fernando paganini ort university, uruguay (on leave from ucla) congestion control with adaptive...
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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
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.