quality of routing congestion games in wireless sensor networks
DESCRIPTION
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. Outline of Talk. Introduction. Price of Stability. Price of Anarchy. - PowerPoint PPT PresentationTRANSCRIPT
1
Quality of Routing Congestion Games in Wireless Sensor Networks
Costas BuschLouisiana State University
Rajgopal KannanLouisiana State University
Athanasios VasilakosUniv. of Western Macedonia
2
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
3
Sensor Network RoutingEach player corresponds to a pair of source-destination
Objective is to select paths with small cost
4
Main objective of each player is to minimize congestion: minimize maximum utilized edge
3 congestion C
iplayer
5
A player may selfishly choose an alternativepath that minimizes congestion
CC 31 congestion
Congestion Games:
6
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
7
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
8
Player cost function for routing :i
iii SCppc )(
p
Congestionof selected path
Cost of respectiverouting class
9
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
11
Results:• Price of Stability is 1
• Price of Anarchy is
)log),(min( ** nSCO
12
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
13
We show:
• QoR games have Nash Equilibriums
(we define a potential function)
• The price of stability is 1
14
],,,,,[)( 21 rk mmmmpM
number of players with cost km k
)( QSNr Size of vector:
Routing Vector
15
Routing Vectors are ordered lexicographically
],,,[)( 21 rmmmpM
],,,[)( 21 rmmmpM
= = = =
],,,,,[)( 11 rkk mmmmpM
],,,,,[)( 11 rkk mmmmpM
< < = =
)()( pMpM
)()( pMpM )( pp
)( pp
16
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
17
kk
iii SCppck )(
iii SCppck )(
Player CostiBefore greedy move:After greedy move:
Since player cost decreases:
18
],,,,,,,[)( 11 rkkk mmmmmpM
Before greedy move player was counted herei
],,,,,,,[)( 11 rkkk mmmmmpM
After greedy moveplayer is counted herei
19
],,,,,,,[)( 11 rkkk mmmmmpM
],,,,,,,[)( 11 rkkk mmmmmpM
> ==No change
Definite Decrease
possibledecrease
possibleincreaseor decrease
Possible increase
>
END OF PROOF IDEA
20
Existence of Nash Equilibriums
Greedy moves give lower order routings
Eventually a local minimum for every playeris reached which is a Nash Equilibrium
21
minp
Price of Stability
Lowest order routing :
*min )( SCpSC
• Is a Nash Equilibrium
• Achieves optimal social cost
1)(Stability of Price *min
SCpSC
22
Introduction
Price of Stability
Price of Anarchy
Outline of Talk
23
We consider restricted QoR games
For any path :p )(|| pSp
Path length Service Cost of path
24
We show for any restricted QoR game:
Price of Anarchy = )log),(min( ** nSCO
25
Path of player
Consider an arbitrary Nash Equilibriump
i
iCedgemaximum congestionin path
26
must have an edge with congestion
Optimal path of player
In optimal routing :*p
i
iC
*SCC i
)(111 *** ppcSCCSSCSCcp iiiiiiii
***)( SCpSC
Since otherwise:
27
C
00
0
edges use that Paths: Congestion of Edges :ECE
In Nash Equilibrium :p SCpSC )(
0 0
28
C *SC *SC
0 0
Edges in optimal paths of 0
29
C *SC *SC
0 01 1
11
*1
edges use that Players:least at Congestion of Edges :E
SCE
30
C *SC *SC *2SC *2SC *2SC *2SC
0 01 1
Edges in optimal paths of 1
31
C *SC *SC *2SC *2SC *2SC
0 01 1
*2SC
2 2
22
*2
edges use that Players:2least at Congestion of Edges :
ESCE
32
In a similar way we can define:
jj
j
E
jSCE
edges use that Players:
least at Congestion of Edges : *
33
,,,,
,,,,
3210
3210
EEEEWe obtain sequences:
There exist subsequence:110
110
,,,,,,,
s
ss EEEE
||2|| 1 jj EEWhere: ||2|| 1 ss EEand1sj
ns log
34
||))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
35
)log( ** nSOCC
)log),(min( Anarchy of Price **** nSCOSCSC
We have:
By considering class service costs, we obtain: