Congestion control algorithms Frank Kelly, Cambridge (joint work - - PowerPoint PPT Presentation

Congestion control algorithms

Frank Kelly, Cambridge (joint work with Ruth Williams, UCSD) Workshop on Algorithmic Game Theory DIMAP, Warwick, March 2007

Fluid model for a network operating under a fair bandwidth- sharing policy K & W Ann Appl Prob 2004 On fluid and Brownian approximations for an Internet congestion control model. W. Kang, K, N.H. Lee & W CDC 2004 State space collapse and diffusion approximation…

• W. Kang, K, N.H. Lee & W forthcoming

End-to-end congestion control

Senders learn (through feedback from receivers)

• f congestion at queue, and slow down or speed

up accordingly. With current TCP, throughput of a flow is proportional to

senders receivers

) /( 1 p T T = round-trip time, p = packet drop probability.

(Jacobson 1988, Mathis, Semke, Mahdavi, Ott 1997, Padhye, Firoiu, Towsley, Kurose 1998, Floyd and Fall 1999)

Model definition

• We want to describe a network model,

with fluctuating numbers of flows

• We first need

– notation for network structure – abstraction of rate allocation

• Then we need to define the random nature
• f flow arrivals and departures

Network structure (J, R, A)

1 = =

jr jr

A A R J

• set of resources
• set of routes
• if resource j is on route r
• otherwise

resource route

Rate allocation

r r r

x n w

• weight of route r
• number of flows on route r
• rate of each flow on route r

Given the vector how are the rates chosen ?

) , ( ) , ( R r x x R r n n

r r

∈ = ∈ =

Optimization formulation

) (n x x =

α

α

−

−

∑

1

1 r r r r

x n w

subject to R r x J j C x n A

r j r r r jr

∈ ≥ ∈ ≤

∑

maximize Suppose is chosen to (weighted -fair allocations, Mo and Walrand 2000)

α

∞ < <α α

α

−

−

1

1 r

x

(replace by if )

) log(

r

x 1 = α

R r n p A w x

j j jr r r

∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ∑

α / 1

) (

) (n p j

• shadow price (Lagrange multiplier)

for the resource j capacity constraint Observe alignment with square-root formula when

∑

≈ = =

j j jr r r r

p A p T w , / 1 , 2

2

α

Solution

• maximum flow
• proportionally fair
• TCP fair
• max-min fair

) 1 ( ) / 1 ( 2 ) 1 ( 1 ) 1 (

2

= ∞ → = = = → = → w T w w w

r r

α α α α

Examples of -fair allocations α

α

α

−

−

∑

1

1 r r r r

x n w

subject to

R r x J j C x n A

r j r r r jr

∈ ≥ ∈ ≤

∑

maximize

R r n p A w x

j j jr r r

∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ = ∑

α / 1

) (

Example

1 1

J j C R r w n

j r r

∈ = ∈ = = 1 , 1 , 1

maximum flow:

1/3 2/3 2/3 1/2 1/2 1/2

max-min fairness: proportional fairness:

∞ → α → α 1 = α

Flow level model

Define a Markov process with transition rates

) ), ( ( ) ( R r t n t n

r

∈ = 1 1 − → + →

r r r r

n n n n

at rate at rate

R r n x n R r

r r r r

∈ ∈ μ ν ) (

• Poisson arrivals, exponentially distributed file sizes
• let

the load on route r

, R r

r r r

∈ = μ ν ρ

Suppose vertical streams have priority: then condition for stability is and not

) 1 ( ) 1 (

2 1

ρ ρ ρ − − < } 1 , 1 min{

2 1

ρ ρ ρ − − <

2

ρ ρ

1

ρ

C=1 C=1

Stability? (i.e. positive recurrence?)

