Fernando Paganini Universidad ORT Uruguay

On congestion control, multipathrouting, and admission control

• Part I (with Enrique Mallada). Combined congestion control and node-based multipath routing: new results on stability since CISS’06.

• Part II (with Andrés Ferragut, in CISS’08 paper).Achieving network stability and user fairness through admission control of TCP connections.

,

)( .

Source-destinationpair ("commodity") input rate (bps)Utility

kk

k

k

U xx

Source s(k)

Destinationd(k)

, ( ). Rate Capacity or cost

l

l l l

yc yφ

ii

Links

,

,

:

{ }. (simplex)

fraction of traffic destined to ,

sent to node

d

i

i jd di i j

d

j

α

α α= ∈∆

Router i j

[Paganini, CISS’06, ECC07Mallada-P’ Netcoop ’07]

Congestion control with multipath routing

Congestion prices

( ): ,

, ,( )

0, , d d dd i i j j

j i j L

di jq q p q i dα

∈

= = + ≠∑

[ ]'( ) .)

:

. (e.g., Define: link congestion price or

average price from node to destina ion t

ll l l l ll

di

dpp y y cdt

p

q i d

φ γ= = −•

•

i

jdiq

,di jα

djq

,i jp( )

( ): , ( )average price seen by source k d ks kq q s k=

{ }( ) { }( ),,, ,::= Routers control based on seen prices

i iL jj L

d di i j

d di i j jp qα α π

∈∈+=

Multipath routing control.

.

The projection keeps

First choice (essentially from Gallager '77):

follow negative price gradient. .di

di

di

di i

di i

E

Eα

α

πα β

α

•

−=

∈∆

w

[ ]E wα

Primal congestion control undergradient control of routing fractions

( ) '( )k k k kx x U x qκ = −

: ( ) ( )

with these dynamics the system convergesglobally to the optimum of the SURPLUS proble

maxm,

Theorem:

k kl lS U x yφ= −∑ ∑

[ ],di

di j

di iEα πα β −= LINKS

SOURCES

Traffic splitting

Node pricerecursionSource prices kq Link prices lp

Source rates kx

Link rates ly

Split ratios d

iα

'( )l lp yφ=

Prices seen diπ

[See P' CISS '06], extends Gallager '77.Also in [Xi & Yeh CISS'06], combined with wireless power control.

Dual congestion control undergradient control of routing fractions

[ ],di

di j

di iEα πα β −= LINKS

SOURCES

Traffic splitting

Node pricerecursionSource prices kq Link prices lp

Source rates kx

Link rates ly

Split ratios d

iαPrices seen d

iπ

Convergence to equilibrium?

arg max ( )k k k kx U x q x = −

[ ]l l l l lpp y cγ• += −

( )solution to optimal W ELFARE p

subject to

roble

m,

maEquilibri

x um:

kk y cU x ≤∑

No! Simple examples exhibit harmonic oscillations.

Dual model of queue is more appropriate. Indeed, it predicts correctly oscillation period.••

[ ]( ?1)

Static

Which model is correct for queuing delay:

or integrator l l l ll

l lpp y c

cp yϕ +−= =

Example of instability,packet simulation in ns2 Single source, two bottlenecks.

12

34s

1 1, c p

2 2,c pd

α

1 α−

x

Queues

Source rate

Solving the problem*Adapt based on anticipated (rather than current) price

In control terms, add derivative action. Same equilibrium. Simulations:

di

d di i

diiα π ππ ν= +

2 , 5ν =0ν =

( ) locally

) is:

asymptot arbitrary network globally network o

ically stablef

p

in an asymptotically stable in a

[P'-Mallada, submitted to ToN, CDC'08]the equilibrium max point (optimTheorem

um

s

k kU x••

∑

arallel links.

variantsPacket i of TCP-mplement FAST andation: RIP.

Fernando Paganini Universidad ORT Uruguay

On congestion control, multipathrouting, and admission control

• Part I (with Enrique Mallada). Combined congestion control and node-based multipath routing: new results on stability since CISS’06.

• Part II (with Andrés Ferragut, in CISS’08 paper).Achieving network stability and user fairness through admission control of TCP connections.

