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Network control
Frank KellyUniversity of Cambridge
www.statslab.cam.ac.uk/~frank
Conference on Information Science and SystemsPrinceton, 22 March 2006
Outline
• Example: end-to-end congestion control in the Internet– square-root formula
• Control of elastic network flows– utilities, fairness and optimization formulation
• Stability, under– delays– noise – fluctuating demand
End-to-end congestion control
Senders learn (through feedback from receivers) of congestion at queue, and slow down or speed up accordingly. With current TCP, throughput of a flow is proportional to
senders receivers
)/(1 pT
T = round-trip time, p = packet drop probability. (Jacobson 1988, Mathis, Semke, Mahdavi, Ott 1997, Padhye, Firoiu, Towsley, Kurose 1998, Floyd and Fall 1999)
Control of elastic network flows
user
resource
flow
• How should available resources be shared between competing streams of elastic traffic?• Conceptual problem: fairness• Pragmatic problem: stability of rate control algorithm
utility,U(x)
rate, x
utility,U(x)
rate, x
Elastic flows
elastic traffic - prefers to share(Shenker)
inelastic traffic - prefers to randomize
Network structure (J, R, A)
0
1
jr
jr
A
A
R
J - set of resources- set of routes - if resource j is on route r- otherwise
resource
route
Notation
J
R
jr
xr
Ur (xr )
C j
Ax C
- set of resources- set of users, or routes - resource j is on route r- flow rate on route r - utility to user r- capacity of resource j- capacity constraints
resource
route
The system problem
0
)(
xover
CAxtosubject
xUMaximize rRr
r
Maximize aggregate utility, subject to capacity constraints
SYSTEM(U,A,C):
The user problem
0
r
rr
rr
wover
ww
UMaximize
User r chooses an amount to pay per unit time, wr ,
and receives in return a flow xr =wr /r
USERr(Ur;r):
0
log
xover
CAxtosubject
xwMaximize rRr
r
As if the network maximizes a logarithmic utility function, but
with constants {wr} chosen by the users
NETWORK(A,C;w):
The network problem
Problem decomposition
Theorem: the system problem
may be solved
by solving simultaneously
the network problem and
the user problemsJohari, Tsitsiklis 2005, Yang, Hajek 2006
Max-min fairness
Rates {xr} are max-min fair if they are feasible:
CAxx ,0
and if, for any other feasible rates {yr},
rssrr xxysxyr ::Rawls 1971, Bertsekas, Gallager 1987
Proportional fairness
Rates {xr} are proportionally fair if they are feasible:
CAxx ,0
and if, for any other feasible rates {yr}, the aggregate of proportional changes is negative:
yr xr
xr
0rR
Weighted proportional fairness
A feasible set of rates {xr} are such that are weighted proportionally fair
if, for any other feasible rates {yr},
w r
yr xr
xr
0rR
Fairness and the network problem
Theorem: a set of rates {xr} solves the network problem,
NETWORK(A,C;w), if and only if the rates are
weighted proportionally fair
Bargaining problem (Nash, 1950)
Solution to NETWORK(A,C;w) with w = 1 is unique point satisfying
• Pareto efficiency
• Symmetry
• Independence of Irrelevant Alternatives
(General w corresponds to a model with unequal bargaining power)
Market clearing equilibrium (Gale, 1960)
Find prices p and an allocation x such that
Rrpxw
AxCp
CAxp
rjjrr
T
,
0)(
,0 (feasibility)(complementary slackness)(endowments spent)
Solution solves NETWORK(A,C;w)
Fairness criteria
We’ve seen three fairness criteria:
• proportional fairness
• max-min fairness
• TCP-fairness (as described by the square-root formula)
Can we unify these fairness criteria within a single framework?
1
1r
rr
xw
subject to
Rrx
JjCxA
r
jrr
jr
0
maximize
Optimization formulation
Suppose the allocation x is chosen to
(weighted -fair allocations, Mo and Walrand 2000)
Rrp
wx
rjj
rr
/1
jp - shadow price (Lagrange multiplier) for the resource j capacity constraint
Observe alignment with square-root formula when
rj
jrrr ppTw ,/1,2 2
Solution
- maximum flow - proportionally fair- TCP fair - max-min fair)1(
)/1(2
)1(1
)1(0
2
w
Tw
w
w
rr
Examples of -fair allocations
1
1r
rr
xw
subject to
Rrx
JjCxA
r
jrr
jr
0
maximize
Rrp
wx
rjj
rr
/1
Example
0
1 1
JjC
Rrwn
j
rr
1
,1,1
maximum flow:
1/3
2/3 2/3
1/2
1/2 1/2
max-min fairness:
proportional fairness:
0
1
Rate control algorithms
• Can rate control algorithms be interpreted within the optimization framework?
• Several types of algorithm possible, based on, e.g.,– congestion indication using increase and
decrease rules at sources– explicit rates determined by shadow prices
estimated by resourcesLow, Srikant 2004, Srikant 2004
Network structure
)(
)(
t
tx
rj
R
J
j
r
- set of resources- set of routes - resource j is on route r- flow rate on route r at time t - rate of congestion indication, at resource j at time t
resource
route
A primal algorithm
sjssjj
rjjrrrrr
txpt
ttxwtxtxdt
d
:
)()(
)()())(()(
xr(t) - rate changes by linear increase, multiplicative decreasepj(.) - proportion of packets marked as a function of flow through resource
Global stability
Theorem: the above dynamical system has a stable point to which all
trajectories converge. The stable point is proportionally fair with respect to
the weights {wr}, and solves the network problem, when
j
jj
Cx
Cxxp
0)(
K, Maulloo, Tan 1998
What’s missing?
