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

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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

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Introduction Price of Stability Price of Anarchy Outline of Talk

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Sensor Network Routing

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

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

3 congestion  C i player

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A player may selfishly choose an alternative path that minimizes congestion

C C     3 1 congestion

Congestion Games:

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We consider Quality of Routing (QoR) congestion games where the paths are partitioned into routing classes:



Q Q Q , , ,

2 1

 ) ( ) ( ) (

2 1 

Q S Q S Q S    

With service costs: Only paths in same routing class can cause congestion to each other

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An example:

• We can have routing classes

) (logn O  

• Each routing class contains paths

with length in range

j

Q ] 2 , 2 [

1  j j 1

2 ) (





j j

Q S

• Service cost:
• Each routing class uses a different

wireless frequency channel

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Player cost function for routing :

i

i i i

S C p pc   ) ( p

Congestion

• f selected path

Cost of respective routing class

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Social cost function for routing :

S C p SC   ) (

p

Largest player cost

We are interested in Nash Equilibriums where every player is locally optimal Metrics of equilibrium quality:

p

Price of Stability

) ( ) ( min

*

p SC p SC

p

Price of Anarchy

) ( ) ( max

*

p SC p SC

p

*

p is optimal coordinated routing

with smallest social cost

* * *)

( S C p SC  

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Results:

• Price of Stability is 1
• Price of Anarchy is

) log ) , (min(

* *

n S C O  

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Introduction Price of Stability Price of Anarchy Outline of Talk

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We show:

• QoR games have Nash Equilibriums

(we define a potential function)

• The price of stability is 1

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] , , , , , [ ) (

2 1 r k

m m m m p M   

number of players with cost



k

m k

) (



Q S N r  

Size of vector:

Routing Vector

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Routing Vectors are ordered lexicographically

] , , , [ ) (

2 1 r

m m m p M   ] , , , [ ) (

2 1 r

m m m p M      

= = = =

] , , , , , [ ) (

1 1 r k k

m m m m p M  



 ] , , , , , [ ) (

1 1 r k k

m m m m p M      

 



<

< = =

) ( ) ( p M p M  

) ( ) ( p M p M   ) ( p p   ) ( p p  

If player performs a greedy move transforming routing to then:

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p p

p p  

i

Lemma: Proof Idea:

Show that the greedy move gives a lower order routing vector

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k k  

i i i

S C p pc k    ) (

i i i

S C p pc k        ) (

Player Cost

i

Before greedy move: After greedy move: Since player cost decreases:

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] , , , , , , , [ ) (

1 1 r k k k

m m m m m p M   

 



Before greedy move player was counted here

i

] , , , , , , , [ ) (

1 1 r k k k

m m m m m p M       

 

  

After greedy move player is counted here

i

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] , , , , , , , [ ) (

1 1 r k k k

m m m m m p M   

 



] , , , , , , , [ ) (

1 1 r k k k

m m m m m p M       

 

  

>

= =

No change Definite Decrease

possible decrease possible increase

• r decrease

Possible increase

>

END OF PROOF IDEA

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Existence of Nash Equilibriums

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

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min

p

Price of Stability

Lowest order routing :

* min)

( SC p SC 

• Is a Nash Equilibrium
• Achieves optimal social cost

1 ) ( Stability

• f

Price

* min 

 SC p SC

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Introduction Price of Stability Price of Anarchy Outline of Talk

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We consider restricted QoR games For any path :

p

) ( | | p S p 

Path length Service Cost

• f path

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

) log ) , (min(

* *

n S C O  

Path of player

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Consider an arbitrary Nash Equilibrium p

i

i

C

edge

maximum congestion in path

must have an edge with congestion Optimal path of player

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In optimal routing :

*

p

i

i

C

*

S C C

i 

 

) ( 1 1 1

* * *

p pc S C C S S C S C c p

i i i i i i i i

             

* * *)

( S C p SC  

Since otherwise:

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C edges use that Paths : Congestion

• f

Edges : E C E 

In Nash Equilibrium :

p

S C p SC   ) (

 

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C

*

S C 

*

S C 

 

Edges in optimal paths of



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C

*

S C 

*

S C 

 

1



1



1 1 * 1

edges use that Players : least at Congestion

• f

Edges : E S C E  

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C

*

S C 

*

S C 

*

2S C 

*

2S C 

*

2S C 

*

2S C 

 

1



1



Edges in optimal paths of

1



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C

*

S C 

*

S C 

*

2S C 

*

2S C 

*

2S C 

 

1



1



*

2S C 

2



2



2 2 * 2

edges use that Players : 2 least at Congestion

• f

Edges : E S C E  

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In a similar way we can define:

j j j

E jS C E edges use that Players : least at Congestion

• f

Edges :

*

 

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  , , , , , , , ,

3 2 1 3 2 1

    E E E E

We obtain sequences: There exist subsequence:

1 1 1 1

, , , , , , ,

 

  

s s s

E E E E  

| | 2 | |

1 



j j

E E

Where:

| | 2 | |

1 



s s

E E

and

1   s j

n s log 

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| | ) ) 1 ( ( | |

1 * 1  

     

s s

E S s C L

   



| | | |

1 * s s

E C

Maximum edge utilization Minimum edge utilization

*

S L 

Maximum path length

) log (

* *

n S O C C    

n s log 

| | 2 | |

1 



s s

E E Known relations

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) log (

* *

n S O C C     ) log ) , (min( Anarchy

• f

Price

* * * *

n S C O S C S C       

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