in Wireless Sensor Networks Costas Busch Louisiana State University - PowerPoint PPT Presentation

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 1

Outline of Talk Introduction Price of Stability Price of Anarchy 2

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

Main objective of each player is to minimize congestion: minimize maximum utilized edge player i  congestion 3 C 4

Congestion Games: A player may selfishly choose an alternative path that minimizes congestion     congestion 1 3 C C 5

We consider Quality of Routing (QoR) congestion games where the paths are partitioned into routing classes:  , , , Q Q Q  1 2 With service costs:     ( ) ( ) ( ) S Q S Q S Q  1 2 Only paths in same routing class can cause congestion to each other 6

An example:   • We can have routing classes (log n ) O • Each routing class contains paths Q j with length in range  1 j j [ 2 , 2 ]   • Service cost: 1 j ( ) 2 S Q j • Each routing class uses a different wireless frequency channel 7

Player cost function for routing : p i   ( ) pc p C S i i i Congestion Cost of respective of selected path routing class 8

Social cost function for routing : p   ( ) SC p C S Largest player cost 9

We are interested in Nash Equilibriums p where every player is locally optimal Metrics of equilibrium quality: Price of Stability Price of Anarchy ( ) SC p ( ) SC p min max * ( ) * SC p p ( ) SC p p p is optimal coordinated routing * with smallest social cost   * ) * * ( SC p C S

Results: • Price of Stability is 1 • Price of Anarchy is    * * (min( , ) log ) O C S n 11

Outline of Talk Introduction Price of Stability Price of Anarchy 12

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

Routing Vector    ( ) [ , , , , , ] M p m m m m 1 2 k r  number of players with cost k m k   Size of vector: ( ) r N S Q  14

Routing Vectors are ordered lexicographically     ( ) ( ) ( ) p p M p M p   ( ) [ , , , ] M p m m m 1 2 r = = = =       ( ) [ , , , ] M p m m m 1 2 r     ( ) ( ) ( ) M p M p p p    ( ) [ , , , , , ] M p m m m m  1 1 k k r < = < =          ( ) [ , , , , , ] M p m m m m 1 1 k k r 15

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

Player Cost i    Before greedy move: ( ) k pc p C S i i i     After greedy move:    ( ) k pc p C S i i i   Since player cost decreases: k k 17

Before greedy move player was counted here i     ( ) [ , , , , , , , ] M p m m m m m   1 1 k k k r           ( ) [ , , , , , , , ] M p m m m m m   1 1 k k k r After greedy move player is counted here i 18

    ( ) [ , , , , , , , ] M p m m m m m   1 1 k k k r > = = >           ( ) [ , , , , , , , ] M p m m m m m   1 1 k k k r possible possible No change increase decrease or decrease Definite Decrease Possible increase END OF PROOF IDEA 19

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

Price of Stability Lowest order routing : p min • Is a Nash Equilibrium • Achieves optimal social cost  * ( min ) SC p SC ( min  ) SC p  Price of Stability 1 * SC 21

Outline of Talk Introduction Price of Stability Price of Anarchy 22

We consider restricted QoR games p  For any path : p | | ( ) S p Path length Service Cost of path 23

We show for any restricted QoR game: Price of Anarchy =    * * (min( , ) log ) O C S n 24

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

* In optimal routing :   p * ) * * ( SC p C S Optimal path of player i must have an edge with congestion   i  * C C S C i Since otherwise:               * * * 1 1 1 ( ) p c C S C S S C C S pc p i i i i i i i i 26

In Nash Equilibrium :   p ( ) SC p C S C   0 0 : Edges of Congestion E C 0  : Paths that use edges E 0 0 27

C  C * C  S * S   0 0  Edges in optimal paths of 0 28

C  C * C  S * S     1 1 0 0  * : Edges of Congestion at least E C S 1  : Players that use edges E 1 1 29

C  C  C * C  * C  * 2 S S C  * 2 S * C  S 2 S * 2 S     1 1 0 0  Edges in optimal paths of 1 30

C  C  C * C  * C  * 2 S S C  * 2 S * C  S 2 S * 2 S       1 1 0 0 2 2  * : Edges of Congestion at least 2 E C S 2  : Players that use edges E 2 2 31

In a similar way we can define:  * : Edges of Congestion at least E C jS j  : Players that use edges E j j 32

We obtain sequences:  , , , , E E E E 0 1 2 3      , , , , 0 1 2 3 There exist subsequence:  , , , , E E E E  0 1 1 s s     , , ,  0 1 1 s  log s n and Where:   | | 2 | | | | 2 | | E E E E   1 j j 1 s s  s  1 j 33

Maximum path length L  * S Maximum edge utilization       * | | ( ( 1 ) ) | | L C s S E   1 1 s s C     * ( log ) O S n Minimum edge utilization * C  | |   * 1 s C   | | E s Known relations   log s n | | 2 | | E E  1 s s 34

C We have:     * ( log ) O S n * C By considering class service costs, we obtain:  C S      * * Price of Anarchy (min( , ) log ) O C S n  * * C S 35