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Capacity Allocation in Networks. Winter 2006. Under Noncooperative Elastic Users. Eliron Amir. Instructor: Ishai Menache. Project Goals. Investigate uniqueness of Nash equilibrium point under non-cooperative routing. Find optimal capacity allocation under non-cooperative routing. - PowerPoint PPT Presentation
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Capacity Allocation in Networks
Under Noncooperative Elastic Users
Instructor: Ishai Menache
Eliron AmirEliron Amir
Winter 2006
Project Goals
• Investigate uniqueness of Nash equilibrium point under non-cooperative routing.
• Find optimal capacity allocation under non-cooperative routing.
Definitions
Set of users:
Set of links:
Flow configuration of user i:
Flow configuration of link l:
System flow configuration:
Cost function:
L {1,2,..., }L
1 2( , ,..., )i i iLf f fif
1 2( , ,..., )If f f f
1 2( , ,..., )Il l lf f flf
( )iJ f
I {1,2,..., }I
Definitions
Type-A cost function:
( ) ( ).
:[0, ) [0, ],a continuous function.
is convex in
Wherever finite, is continuously differeniable in . .
is a function of two arguments, i.e. ( ) (
i il
l L
i I
i il
ii i i ll l l i
l
i i il l l
J J
J
J f
JJ f K
f
J J J
l
l
f f
f , ).
is increasing in each of its two arguments.
( , ) is strictly increasing in each of it's two arguments.
il l
il
i i il l l l
f f
J
K K f f
Nash Equilibrium
A reasonable model for a working point in a multiple user game.
Occurs when no player has anything to gain by changing his own strategy unilaterally.
NEP – Nash Equilibrium Point.1 1 1 1 1 1
: ( ) ( ,..., , , ,..., ) min ( ,..., , , ,..., )i
i i i I i i Ii i i ii J J J
f
f f f f f f f f f f f
Uniqueness of NEP
Proved for parallel networks in:
Ariel Orda, Raphael Rom and Nahum Shimkin, "Competitive routing in multi-
user communication networks“,1993.
Uniqueness of NEP•Multiple NEPs for general topologies under type-A assumption.•2005, topology-dependant proof for uniqueness, “Richman and Shimkin”.•It’s unknown whether under type-B cost functions, a uniqueness exists for general topologies.
Type-B:
( , ) .
:[0, ) (0, ].
is positive, strictly increasing and convex.
is continuously differentiable.
i i il l l l l l
l
l l
l l
J f f f T f
T
T f
T f
Uniqueness of NEP
Directions investigated during the project:
•Proving uniqueness through convex potential functions, Ramesh Johari and John N. Tsitsiklis, "Efficiency loss in a
network resource allocation games".
•Finding monotone properties of parallel components, in order to create reduction to the original parallel-link proof.
Elastic Users
• We concentrate on parallel link networks
• Throughput demand is not constant, and depends on network congestion.
• Additional term to the cost function:
( ) wherei i i i
ll L
U r r f
Uniqueness of NEP for Elastic Users - Proof
Idea: Reduction to the plastic problem.
• Add another link, , to the network. • Assign it with a cost function, . Where ,
is a dummy parameter, set higher than the typical flow of the network.
( )i i i
lU r f
l
lir
Capacity Allocation
• Type-C functions:
• Yannis A. Korilis, Aurel A. Lazar and Ariel Orda, "Capacity allocation under noncooperative routing”.
• Transferring capacity from any link, to a link with initially higher capacity reduces the cost of all users.
• Best capacity allocation in term of overall cost, is achieved when we put all the capacity in one link.
1l l
l ll
l l
f CC fT
f C
Capacity Allocation
• Network provider goal is to maximize its profit,
, with the right capacity allocation.• Prices are static and must not be
modified.• Users are elastic, with M/M/1 latencies. • An added term to the users cost function:• The proof we uniqueness allows injective
mapping between capacity configuration and flow configuration.
l ll L
f P
lP P
Capacity allocation
• New cost function:
• We shell consider two cases: – Symmetrical users, .
– Non symmetrical users (type-A cost function).
( ) ( ) ( )i i i i i i il l
l L
J J U r f P
f f
,i i i I
Simulations
• Matlab based script.
• Best response method – Each user in it’s turn minimizes his own cost function until convergence is achieved.
• We check if the flow configuration was indeed a NEP by checking the KKT conditions.
• Although no theoretical proof for convergence of (synchronized) dynamics exists, in practice all experiments converge to the (unique) NEP.
Experimental Results
Symmetrical users:1 1
2 2
3 3
1 2
69
3 - 0.1
3 - 0.1
3 - 0.1
0.035 - 0.11
C
P P
Experimental Results
Symmetrical users:
Experimental Results
• Best capacity allocation for network provider is achieved when one link only is active.
• Users throughput is higher when all the capacity is allocated to only one link.
• Larger throughput larger profit.
• A discontinuity in derivative of flow/cost occurs when the users switch from one link to the other
Experimental Results
Non symmetrical users:1 1
2 2
3 3
1 2
69
0.01 - 0.1
4 - 0.15
2 - 0.1
0.035 - 0.11
C
P P
Experimental Results
Non symmetrical users:
Experimental Results
Non symmetrical users:
Experimental Results
• Different sensitivity of users to delay costs results in different behavior of users flow functions.
• Non symmetry allows us to find an example where the “peak” is in the center.
• Sometimes it is worthy to split the capacity between several links.
Experimental Results
• Three links – Three users
00.2
0.40.6
0.81
0
0.5
1
0
0.1
0.2
0.3
0.4
0.5
c2/C
3 Links - 3 Users
c1/C
Gai
n
Experimental Results
• Multiple 3 links/3 users simulations to check if splitting capacity is beneficial.
• Low variance in utility parameters (i.e., users close to symmetrical), only in 5% of cases it is worthy to split capacity.
• High variance in utility parameters (i.e. users close to non symmetrical), in 45% of cases it is worthy to split the capacity.
• Larger networks might increase probability.
Conclusions and future work
• Uniqueness of the NEP in parallel networks with elastic users.
• We have seen that in some cases, splitting the capacity is beneficial for the network administrator.
• Find an analytical solution (e.g., optimization based) for the capacityallocation problem.