Download presentation

Presentation is loading. Please wait.

Published byZechariah Grissom Modified over 3 years ago

1
1 Atomic Routing Games on Maximum Congestion Costas Busch Department of Computer Science Louisiana State University Collaborators: Rajgopal Kannan, LSU Malik Magdon-Ismail, RPI

2
2 Introduction Price of Stability Price of Anarchy Outline of Talk Bicriteria Game

3
3 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 Player cost function for routing : Congestion of selected path Social cost function for routing : Largest player cost

7
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

8
8 Results: Price of Stability is 1 Price of Anarchy is Maximum allowed path length

9
9 Introduction Price of Stability Price of Anarchy Outline of Talk Bicriteria Game

10
10 We show: QoR games have Nash Equilibriums (we define a potential function) The price of stability is 1

11
11 number of players with cost Routing Vector

12
12 Routing Vectors are ordered lexicographically = = == < <= =

13
If player performs a greedy move transforming routing to then: 13 Lemma: Proof Idea: Show that the greedy move gives a lower order routing vector

14
14 Player Cost Before greedy move: After greedy move: Since player cost decreases:

15
15 Before greedy move player was counted here After greedy move player is counted here

16
16 > == No change Definite Decrease possible decrease possible increase or decrease Possible increase > END OF PROOF IDEA

17
17 Existence of Nash Equilibriums Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium

18
18 Price of Stability Lowest order routing : Is a Nash Equilibrium Achieves optimal social cost

19
19 Introduction Price of Stability Price of Anarchy Outline of Talk Bicriteria Game

20
20 We show for any restricted QoR game: Price of Anarchy =

21
Path of player 21 Consider an arbitrary Nash Equilibrium edge maximum congestion in path

22
must have an edge with congestion Optimal path of player 22 In optimal routing : Since otherwise:

23
23 In Nash Equilibrium social cost is:

24
24 Edges in optimal paths of

25
25

26
26 Edges in optimal paths of

27
27

28
28 In a similar way we can define:

29
29 We obtain sequences: There exist subsequence: Where: and

30
30 Maximum edge utilization Minimum edge utilization Maximum path length Known relations

31
31 Worst Case Scenario:

32
32 Introduction Price of Stability Price of Anarchy Outline of Talk Bicriteria Game

33
33 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

34
34 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

35
35 Player cost function for routing : Congestion of selected path Cost of respective routing class

36
36 Social cost function for routing : Largest player cost

37
37 Results: Price of Stability is 1 Price of Anarchy is

38
38 We consider restricted QoR games For any path : Path lengthService Cost of path

39
39 We show for any restricted QoR game: Price of Anarchy =

40
Path of player 40 Consider an arbitrary Nash Equilibrium edge maximum congestion in path

41
must have an edge with congestion Optimal path of player 41 In optimal routing : Since otherwise:

42
42 In Nash Equilibrium:

43
43 Edges in optimal paths of

44
44

45
45 Edges in optimal paths of

46
46

47
47 In a similar way we can define:

48
48 We obtain sequences: There exist subsequence: Where: and

49
49 Maximum edge utilization Minimum edge utilization Maximum path length Known relations

50
50 We have: By considering class service costs, we obtain:

Similar presentations

Presentation is loading. Please wait....

OK

Clock will move after 1 minute

Clock will move after 1 minute

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on panel discussion script Ppt on store design Ppt on school annual function Ppt on project management consultancy Ppt on autonomous car news Ppt on crop production management Ppt on ms excel tutorial Elementary ppt on cells Ppt on periscope tv Ppt on sources of energy for class 8th december