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.
