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Resource sharing in networks
Frank KellyUniversity of Cambridge
www.statslab.cam.ac.uk/~frank
IMA MATHEMATICS 2010London, April 2010
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Outline
• The processor sharing queue• Sharing in networks –
proportional fairness• Multipath routing• Markov chain description,
and heavy traffic• Ramp metering• Resource pooling
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Processor sharing discipline
• Often attractive in practice, since gives – rapid service for
short jobs– the appearance of a processor continuously
available (albeit of varying capacity)• Tractable analytically –
a symmetric discipline.
E.g. for M/G/1 PS
[ ]ρ−
=C
xxSE ,sizejob,timesojourn
Kleinrock, 1967, 1976; Boxma tutorial, informs 2005
(similar tractability for LCFS, Erlang loss system, networks of
symmetric queues)
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The M/G/1 processor sharing queue
t
number in system
remaining service requirement
t
x
n
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The M/G/1 processor sharing queue
[ ]
[ ] )(1.
)/1(
xoC
nxxS
xoC
xxS
++
≅
+−
≅ρ
t
number in system
remaining service requirement
t
x
if x is large;
if x is small, where nis a geometric random variable.
n
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The M/G/1 processor sharing queue
[ ]
[ ] )(1.
)/1(
xoC
nxxS
xoC
xxS
++
≅
+−
≅ρ
t
number in system
remaining service requirement
t
x
if x is large;
if x is small, where nis a geometric random variable.
n
n
CCn ⎟
⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −=
ρρπ 1)(
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The M/G/1 processor sharing queue
[ ]
[ ] )(1.
)/1(
xoC
nxxS
xoC
xxS
++
≅
+−
≅ρ
t
number in system
remaining service requirement
t
x
if x is large;
if x is small, where nis a geometric random variable.
n
in both cases, of course![ ]ρ−
=C
xxSE
n
CCn ⎟
⎠⎞
⎜⎝⎛⎟⎠⎞
⎜⎝⎛ −=
ρρπ 1)(
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What is the network equivalent?
0
1
=
=
jr
jr
A
ARJ - set of resources
- set of routes - if resource j is on route r- otherwise
resource
route
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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
∈=∈=
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Optimization formulation
)(nxx =
α
α
−
−
∑ 11
rr
rr
xnw
subject to
Rrx
JjCxnA
r
jrrr
jr
∈≥
∈≤∑0
maximize
Suppose is chosen to
(weighted -fair allocations, Mo and Walrand 2000)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
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- 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
−
−
∑ 11
rr
rr
xnw
subject to
Rrx
JjCxnA
r
jrr
rjr
∈≥
∈≤∑0
maximize
RrnpA
wx
jjjr
rr ∈
⎟⎟⎟
⎠
⎞
⎜⎜⎜
⎝
⎛=∑
α/1
)(
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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
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Source: CAIDA,Young Hyun
http://mappa.mundi.net/maps/maps_020/index.html#walrus
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Source: CAIDA -Young Hyun,Bradley Huffaker(displayed at
MOMA)
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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
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If JjCA jrr
jr ∈
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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
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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
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First cut constraint
1C
2C
3C
3C1n
2n 3n
212211 CCxnxn +≤+
Cut defines a single pooled resource
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Second cut constraint
1C
2C
3C
3C1n
2n 3n
3331121 Cxnxn ≤+
Cut defines a second pooled resource
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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
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Outline
• The processor sharing queue• Sharing in networks –
proportional fairness• Multipath routing• Markov chain description,
and heavy traffic• Ramp metering• Resource pooling
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What we've learned about highway congestion P. Varaiya, Access
27, Fall 2005, 2-9.
http://paleale.eecs.berkeley.edu/~varaiya/papers_ps.dir/accessF05v2.pdf
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Data, modelling and inference in road traffic networks R.J.
Gibbens and Y. Saatci Phil. Trans. R. Soc. A366 (2008),
1907-1919.
http://rsta.royalsocietypublishing.org/content/366/1872/1907.abstracthttp://rsta.royalsocietypublishing.org/content/366/1872/1907.abstracthttp://rsta.royalsocietypublishing.org/content/366/1872/1907.abstract
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A linear network
0,d))(()()0()(0
≥Λ−+= ∫ tssmtemtmt
iiii
cumulative inflow
queue size
metering rate K and Williams 2010
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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
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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
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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
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Outline
• The processor sharing queue• Sharing in networks –
proportional fairness• Multipath routing• Markov chain description,
and heavy traffic• Ramp metering• Resource pooling
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Published by AAAS
F. Schweitzer et al., Science 325, 422 -425 (2009)
Fig. 2 A sample of the international financial network, where
the nodes represent major financial institutions and the links are
both directed and weighted and represent the strongest existing
relations
among them
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Chart 3.2 Network of large exposures between UK
banks(a)(b)(c)
Source: FSA returns.
