wth length n range [2 contans aths j 2 j1 ] Servce cost: S( Q j 2 j 1 Each routng class uses a dfferent wreless frequency channel 7

8 Player cost functon for routng : c ( S ongeston of selected ath ost of resectve routng class 8

9 Socal cost functon for routng : S( S Largest layer cost 9

10 We are nterested n Nash Equlbrus where every layer s locally otal Metrcs of equlbru qualty: Prce of Stablty S( n S( Prce of Anarchy S( ax S( s otal coordnated routng wth sallest socal cost S ( S

11 Results: Prce of Stablty s 1 Prce of Anarchy s O(n( S log n 11

12 Outlne of Tal Introducton Prce of Stablty Prce of Anarchy 12

13 We show: QoR gaes have Nash Equlbrus (we defne a otental functon The rce of stablty s 1 13

14 Routng Vector M ( [ 1 2 r ] nuber of layers wth cost Sze of vector: r N S ( Q 14

15 15 Routng Vectors are ordered lexcograhcally ] [ ( 2 1 r M ] [ ( 2 1 r M = = = = ] [ ( 1 1 r M ] [ ( 1 1 r M < < = = ( ( M M ( ( M M ( (

16 Lea: If layer erfors a greedy ove transforng routng to then: Proof Idea: Show that the greedy ove gves a lower order routng vector 16

17 Player ost Before greedy ove: After greedy ove: c ( c ( S S Snce layer cost decreases: 17

18 18 ] [ ( 1 1 r M Before greedy ove layer was counted here ] [ ( 1 1 r M After greedy ove layer s counted here

19 19 ] [ ( 1 1 r M ] [ ( 1 1 r M > = = No change Defnte Decrease ossble decrease ossble ncrease or decrease Possble ncrease > END OF PROOF IDEA

20 Exstence of Nash Equlbrus Greedy oves gve lower order routngs Eventually a local nu for every layer s reached whch s a Nash Equlbru 2

21 Prce of Stablty n Lowest order routng : Is a Nash Equlbru Acheves otal socal cost S( S n S( Prce of Stablty S n 1 21

22 Outlne of Tal Introducton Prce of Stablty Prce of Anarchy 22

23 We consder restrcted QoR gaes For any ath : S( Path length Servce ost of ath 23

24 We show for any restrcted QoR gae: Prce of Anarchy = O(n( S log n 24

25 onsder an arbtrary Nash Equlbru Path of layer edge axu congeston n ath 25

26 ust have an edge wth congeston Otal ath of layer 26 In otal routng : S ( c S S S S c ( S S Snce otherwse:

27 In Nash Equlbru : S( S E : Edges : Paths of ongeston that use edges E 27

28 S S Edges n otal aths of 28

29 S S 1 1 E 1 : Edges of ongeston at least S 1 : Players that use edges E 1 29

30 2S 2S S S 2S 2S 1 1 Edges n otal aths of 1 3

31 2S 2S S S 2S 2S E 2 2 : Edges of ongeston at least 2S : Players that use edges E 2 31

32 In a slar way we can defne: E j : Edges of ongeston at least js j : Players that use edges E j 32

33 We obtan sequences: E E 1 1 E 2 2 E 3 3 There exst subsequence: E E 1 1 E E s 1 s s1 s log n and Where: E j 2 E j1 E s 2 Es 1 j s 1 33

34 Maxu ath length L S Maxu edge utlzaton s1 L ( ( s 1 S Es 1 Mnu edge utlzaton s log n s1 E Known relatons s E s 2 E s1 O( S log n 34

35 We have: O( S log n By consderng class servce costs we obtan: Prce of Anarchy S O(n( S log n S 35