Optimization-based approach to congestion control Resource - - PowerPoint PPT Presentation

Optimization-based approach to congestion control

Resource allocation as optimization problem:

how to allocate resources (e.g., bandwidth) to

• ptimize some objective function

may not possible that optimality exactly obtained

but…

• ptimization framework as means to explicitly

steer network towards desirable operating point

practical congestion control as distributed

asynchronous implementations of optimization algorithm

systematic approach towards protocol design

Model

network:

links {l}, capacities {cl}

sources S: (L(s), Us(xs)), s∈S

L(s) - links used by source s Us(xs) – utility, strictly concave function

• f source rate xs

Us(xs) xs

User 0 User 1 User 2

) ( 1

1 x

U ) ( 0

0 x

U ) ( 2

2 x

U

cA cB

Kelly’s System Problem

subject to

Optimization Problem

L l c x x U

l S s l s s s s xs

∈ ∀ ≤

∑ ∑

∈ ≥

, subject to ) ( max

) (

“system” problem

maximize system utility (note: all sources “equal) constraint: bandwidth used less than capacity centralized solution to optimization impractical

must know all utility functions impractical for large number of sources we’ll see: congestion control as distributed

asynchronous algorithms to solve this problem

Issues

will users truthfully reveal their utility

functions?

if not, can we design a pricing scheme

(mechanism) to induce truth-telling?

is there a distributed algorithm to compute

the prices?

what are good choices for utilities?

Max-min Fairness

rates {xr} max-min fair if for any other feasible rates {yr}, if ys > xs, then ∃ p, such that xp ≤xs and yp < xp

Proportional fairness

rates {xr} are proportionally fair if for any

feasible {yr},

corresponds to Ur (xr) = log xr weighted proportional fairness if Ur (xr) =

wr log xr

≤ −

∑

∈S r r r r

x x y ≤ −

∑

∈S r r r r r

x x y w

Minimum potential delay fairness

rates {xr} are minimum potential delay fair

if Ur (xr) = -wr/xr Interpretation: if wr is file size, then wr/xr is transfer time; optimization problem is to minimize sum of transfer delays

Max-min Fairness

rates {xr} max-min fair if for any other feasible rates {yr}, if ys > xs, then ∃ p, such that xp ≤xs and yp < xp What is corresponding utility function?

α

α α

− =

− ∞ → 1

lim ) (

1 r r r

x x U

Computing Source Rates

Remove constraints

consider following problem fl(y) – penalty function

fl () non decreasing, continuous and

∑∫ ∑

∈

∑ − =

L l x l r r r

s

dy y f x U V ) ( ) ( ) (x ∞ → ∞ →

∫

y as x f

y l

) (

max V(x)

L l x y S r y f x U S r x V

s l s s l r l l l l r r r

∈ = ∈ = − ∈ = ∂ ∂

∑ ∑

∈ ∈

, , ) ( ) ( ' ,

: :

pl (t) price of link l at time t

pl (t) = fl(yl(t)) U’r(xr) – qr = 0, r∈S

Source Algorithm

source needs only its path price: kr() nonnegative nondecreasing function above algorithm converges for any initial

condition to unique solution

example: qr – loss/marking probability

) ) ( ' )( (

r r r r r r

q x U x k x − = &

Proportionally-Fair Controller

If utility function is

then a controller that implements it is given by

Proportionally-Fair Controller

If utility function is

then a controller that implements it is given by

Computing Source Rates

Computing Lagrange Multipliers

define dual problem:

Dual Algorithm

• pl delay at link l
• TCP-Vegas: modify source rates in

response to measured delay

Dual algorithm

converges to optimum rates

Primal-Dual Algorithm

source can be TCP-Reno feedback generated by active queue management

algorithms

Active Queue Management

feedback function of queue length bl

Random Early Detection (RED)

Simplified view:

Random Early Marking (REM)

Exponential-RED

Pricing

can network choose pricing scheme to

achieve fair resource allocation?

suppose network charges price qr ($/bit)

where qr=∑ pl

user’s strategy: spend wr ($/sec.) to

maximize

Optimal User Strategy

equivalently,

Distributed Computation

with optimal choice of wr, controller

becomes

We have already seen that this solves

Price Takers vs. Strategic Users

Kelly Mechanism: users are price takers,

i.e., user does not know the impact of its action on the price

strategic users:

Efficiency and Competition

price takers: selfish users can maximize

social welfare

strategic users: competition leads to loss

• f efficiency, i.e., social welfare is not

maximized

TCP-Reno

condition for optimality is

• r

if we have an expression for xi, we can use to

• btain Ui

) ( ' = − p x U

i i

) (

1 '

p U x

i i −

=

TCP-Reno

TCP-Reno in equilibrium: utility function:

Simplified TCP-Reno

suppose then, interpretation: minimize (weighted) delay

Price versus Probabilistic Feedback

price: qi = ∑l2 i pl loss probability: qi=1-∏l2 i(1-pl). If the pl’s

are small, they are approximately equal

else, TCP solves a modified version of the

resource allocation problem

Explicit Congestion Notification (ECN)

Instead of dropping packets, mark packets

to indicate incipient congestion

Marking: Router flips a bit in the packet

header from 0 to 1 to indicate congestion

Destination echoes the ECN bit back to

the source in the ack

Multicast

single rate

sender determines rate U(x) = |R|V(x) preferred

multirate

each rcvr determines rate each rcvr has own utility function

Other issues

joint congestion control and routing high performance environments

Optimization-based congestion control: summary

bandwidth allocation as optimization problem: practical congestion control (TCP) as distributed

asynchronous implementations of optimization algorithm

• ptimization framework as means to explicitly steer

network towards desirable operating point

systematic approach towards protocol design