Brownian network models of ramp metering

Frank KellyUniversity of Cambridge

www.statslab.cam.ac.uk/~frank

Conference on Stochastic Processing Networks in Honor of J. Michael Harrison

Outline

• Rate control in communication networks (relatively well understood)

• Philosophy: optimization vs fairness

• Ramp metering (very preliminary)

Network structure

0

1

=

=

jr

jr

A

ARJ - set of resources

- set of routes - if resource j is on route r- otherwise

resource

route

Rate allocation

r

r

xn - number of flows on route r

- rate of each flow on route r

Given the vector how are the rates chosen ?

),(),(

RrxxRrnn

r

r

∈=∈=

Optimization formulation

)(nxx =

α

α

−

−

∑ 1

1r

rr

rxnw

subject to

Rrx

JjCxnA

r

jrrr

jr

∈≥

∈≤∑0

maximize

Suppose is chosen to

(weighted -fair allocations, Mo and Walrand 2000)α

∞<<α0α

α

−

−

1

1rx(replace by if ) )log( rx 1=α

RrnpA

wx

jjjr

rr ∈

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=∑

α/1

)(

)(npj - shadow price (Lagrange multiplier) for the resource j capacity constraint

Solution

JjxnACnp

Jjnp

RrxJjCxnA

rr

rjrjj

j

rjrr

rjr

∈≥⎟⎠

⎞⎜⎝

⎛−

∈≥

∈≥∈≤

∑

∑

0)(

0)(

0;where

KKT conditions

- maximum flow - proportionally fair- TCP fair - max-min fair)1(

)/1(2

)1(1)1(0

2

=∞→==

=→=→

wTw

ww

rr

αα

αα

Examples of -fair allocations α

α

α

−

−

∑ 1

1r

rr

rxnw

subject to

Rrx

JjCxnA

r

jrr

rjr

∈≥

∈≤∑0

maximize

RrnpA

wx

jjjr

rr ∈

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=∑

α/1

)(

Example

0

1 1

JjCRrwn

j

rr

∈=∈==

1,1,1

maximum flow:

1/3

2/3 2/3

1/2

1/2 1/2

max-min fairness:

proportional fairness:

∞→α

0→α

1=α

Source: CAIDA,Young Hyun

Source: CAIDA -Young Hyun,Bradley Huffaker(displayed at MOMA)

Flow level model

Define a Markov processwith transition rates

)),(()( Rrtntn r ∈=

11

−→+→

rr

rr

nnnn at rate

at rate RrnxnRr

rrr

r

∈∈

μν

)(

- Poisson arrivals, exponentially distributed file sizes

Roberts and Massoulié 1998

If JjCA jrr

jr ∈<∑ ρ

Stability

De Veciana, Lee & Konstantopoulos 1999; Bonald & Massoulié 2001

then the Markov chainis positive recurrent

)),(()( Rrtntn r ∈=

Rrr

rr ∈=

μνρLet

Heavy traffic: balanced fluid model

The following are equivalent:• n is an invariant state • there exists a non-negative vector p with

RrpAnj

jjrr

rr ∈= ∑μ

ν

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

1=α

Example

Rrr ∈= ,1μ

slope0

02

ρρρ +

01

0

ρρρ+slope

Each bounding face corresponds to a resource not working at full capacityEntrainment: congestion at some resources may prevent other resources from working at their full capacity. 1W

2W

02 =p

01 =p

Stationary distribution?

1W

2W

02 =p

01 =p

1p

2p

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

⎟⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜⎜

⎝

⎛

=

1000001000001000001000001

.....

.....

.....

.....

.....

.....

.....

.....

.....

.....

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

Rrwr ∈== ,1,1αUnder the stationary distribution for the reflected Brownian motion, the (scaled) components of pare independent and exponentially distributed.The corresponding approximation for n is

where

RrpAnj

jjrrr ∈≈ ∑ρ

JjACpr

rjrjj ∈−∑ )Exp(~ ρ

Dual random variables are independent and exponential

Kang, K, Lee and Williams 2009

What we've learned about highway congestionP. Varaiya, Access 27, Fall 2005, 2-9.