( )

User rate

Utility

k

k kU

ρ

ρ

Link rate Capacity

l

l

yc

Stability and user-level fairness

': ,max ( )

LINK CAPACITYCONSTRAINTS

USER UTILITY FUNCTION

subject to k k

k

KELLY s SYSTEMPROBLEM

U R cρ ρ ρ ≤∑

1 if user uses link 0 otherwise lk

k lR

=

Single path routing matrix:

y Rρ=

( )Rate per flowUtility

flows per user.

k

TCPk k

k

xU x

n

→

Contrast with flow-level fairness of TCP

:: ( )max

TCP UTILITY FUNCTION

subject to k

k TCPk kx lk k k lk

TCP NETWORKn U x

PROBLEMR n x c

ρ

≤∑ ∑

' ( )

TCP utility (per flow) determined by the protocol, e.g., fo

=r 2.

TCPk

TCPk k

U

U x x ακ α−=

lc

Without control of number of connections, fairness per flow is moot (Briscoe'07). Incentives to employ many TCP flows (e.g., p2p) . Tragedy of the commons? If we could control , can we induce wkn

••• ith TCP the SYSTEM problem allocation? (similar to "user problem" setting weight parameter in Kelly-Maulloo-Tan '98)

User: Poisson ( ) arrivals, exp( ) workloads.

k

k

λµ →

On stochastic stability of a network served by TCP[deVeciana, Lee, Konstantopoulos ’99, Bonald-Massoulié ’01]

{ }' ( ) 0.

For each fixed service rates determined by

TCP congestion control fo

r

,

k k

TCPk k

n x

U x x ακ α−= >

lc

.

Both stability, and resource allocation depend solely on users'

Remark: congestion control ensures

"open loop" deman

neither stability nor fa

ds

Fairness choice per flow

i

rness.

(e.g., value

k

k

λµ

•

• ),

of has minimal impact. A heavy user will compensate a low TCP rate by increasing

until serves demand, if feasible. If not 's grow without bo s. undk

k k

nn

α

ρ

{ } .Result: Markov chain stable i f and only if klk l

k kkk R c lnn λ

µ< ∀∑

1

,( )

( ' ) .

Assume that for fixed the flow rate is determined by TCP: where is the congestion price seen by the source, and TCP demand curve. The user rate is

, k k

k TCPk k

TCPk TCPk

k

k k k

n xx f q

f Uq

n xρ−=

= =

Closing the loop on for user-level fair sneskn

( )1 .' ( )

Control law forcontinuous :

k

k k k kn U q

n

β ρ−= −

, ,max ( )Objective: control so that the system converges to an equilibrium where

withsolv utie lities defined by users. s s.t.k

k k k k kk

n

n x R cUρρ ρρ= ≤∑

1'kU −

∫kn

+

−

TCPkfkq

kρ

kx

kn ×

kq

NWKOther recent work oncontrolling no. of flows: Chen - Zakhor '06(for TCP over wireless).

Analysis using dual TCP congestion control,

[ ]l l l l lpy cp γ += −LINKS

R

TR

y

q p

ρ

USERS

( )1

.

' ( ) ;;( )

k k k k

k k k

k TCPk k

n U qn x

x f q

β ρρ

−= −

==

max ( ) , subject to , and is locally asymptotically stable. Proof: passivity argument (as in Wen-Arcak '03).

Theorem 1 (arbitrary netwThe equilibrium satisfies

ork). k k

kU R cρ ρ ρ ≤∑

{ } ˆ( ), ( ),

ˆ ˆ( ) ( )

ˆlet

be the and hen the "slow" dynamics

Assume time-scale separation: for fixed equilibrium values from dual congestion control, .

T

Theorem 2 (single bottleneck).

k k

k k k

k n x n

n x nnn n q

ρ =

=

( )*

1

* ˆ ( )ˆˆ' ( ( )) ( ) are globally convergent

to a point where the corresponding are at the optimum welfare point.

k k k k

k nn U q n n

n ρβ ρ−= −

sources'In practice, is discrete (number of TCP connections). Furthermore: Real-time control at ap plication layer) is impractical, incentives? Killing an ongoing TCP connection to reduce is

(k

k

n

n•• undesirable.

