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