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.