From fluid control to admission control.

k

More practical alternative: Control increase of (admit new connections), rely on natural termination.

Admission control carried out by edge rout

( )

er.

User utility describes the SLA: admit k

k

U

n

ρ

•

•

• 1' ( )new connection k k kU q ρ−⇔ >

)Stochastic model. Poisson( ) arrivals, exp( workloads.Active sessions served with rate obtained from the network.

k k

kxλ µ

{ }{ }1

, ,

.

ˆ ˆˆ' ( ( )) ( ) ; ( )1Continuous time Markov chain with state

Transition rates:

q qk

k k k k k kk kn n e n n e

n n

U q n n nλ ρ ρµ−+ −

=

> ==

( ) log( ).Single bottleneck, two users, utility State space and transition rates:

k k kU Kρ ρ=

12 2 2ˆˆ' ( ( )) ( )U q n nρ− =

12 2 2ˆˆ' ( ( )) ( )U q n nρ− =

0

,

Irreducible set around is bounded

Stability,Independentlyof k k

n

λ µ

⇒

=

Fairness? Simulations: JAVA-based tool with random arrivals and workload, simulated dual congestion control.

2

5

' ( )

' ( ) ' .

TCP utility User utility emulate

.

s max-min fairness.

TCPk k

k

U x x

U x x

κ

κ

−

−

=

=

21 3,, ρ ρρ 21 3,, n nn

*

Simulation results show admission control achieves optimal allocation,

provided the offered loads are larger than the equilibrium fairshare

We seek analytical proof, and also understanding of

.kk

k ρλµ

the non-greedy case.

Fluid modeling of admission control.

Optimal point

( )( )( )

0(1)

0( ) .,

Try a large network asymptotic, scaling capacity and user demand curve.

Rescaled simulation plot k k

LLL kc c L

nU U L L

ρρ ρ

= =

Fluid modeling of admission control.

Optimal point

Fluid limit.

Optimal point

*

1' ( ( )) ( )ˆˆˆ ( )1

For (optimal fairshare) fluid simulations converge to optimal point.

We can prove it in simple cas

>

e

s.

kk

k

k k k kk k kU q n n

n nρ

ρλ

λ ρ

µ

µ

− >−=

Fluid limit for the case of non-greedy users:*

1' ( ( )) ( )ˆˆˆ ( );1 assume a certain user has < k

k

kk k k k

k k kU q n nn n

ρρ

λλ ρ µµ

− >−=

1 2( , )n n n=

1

1 0.3,1,Example: user 1 with greedy user 2.c λµ ==

( )

.

, ,max ( )

k

Conjecture (verified in simulations so far): converges to solution of

s.t. where corresponds to a demand curve

saturated at rate Non-greedy user is protected

.

k

k

k kk

k

UR cUρ ρ

ρλ

ρ

µ

≤

=

∑ i

1 2( , )ρ ρ

min ' ( ),

Suppose: edge router can choose in which path to route a new flow. Given: prices of the various candidate routes, a natural policy is:

Admit new connection where

If

ad

rr k k k k

r

rk

rk U

qq ρ ρ ρ⇔

•

< =• ∑mitted, select cheapest path.

Back to multipath, work in progress.

Example (symmetric users):

Multipath admission control Simulations 12ρ

1211 ),( ρρ 2322( , )ρρ

23ρ

22ρ

11ρ

, ,max ( ) subject to

Conjecture: converges to optimal multipath welfare allocation,

.rk k l

r

rk k k

k k r lcUρ ρ ρρ ρ

∋= ≤∑∑ ∑∑

Conclusions and Future Work• We studied two cross-layer resource allocation problems:

I. Congestion control and multipath routing.II. Congestion control and admission control

• Objective: welfare optimization. We designed decentralizedcontrol laws based on prices, achieve these equilibria.

• Dynamic analysis: local stability proofs in arbitrary networks, global results for simpler cases.

• Beware on simplistic models for delay! • Simulation studies confirm and generalize the above theory. • Future work:

– Part I: Global proof in arbitrary networks, loss-based implementation.– Part II: Stability proofs for fluid limit model, ns2 implementation.– Combination: multipath admission control.