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Resource sharing in networks
Frank KellyUniversity of Cambridge
www.statslab.cam.ac.uk/~frank
Chinese University of Hong Kong8 February 2010
Outline
• Fairness in networks
• 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
Notation
JRj∈rxr
Ur(xr )Cj
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
)(
≥≤
∑∈
xoverCAxtosubject
xUMaximize rRr
r
Maximize aggregate utility, subject to capacity constraints
SYSTEM(U,A,C):
The user problem
0≥
−⎟⎟⎠
⎞⎜⎜⎝
⎛
r
rr
rr
wover
wwUMaximizeλ
User r chooses an amount to pay per unit time, wr ,
and receives in return a flow xr =wr /λr
USERr(Ur;λr):
0
log
≥≤
∑∈
xoverCAxtosubject
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 problemsK 1997,Johari, Tsitsiklis 2005, Yang, Hajek 2006
Max-min fairnessRates {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 fairnessRates {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
≤ 0r∈R∑
Weighted proportional fairness
A feasible set of rates {xr} are such that are weighted proportionally fair
if, for any other feasible rates {yr},
wr
yr − xr
xr
≤ 0r∈R∑
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) withw = 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
RrpxwAxCp
CAxp
rjjrr
T
∈=
=−
≤≥
∑∈
,0)(
,0 (feasibility)(complementary slackness)(endowments spent)
Solution solves NETWORK(A,C;w)
r
r
xn - number of flows on route r
- rate of each flow on route rGiven the vector how are the rates chosen ?
),(),(
RrxxRrnn
r
r
∈=∈=
Optimization formulationof rate control
Various forms of fairness, can be cast in an optimization framework. We’ll illustrate, for the rate control problem.
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.
1,1Henceforth
== wα
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
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
Multipath routingSuppose a source-destination pair has access to several routes across the network:
resource
routesource
destination
srS
∈- set of source-destination pairs- route r serves s-d pair s
Example of multipath routing
1C
2C
3C
3C1n
2n 3n
Routes, as well as flow rates, are chosen to optimize
over source-destination pairs s)log( ss
s xn∑
213 ,CCC <
First cut constraint
1C
2C
3C
3C1n
2n 3n
212211 CCxnxn +≤+
Cut defines a single pooled resource
Second cut constraint
1C
2C
3C
3C1n
2n 3n
3331121 Cxnxn ≤+
Cut defines a second pooled resource
Product formRrwr ∈== ,1,1α
In heavy traffic, and subject to some technical conditions, the (scaled) components of the shadow prices p for the pooled resources are independent and exponentially distributed. The corresponding approximation for n is
where SsApn
jjsjss ∈≈ ∑ρ
Dual random variables are independent and exponential
JjACps
sjsjj ∈− ∑ )Exp(~ ρ
Kang, K, Lee and Williams 2009
Outline
• Fairness in networks
• Rate control in communication networks (relatively well understood)
• Philosophy: optimization vs fairness
• Ramp metering (very preliminary)
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
Final remarks
• Often the networks we design are part of a larger system, where agents are optimizing their own actions
• Sharing resources fairly may, in certain circumstances, lead to near optimal behaviour of the larger system, if it exposes agents to appropriate shadow prices
• The proportional fairness criterion can give the appropriate shadow prices