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Quality of Routing Congestion Games in Wireless Sensor Networks
Costas BuschLouisiana State University
Rajgopal KannanLouisiana State University
Athanasios VasilakosUniv. of Western Macedonia
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Introduction
Price of Stability
Price of Anarchy
Outline of Talk
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Sensor Network RoutingEach 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
iplayer
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A player may selfishly choose an alternativepath that minimizes congestion
CC 31 congestion
Congestion Games:
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We consider Quality of Routing (QoR) congestion games where the pathsare partitioned into routing classes:
QQQ ,,, 21
)()()( 21 QSQSQS
With service costs:
Only paths in same routing class can causecongestion to each other
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An example:
•We can have routing classes)(lognO
•Each routing class contains paths with length in range
jQ]2,2[ 1jj
12)( jjQS•Service cost:
•Each routing class uses a different wireless frequency channel
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Player cost function for routing :i
iii SCppc )(
p
Congestionof selected path
Cost of respectiverouting class
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Social cost function for routing :
SCpSC )(
p
Largest player cost
We are interested in Nash Equilibriumswhere every player is locally optimal
Metrics of equilibrium quality:
p
Price of Stability
)()(min *pSCpSC
p
Price of Anarchy
)()(max *pSCpSC
p
*p is optimal coordinated routingwith smallest social cost ***)( SCpSC
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Results:• Price of Stability is 1
• Price of Anarchy is
)log),(min( ** nSCO
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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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],,,,,[)( 21 rk mmmmpM
number of players with cost km k
)( QSNr Size of vector:
Routing Vector
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Routing Vectors are ordered lexicographically
],,,[)( 21 rmmmpM
],,,[)( 21 rmmmpM
= = = =
],,,,,[)( 11 rkk mmmmpM
],,,,,[)( 11 rkk mmmmpM
< < = =
)()( pMpM
)()( pMpM )( pp
)( pp
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If player performs a greedy movetransforming routing to then:p p pp
iLemma:
Proof Idea:Show that the greedy move gives a lower order routing vector
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kk
iii SCppck )(
iii SCppck )(
Player CostiBefore greedy move:After greedy move:
Since player cost decreases:
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],,,,,,,[)( 11 rkkk mmmmmpM
Before greedy move player was counted herei
],,,,,,,[)( 11 rkkk mmmmmpM
After greedy moveplayer is counted herei
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],,,,,,,[)( 11 rkkk mmmmmpM
],,,,,,,[)( 11 rkkk mmmmmpM
> ==No change
Definite Decrease
possibledecrease
possibleincreaseor 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 playeris reached which is a Nash Equilibrium
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minp
Price of Stability
Lowest order routing :
*min )( SCpSC
• Is a Nash Equilibrium
• Achieves optimal social cost
1)(Stability of Price *min
SCpSC
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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 )(|| pSp
Path length Service Cost of path
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We show for any restricted QoR game:
Price of Anarchy = )log),(min( ** nSCO
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Path of player
Consider an arbitrary Nash Equilibriump
i
iCedgemaximum congestionin path
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must have an edge with congestion
Optimal path of player
In optimal routing :*p
i
iC
*SCC i
)(111 *** ppcSCCSSCSCcp iiiiiiii
***)( SCpSC
Since otherwise:
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C
00
0
edges use that Paths: Congestion of Edges :ECE
In Nash Equilibrium :p SCpSC )(
0 0
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C *SC *SC
0 0
Edges in optimal paths of 0
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C *SC *SC
0 01 1
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*1
edges use that Players:least at Congestion of Edges :E
SCE
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C *SC *SC *2SC *2SC *2SC *2SC
0 01 1
Edges in optimal paths of 1
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C *SC *SC *2SC *2SC *2SC
0 01 1
*2SC
2 2
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*2
edges use that Players:2least at Congestion of Edges :
ESCE
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In a similar way we can define:
jj
j
E
jSCE
edges use that Players:
least at Congestion of Edges : *
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,,,,
,,,,
3210
3210
EEEEWe obtain sequences:
There exist subsequence:110
110
,,,,,,,
s
ss EEEE
||2|| 1 jj EEWhere: ||2|| 1 ss EEand1sj
ns log
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||))1((|| 1*
1 ss ESsCL
|||| 1*
s
s
EC
Maximum edge utilization
Minimum edge utilization
*SLMaximum path length
)log( ** nSOCC
ns log ||2|| 1 ss EEKnown relations
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)log( ** nSOCC
)log),(min( Anarchy of Price **** nSCOSCSC
We have:
By considering class service costs, we obtain:
