Fernando PaganiniORT University, Uruguay

(on leave from UCLA)

Congestion control with adaptive multipath routing

based on optimization

Collaborator: Enrique Mallada, ORT University, Uruguay.

, ( )

s

Market based-decentralized solution: link generates price

ource sees tries to maximize profit

k k k k k

lk l

lp x

l pq R U x qk

[Kelly-Maulloo-Tan ’98, Low-Lapsley ’99, many others] Book by Srikant, 2004.

Source rate x(t)

Optimization on the demand side: congestion control

max ( )

.subject to

Kelly's system problem

x i iiy

U x

Rx c

, )

1

( .

has total rate :,

Source has rate receives utility

Single route case: iff route contains link .

Link and capacity

.

k k k

lk

lil i l

R k l

k U x

l Ry

x

x c

Pricefeedback

Optimization on the supply side

( )( ) ( ),back to Pigou, Wardrop, transportation networks; see Roughgarden '05.

where is a link latency, compare optimum with the Wardrop

equilibrium where traffic units r

Selfish routing:

ll l l l yy y y '( ).outes selfishly. Natural prices: l l lp y

min .

( )( )

Given: a demand matrix of traffic between source-destination pairs (commodities)

and a convex cost per link, find routes through the

optimization Problem is convex if m

multicommodity l l

l l

yy

ultiple paths are allowed.

Network operator solves optimization offline,

implements (or approximates) via IP weights or MPLS tunnels.

[Fortz-Thorup '00, Sridharan-Guerin-Diot '05, many othes]

Traffic engineering in IP networks:

,

Adaptive version: MATE, Elwalid-Jin-Low-Widjaja '01.

Combining demand and supply?

( ) .

Two natural optimization problems we will work on

(both allowing multiple routes per source-destination pair):

subject to flow balance at each node, and

SYSTEM problem: max

BARRIER or S

k kl lU x y c

( ) ( ): subject to flow balance .

urplus problem:

m aax t each ode nk kl lU x yS

,

,

path variables: Decentralized solutions? Most previous work uses

for each end-to-end path available to commodity let be the rate.

Adapting each based on path c

o

nge

.kl

p k

kp

kp

k kp p

p l

p k

x y

z

z

z z

stion prices: (Kelly-Maullo-Tan '98,

Han-Shakkottai-Hollot-Srikant-Towsley '03, Lestas-Vinnicombe '04 , Voice '04).

Difficulties with the path formulation

• An exponential number of paths! How do we limit size?• Sources do not have the path information, nor is it

reasonable to add all this complexity to them.• Overlay with the edge router doing rate control? but even

routers don’t know end-to-end paths.

A better set of control variables.

As feedback information, the source receives a single price signalthat reflects the overall congestion state of all paths available to it.

kq

,

Routers control the traffic split of each commodity among their

outgoing links.The splits depend only on destination (as in IP routing)

: fraction of traffic destined to that router se

:

di j d i

( , )nds through link .

i j

Routers update their split ratios gradually based on prices reported by

neighboring nodes. Idea already developed in Gallager '78, for the

"supply only" problem (optimization of delay cost).

Everyone (source or router) uses the same "congestion currency".

ource controls only the total rate into the network. Each s kx

Other related work, wireless ad-hoc context: Chen-Low-Chiang-Doyle, Infocom '06.

More detailed notation:

1s

2s

d, Nodes (sources, routers).i j

( , ). Links l i j

, ( ) ( ) Commodities (source , des t ). k s k d k

( )

,

, ,

.

.

( , ).

:

:

:

:

Total rate of commodity reaching node

Source rate of commodity

Rate of commodity going through link

Total rate going through link

ki

k ks k

ki j

ki j i j

k

x k i

x x k

y k l i j

y y

( , ). l i j

, ,( , ) ( , )

, ( ). , ( ). Mass balance: k k k ki i j j i j

i j L i j L

y i d k y j s kx x

,( ) ( )

( ), ,

,

Traffic split (per commodity)

Alternatively, (per destination)

k ki j i

d k d d k d

d kk ki j i j i

di jy x

y x

Price information:

,

, ,( , )

.

,

( , ).

0, ,

Recursion can be solved i

:

f th

Congestion price of link

Average price of reaching destination from node with

current routing

:

.

i j

di

d di j i j j

i j L

d did

p l i j

d i

q i d

q

pq q

( )( )

ere's a path between every node and

the destination. We do not mode

l these dynamics.

Average price seen by source. :k d ks kqq

,Under routing splits , we have a congestion control picture, analogous to the stand

fa

ixed rd one.

di j

, .

The matrix can bedetermined from the set

of routing spli

ts di j

R

LINKSSOURCES

R

TRSource

prices kqLink prices lp

Source rates kx Link

rates ly

Adaptation of router traffic splits

,

,( , )

, ,

,

0 (maintain mass balance).

Use price information from links and neighbor nodes to change

shifting traffic in the direction of cheaper route

gradually

s.

Assume:

di j

di j

i j L

d ki j i j jqp

,

,

,( , )

,

}

( , )

0

0, ,

0 (changes in have negative correlation with prices).

Equality above only if and this happens only if

for each either or

nd

{

a

:

= ,

di j

di j

di j

i j L

d d di i j j ii j L q p q q

, .< di j jp q

LINKS

SOURCES

Traffic splitting

Node price recursionSource prices kq Link prices lp

Source rates kx Link rates ly

Node prices diq

Adapt splits,

Split ratios d

i j

Primal congestion control under adaptive multipath routing:

( ) '( )

'( )

Assume the sources run the control law

and links set prices as marginal costs:

k k k k

l l

x x U x q

p y

: ( ) ( )

with these dynamics, and under the earlier assumptions on split adaptation, the system converges globally to the optimum of the BARRIER problem

max

,

Theorem:

k kl lS U x y

0.the surplus increases along trajectories, It could "stall" while the system searches for a cheap route, but will only "settle" at the global optimum.The proof involves Lasalle's princip

Pr

l

oof : S

e, and invoking dualityon the barrier problem wrt. to the mass balance constraints.

Dual congestion control under fixed multipath routing:

arg max ( )Sources set rates instantaneously to

and links set prices with the dynamic rule:

k k k k

l l l llp

x U x q x

p y c

,

: ( , ) 0.

. for fixed the dual algorithm{ },

as in single

find

path case, Pf

s Proposition: di j

W p

( , , ) ( ) ( ) [ ( ) ]

The Lagrangian of the System Problem w.r.t. the capacity constraints,k k k k k k

l l l l lk l k l

L p x U x p c y U x q x p c

{ }( , )

( )

( )

max max min max ( , , )Then from duality we have

k k

kxy c p

W p

U x L p x

Dual congestion control under adaptive multipath routing:

, ( , ) the adaptation of pushes in the direction.

So its behavior over time is inconclusive. Indeed, if routes are adapted too

fast relative to price d

ynamics, the system

increasing

could

,Issue: di j W p

oscillate.

{ }( , )

( )

( )

max max min max ( , , ) k k

kxy c p

W p

U x L p x

,

,

are adapted at a slower time-scale than prices.

For each assume prices and rates take instantaneously

their equilibrium values. Then adapting as described earlier, t

Assumption:

Theorem:

di j

he

dynamics asymptotically reach a solution of the SYSTEM problem.

EXAMPLE

Source 1

Source 2

Destination

1i

2i

3i4i

14 10c

24 1c

Links in light blue havevery high capacity.

1,3 1,4 3,2 3,40, 1, 0.5, 0.5

Initially, take the traffic split variables at nodes 1 and 3 to be at

1,4 2,40.1, 1.Under primal or dual congestion control, the system converges to so

bottlenecme

rates and link prices at the say,ks p p

1 2 30.1, 1, 0.5,This yields the following node prices : q q q

1,3 1,4

For node 1, the route through node 3 is more expensive, so there is no incentive to change , all flows remain constant for a while.

3,2 3,4 3 , causing to drop.

.

However, node 3 will adapt Eventually, this route becomes cheaper and node 1 starts using it

, q

1,3

1,4

3,2

3,4

1,4p

3q

SEXAMPLE (cont)

1i

2i

3i4i

14 10c

24 1c

Fluid-flow simulationUsing SCILAB

Implementation issues

, ,( , )

.

: , and then make its

Router receives annoucements from its neighbors indicating they have

a route to a certain destination, and the corresponding prices

It can updat e

d di j i j j

i j L

dj

di q

i j

p

qq

own annoucement.

, ,The above iteration converges, for fixed under mild assumptions. di j

Now, for the theory to be relevant, this convergence must be faster than thedynamics we modeled (link prices, rates, and - adaptation).

In particular, source rates, that adapt quickly with TCP-like algorithms, would not wait for the node to form the price with IP-style routing updates. This implies either: Interpret source demand as aggregate, long term. Use another method to form source node price. In particular ECN marking

proportional to link prices will do this, to first order.

Conclusions• We presented natural optimization problems that combine

multipath routing with elastic demands, using variables which are local to sources and routers.

• We introduced congestion prices for nodes that use multipath routing, and a slow adaptation of traffic split ratios at routers. Combined with standard congestion control, this strategy yields decentralized solutions to the optimization problems.

• The algorithms fit with the TCP/IP philosophy (end-to-end control of source rate, local control of routing based on neighbor information).

• Open question: what happens if we remove time-scale separations?

• We are starting to look at implementation issues, in particular combining explicit and implicit methods to propagate prices.