(Bonald and Massoulie 2001)

Fairness leads to stability

Suppose

J j C A

j r r jr

∈ <

∑

ρ

is positive recurrent (De Veciana, Lee and Konstantopoulos 1999; Bonald and Massoulie 2001). Then the Markov process

) ), ( ( ) ( R r t n t n

r

∈ =

and resource allocation is weighted -fair.

α

Heavy traffic

We’re interested in what happens when we approach the edge of the achievable region, when

J j C A

j r r jr

∈ ≈

∑

ρ

Balanced fluid model

J j C A

j r r jr

∈ =

∑

ρ

Suppose and consider differential equations R r n n x n t t n

r r r r r r

∈ > − = ) ( ) ( d ) ( d μ ν First let’s substitute for the values

• f , to give:

R r n xr ∈ ), (

R r n p A w n t t n

j j jr r r r r r

∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ − =

∑

α

μ ν

/ 1

) ( d ) ( d =

r

n

( care needed when ). Thus, at an invariant state,

R r w n p A n

r j j jr r r r

∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ =

∑

α

μ ν

/ 1

) (

State space collapse

The following are equivalent:

• n is an invariant state
• there exists a non-negative vector p

with R r w p A n

r j j jr r r r

∈ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎝ ⎛ =

∑

α

μ ν

/ 1

Thus the set of invariant states forms a J dimensional manifold, parameterized by p.

Workloads

∑

=

r r r jr j

n A n W μ ) ( Let the workload for resource j, and let Define diagonal matrices ) , ( ), , / ( R r w diag w R r diag

r r r

∈ = ∈ = μ ν ρ Then W lies in the polyhedral cone } , : {

1 1

≥ =

− −

p p A w A W W

T

ρ μ

1 = α

Example

R r wr

r

∈ = = ∞ < < , 1 , 1 μ α

slope

2

ρ ρ ρ +

1

ρ ρ ρ +

slope

Each bounding face corresponds to a resource not working at full capacity Entrainment: congestion at some resources may prevent other resources from working at their full capacity.

1

W

2

W

2 =

p

1 =

p

Stationary distribution?

1

W

2

W

2 =

p

1 =

p

1

p

2

p

Look for a stationary distribution for W, or equivalently, p. Williams (1987) determined sufficient conditions, in terms of the reflection angles and covariance matrix, for a SRBM in a polyhedral domain to have a product form invariant distribution – a skew symmetry condition

Local traffic condition

⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ ⎛ = 1 1 1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Assume the matrix A contains the columns of the unit matrix amongst its columns: i.e. each resource has some local traffic -

Product form under proportional fairness

R r wr ∈ = = , 1 , 1 α

Under the stationary distribution for the reflected Brownian motion, the (scaled) components of p are independent and exponentially distributed. The corresponding approximation for n is where

R r p A n

j j jr r r

∈ ≈

∑

ρ J j A C p

r r jr j j

∈ −∑ ) Exp( ~ ρ

Dual random variables are independent and exponential!

Multipath routing

1

C

2

C

3

C

3

C

1

ν

2

ν

3

ν

1

μ

2

μ

3

μ

Routes, as well as flow rates, are chosen to optimize

α

α

−

−

∑

1

1 s s s s

x n w

• ver source-sink pairs s

First cut constraint

1

C

2

C

3

C

3

C

1

ν

2

ν

3

ν

1

μ

2

μ

3

μ

2 1 2 1

C C + ≤ + ρ ρ

Second cut constraint

1

C

2

C

3

C

3

C

1

ν

2

ν

3

ν

1

μ

2

μ

3

μ

3 3 1

2 1 C ≤ + ρ ρ

Generalized cut constraints

In general, stability requires

J j C A

j s s js

∈ <

∑

ρ

• a collection of generalized cut constraints.

Provided contains a unit matrix, we again have the approximation where

A

S s p A n

J j j js s s

∈ ≈

∑

∈

ρ

J j A C p

s s js j j

∈ −∑ ) Exp( ~ ρ

Again independent dual random variables, now

• ne for each generalized cut constraint!