The global stability results ignore:
• time delays
• stochastic effects
General TCP-like algorithm
Source maintains window of sent, but not yet acknowledged, packets - size cwnd
On route r,
• cwnd incremented by ar cwnd n on positive acknowledgement
• cwnd decremented by br cwnd m for each congestion indication (m>n)
• ar = 1, br = 1/2, m=1, n= -1 corresponds to Jacobson’s TCP
xTcwnd
Differential equations with delays
)()(
)(11)(
)())(())(1())((.
)()(
:rj
rjrrjj
rjjrjr
rm
rrrrn
rrr
r
rrr
Ttxpt
Ttt
tTtxbtTtxa
T
Ttxtx
dt
d
rjrrj TTT r
j
Equilibrium point
Rrb
a
Tx
nm
r
r
r
r
rr
/111
• ar = 1, br = 1/2, m=1, n= -1 corresponds to Jacobson’s TCP, and recovers square root formula
• But what is the impact of delays on stability? Can we choose m, n,… arbitrarily?
Delay stability
2
)(),()( nrrrjj Txaxpxpx
Equilibrium is locally stable if there exists a global constant such that
condition on sensitivity for each resource j
condition on aggressiveness for each route r
Johari, Tan 1999, Massoulié 2000, Vinnicombe 2000,Paganini, Doyle, Low 2001
Consequences?
2
)( nrr cwnda
• n = -1 (Jacobson’s TCP)instability if congestion windows are too small, sluggishness if congestion windows are too big
• n = 0 (Scalable TCP, Vinnicombe, T. Kelly)condition independent of size of congestion window,and choice of br can remove round-trip time bias
Delay stability condition:
Stochastic stability
The signalling of congestion is noisy:for example, the receiver might see---1--0--0-----0------0----------1------0--
receiver
Variance
)1)((2))((Var
11
r
mr
mrr
r nm
Txbtx
If m=1, then: • variance does not depend on Tr • coefficient of variation (= standard deviation/mean)does not depend on xr - scale invariance
Ott 1999, K 2003,Baccelli, McDonald, Reynier 2002
ar = a, n = 0, m = -1 is attractive, giving both delay stability and stochastic stability
Suggestion on single path congestion control?
Routing? Can we extend the algorithm to allow load
balancing across routes? Danger: route flap.
resource
routesource
destinationsr
S
- set of source-destination pairs- route r serves s-d pair s
Combined rate control and routing algorithm
On route r • xr(t) increased by a / Tr
on positive acknowledgement
• xr(t) decreased by br ys(r)(t) / Tr for each congestion indication, where
is rate of returning acknowledgements for
s-d pair s at time t• s = {r} corresponds to Scalable TCP
sr
rrs Ttxty )()(
Delay stability
2)1(),()(
axpxpx jj
Equilibrium is locally stable if there exists a global constant such that
condition on sensitivity for each resource j
condition on aggressiveness of sources
Han, Shakkottai, Hollot, Srikant, Towsley 2003, K, Voice 2005
Delay stability
2)1(),()(
axpxpx jj
Equilibrium is locally stable if there exists a global constant such that
condition on sensitivity for each resource j
condition on aggressiveness of sources
Han, Shakkottai, Hollot, Srikant, Towsley 2003, K, Voice 2005
impact of routing
Suggestion on routing?
• Stable, scalable load balancing across paths, based on end-to-end measurements, can be achieved on the same time-scale as rate control
• For load balancing, the key constraint on the responsiveness of each route is the round-trip time of that route.
• While it is natural for structural information to be provided by the network layer, load balancing is more naturally part of the transport layer.
Han, Shakkottai, Hollot, Srikant, Towsley 2003, K, Voice 2005Key, Massoulié, Towsley 2006
Chiang, Low, Calderbank, Doyle 2003
Example: fair dual algorithm
rj jrj
rr
jsjsjs
sjjj
Tt
wtx
CTtxttdt
d
)()(
)()()(:
A sufficient condition for delay stability:
jT�
average round-trip time of packets through resource j
2
jjj TC�
Low, Lapsley 1999Katabi, Handley, Rohrs 2002Liu, Basar, Srikant 2003, K 2003
Stochastic stability
For a single resource
• neither coefficient of variation depends on • scale invariance with respect to• promising for ad-hoc networks, where there are no natural scalings for prices (of battery power, bandwidth, etc )
22
2
1))((Var
2
1))((Var rjrjjj xtxt
jrx
Flow level model
Define a Markov processwith transition rates
)),(()( Rrtntn r
1
1
rr
rr
nn
nn at rateat rate Rrnxn
Rr
rrr
r
)(
- Poisson arrivals, exponentially distributed file sizes - model originally due to Roberts and Massoulié 1998
where is an -fair allocation)),(( Rrnxr
Suppose JjCA jr
r
rjr
is positive recurrent
Stability
Then Markov process )),(()( Rrtntn r
De Veciana, Lee, Konstantopoulos 1999, Bonald, Massoulié 2001, Ye 2003,Kang, K, Lee, Williams 2004,Lin, Shroff, Srikant 2006, Massoulié 2006
Suppose vertical streamshave priority: then condition for stability is
and not
)1()1( 210
}1,1min{ 210
2
01
C=1C=1
What goes wrong without fairness?
(Bonald and Massoulié 2001)
Conclusions
• End-to-end congestion control can be viewed as a distributed algorithm that:
solves an optimization problem computes a fair allocation finds a market clearing equilibrium• Choices of distributed algorithm are limited by
the requirement of stability under delays noise fluctuating load
Further information and references are available at:
www.statslab.cam.ac.uk/~frank