Data, modelling and inference in road traffic networksR.J. Gibbens and Y. SaatciPhil. Trans. R. Soc. A366 (2008), 1907-1919.

A linear network

0,d))(()()0()(0

≥Λ−+= ∫ tssmtemtmt

iiii

cumulative inflow

queue size

metering rate

Metering policy

0,0d))(()()0()(0

≥≥Λ−+= ∫ tssmtemtmt

iiii

and such that

0,0,0

,

==Λ∈≥Λ

∈≤Λ∑

ii

i

jii

ji

mIi

JjCA

Suppose the metering rates can be chosen to be any vector satisfying)(mΛ=Λ

Optimal policy?

For each of i = I, I-1, …… 1 in turn choose

0d))((0

≥Λ∫t

i ssm

to be maximal, subject to the constraints.

This policy minimizes

)(tmi

i∑for all times t

Proportionally fair metering

ii

im Λ∑ log

subject to

0,0,0

,

==Λ∈≥Λ

∈≤Λ∑

ii

i

jii

ji

mIi

JjCA

maximize

Suppose is chosen to)),(()( Iimm i ∈Λ=Λ

IiAp

mm

jjij

ii ∈=Λ

∑,)(

jp - shadow price (Lagrange multiplier) for the resource j capacity constraint

JjACp

Jjp

JjCAIi

ii

jijj

j

jii

ji

i

∈≥⎟⎠

⎞⎜⎝

⎛Λ−

∈≥

∈≤Λ

∈≥Λ

∑

∑

,0

,0

,,0where

KKT conditions

Proportionally fair metering

Brownian network model

Then is a J-dimensional Brownian motion starting from the origin

with drift

and variance

)),(()( JjtXtX j ∈=

Suppose that is a Brownian motion, starting from the origin, with drift and variance . Let

CA −=− ρθ

)0),(( ≥ttei

2σρi

iρ

tCteAtX jii

jij −=∑ )()(

'][2 AA ρσ=Γ

Brownian network model

.0 allfor )(d})({)()(

,0)0( with ,decreasing-non and continuous is )(

such that process ldimensiona-one a is ,each for )()( paths, continuous has)(

0allfor )()()()(:ipsrelationsh following by the )( Define

}.0,:'][{ and

'][ Let

0≥∈=

=

∈∈≥+=

=∈=

=

∫

+

+

tsUsWItUb

UUa

UJjiiitWWiittUtXtWi

tW

qqAA

AA

j

t jj

jj

j

jJj

J

W

W

RW

RW

ρ

ρ

Brownian network model

Jjj ∈,2

2

θσ

If then there is a unique stationary distribution W under which the components of

are independent, and is exponentially distributed with parameter

and queue sizes are given by

,,0 Jjj ∈>θ

WAAQ 1)'][( −= ρ

jQ

QAM '][ρ=

Delays

Let

- the time it would take to process the work in queue i at the current metered rate. Then

)()(

mmmDi

ii Λ

=

∑=j

jiji AQMD )(

1Q21 QQ +

321 QQQ ++

A tree network

A tree network

541 QQQ ++

Route choices

Route choices

4Q

43 QQ +

Route choices

4Q

43 QQ +

432 QQQ ++

4321 QQQQ +++

Route choices

4Q

43 QQ +

432 QQQ ++

4321 QQQQ +++

)Exp(2

~ 2121

2

2 ρρσ−−+CCQ

Brownian network model

• As Mike has cogently argued, for many applications it may be easier to describe the workload arrival process in terms of the mean and variance of a Brownian motion.

• If relevant time periods long enough, negative increments less likely.

• The Brownian model exposes structure in a way that more detailed models (e.g. MDP models) do not.