(a) A large exposure is one that exceeds 10% of a lending bank’s
eligible capital at the end of a period. Eligible capital is
defined as Tier 1 plus Tier 2 capital, minus regulatory
deductions.
(b) Each node represents a bank in the United Kingdom. The size
of each node is scaled in proportion to the sum of (1) the total
value of exposures to a bank, and (2) the total value of exposures
of the bank to others in the network. The thickness of the line is
proportional to the value of a single bilateral exposure.
(c) Based on 2009 Q2 data.
http://www.bankofengland.co.uk/financialstability/
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Schematic model for a ‘node’ in the IB network.
May R M , Arinaminpathy N J. R. Soc. Interface
doi:10.1098/rsif.2009.0359
©2009 by The Royal Society(Also: Gai and Kapadia –Contagion in
Financial Markets)
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Stylised bank balance sheetAssetsLiabilities
capital
deposits
inter-bank borrowing
inter-bank loans
externalassets
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Stylised bank balance sheetAssetsLiabilities
capital
deposits
inter-bank borrowing
inter-bank loans
externalassets
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Erlang’s formula
• calls arrive randomly, at rate a
• resource has C circuits
• accepted calls hold a circuit for a random holding time, with
unit mean
• blocked calls are lost
• proportion of calls lost is:
∑C n nCCC
0)!/a(
!/a = )E(a, a
C
-
C(1)
A loss network
A( j, r)r∑ n(r)≤ C( j)Link constraint:
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Aims:• respond robustly to failures and overloads• lessen the
impact of forecasting errors• make use of spare capacity in the
network• permit flexible use of network resources
Problems:• instability• complexity
Resource pooling
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Example: alternative routing
• Complete graph• All links have
capacity C• Call routed directly
if possible; otherwise one randomly chosen alternative route may
be tried
C
Marbukh 1984, Gibbens, Hunt, K 1990, Crametz, Hunt 1991, Graham,
Méléard 1993, 1994
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alternative routing
• Arrival rate per link - a• Capacity per link - C• Let B be the
link blocking probability• Then as the number of nodes grows, the
blocking
probability B approaches a solution of:
))],-1(2+1a[E( = CBBB
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instability, and hysteresis
00.050.1
0.150.2
0.250.3
0.350.4
0.45
0.8 0.85 0.9 0.95 1 1.05 1.1
link blockingprobability, B
load, a / C
C = 1000C = 100
C = ∞
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bistability
0 100call holding times
50
45
calls inprogress(‘000s)
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Sudden impact of capacity
00.050.1
0.150.2
0.250.3
0.350.4
0.450.5
0.8 0.85 0.9 0.95 1 1.05 1.1
Feedback signal (loss, delay, price,…)
load, a / C
some poolingno pooling full pooling
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Open questions on resource pooling
• Resource pooling does indeed– respond robustly to failures and
overloads– lessen the impact of forecasting errors– make use of
spare capacity in the network– permit flexible use of network
resources
• But– can produce phase transitions if load amplified– obscures
the approach of capacity overload
• Can decentralised control take account of system-wide
risks?
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The future?
• Many mathematical challenges, associated with the combination
of network flow and stochastic models of resource possession
• Applications to controlled motorways, router design, systemic
risk.....
Resource sharing in networks�OutlineProcessor sharing
disciplineThe M/G/1 processor sharing queueThe M/G/1 processor
sharing queueThe M/G/1 processor sharing queueThe M/G/1 processor
sharing queueWhat is the network equivalent?Rate
allocationOptimization formulationSolution Examples of -fair
allocations ExampleSlide Number 14Slide Number 15Flow level
modelStabilityMultipath routingExample of multipath routingFirst
cut constraintSecond cut constraintProduct formOutlineSlide Number
24Slide Number 25A linear networkMetering policyOptimal
policy?Proportionally fair meteringSlide Number 30OutlineSlide
Number 32Slide Number 33Slide Number 34Stylised bank balance
sheetStylised bank balance sheetErlang’s formulaA loss
networkResource poolingExample: alternative routingalternative
routinginstability, and hysteresisbistabilitySudden impact of
capacity Open questions on resource poolingThe future?
