Some Scaling Properties of Traffic

in Communication Networks

Paulo GonçalvesDANTE - INRIA - ENS Lyon

Patrick Loiseau (PhD, 2006-2009)Shubhabrata Roy (PhD, 2010-2013)

M. Sokol (PhD, 2010- )B. Girault (PhD, 2012-2015)

Seminars Complex Networks, LIP6, UPMC – march 7, 2013

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 1 / 25

Scaling Properties of Traffic

Historical perspective

Mar

kov

2004

Man

djes

: QoS

ON/O

FF(th

eo.)

1997

Taqq

u:ON

/OFF

mod

el

1988

Van

Jaco

bson

: AIM

D- T

CP19

94Pa

xson

Floy

d:LR

DW

AN

MandelbrotLRD (fBm)heavy tails

1917

Erlan

g:cir

cuit

switc

hing

netw

orks

1969

Klein

rock

: pac

ket s

witc

hing

netw

orks

1992

Tim

Bern

ers L

ee: W

eb19

94No

rros:

queu

esan

dLR

D

1997

Crov

ella:

heav

yta

ils

1997

Park

Crov

ella:

QoS

degr

adat

ion

1998

Padh

ye: M

arko

v- 1

TCP

sour

ce

2004

Robe

rts: Q

oSins

ensit

ivity

1993

Lelan

dW

illing

erTa

qqu:

LRD

LAN

1968-69exponentialPoissonMarkov

Mar

kov

Some open questions:Long Range Dependence / Heavy Tailed distributions impact on QoS ?Existing models (e.g. Padhye) only predict mean metrics (e.g. throughput) :what about variability?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 2 / 25

Scaling Properties of Traffic

Historical perspective

Mar

kov

2004

Man

djes

: QoS

ON/O

FF(th

eo.)

1997

Taqq

u:ON

/OFF

mod

el

1988

Van

Jaco

bson

: AIM

D- T

CP19

94Pa

xson

Floy

d:LR

DW

AN

MandelbrotLRD (fBm)heavy tails

1917

Erlan

g:cir

cuit

switc

hing

netw

orks

1969

Klein

rock

: pac

ket s

witc

hing

netw

orks

1992

Tim

Bern

ers L

ee: W

eb19

94No

rros:

queu

esan

dLR

D

1997

Crov

ella:

heav

yta

ils

1997

Park

Crov

ella:

QoS

degr

adat

ion

1998

Padh

ye: M

arko

v- 1

TCP

sour

ce

2004

Robe

rts: Q

oSins

ensit

ivity

1993

Lelan

dW

illing

erTa

qqu:

LRD

LAN

1968-69exponentialPoissonMarkov

Mar

kov

Some open questions:Long Range Dependence / Heavy Tailed distributions impact on QoS ?Existing models (e.g. Padhye) only predict mean metrics (e.g. throughput) :what about variability?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 2 / 25

Scaling Properties of Traffic

Historical perspective

Mar

kov

2004

Man

djes

: QoS

ON/O

FF(th

eo.)

1997

Taqq

u:ON

/OFF

mod

el

1988

Van

Jaco

bson

: AIM

D- T

CP19

94Pa

xson

Floy

d:LR

DW

AN

MandelbrotLRD (fBm)heavy tails

1917

Erlan

g:cir

cuit

switc

hing

netw

orks

1969

Klein

rock

: pac

ket s

witc

hing

netw

orks

1992

Tim

Bern

ers L

ee: W

eb19

94No

rros:

queu

esan

dLR

D

1997

Crov

ella:

heav

yta

ils

1997

Park

Crov

ella:

QoS

degr

adat

ion

1998

Padh

ye: M

arko

v- 1

TCP

sour

ce

2004

Robe

rts: Q

oSins

ensit

ivity

1993

Lelan

dW

illing

erTa

qqu:

LRD

LAN

1968-69exponentialPoissonMarkov

Mar

kov

Some open questions:Long Range Dependence / Heavy Tailed distributions impact on QoS ?Existing models (e.g. Padhye) only predict mean metrics (e.g. throughput) :what about variability?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 2 / 25

Scaling Properties of Traffic

Our approach

To combine theoretical models with controlled experiments in realisticenvironments and real-world traffic traces

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 3 / 25

Scaling Properties of Traffic

Simplified System

Access Point Core NetworkCongestion Overdimensioned

– Congestion essentially arises at the access points→ Simplified System : single bottleneck

– Users’ behavior : ON/OFF source model

– MetroFlux : a probe for traffic capture at packet level (O. Goga,. . . )

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 4 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Long memory in aggregated traffic: the Taqqu model

Heavy-tailed distributed ON periods: heavy tail index αON > 1

Theorem (Taqqu, Willinger, Sherman, 1997)

In the limit of a large number of sources Nsrc, if:

flow throughput is constant,

same throughput for all flows ;

aggregated bandwidth B(∆)(t) is long range dependent, with parameter:

H = max(3− αON

2,12

)Long memory: long range correlation (H > 1/2)

CovB(∆) (τ) = E{B(∆)(t)B(∆)(t + τ)

}∼

τ→∞τ (2H−2)

Variance grows faster than ∆: Var{B(∆)(t)

}∼ ∆2H

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 5 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Theorem validation on a realistic environment

Controlled experiment: MetroFlux 1 Gbps, 100 sources, 8 hours traffic

UDP/TCP: throughput limited to 5 Mbps (no congestion)

ON Distribution Log-diagram Taqqu Prediction(source) (aggregated traffic)

distribution

10−2

100

102

10410

−10

10−5

100

µON

αON

=1.5

logVar{B

(∆) }

0.1ms 1ms 10ms 100ms 1s 10s 100s

+ : TCPo : UDP

RTT µON

H

1 2 3 4

0.4

0.6

0.8

1 + : TCP

o : UDP

ON duration scale ∆ αON

⇒ Protocol has no influence at large scales

⇒ Long memory shows up beyond scale ∆ = µON (mean flow duration)

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 6 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Influence of flow mean throughput / duration correlation

Web traffic acquired at in2p3 (Lyon) with MetroFlux 10 Gbps

ON Distribution Size Distribution

Mean throughput

distribution

0.01ms0.1ms1ms10ms0.1s 1s 10s 100s1000s

10−10

10−5

100

αON

=1.2

distribution

100

105

1010

10−10

10−5

100

αSI

=0.85

E{thr.|d

ur.}

0.1s 1s 10s 100s 1000s10

4

105

106

107

β−1=0.4

ON duration size

duration

Heavy-tailed ON periods, αON = 1.2

Heavy tailed flow sizes, αSI = 0.85

Flow throughput and duration are correlated:

E{thr.|dur.} ∝ (dur.)β−1, β = αON/αSI (= 1.4)

⇒ Which heavy tail index does control LRD ? (αON , αSI ) ?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 7 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Influence of flow mean throughput / duration correlation

Web traffic acquired at in2p3 (Lyon) with MetroFlux 10 Gbps

ON Distribution Size Distribution Mean throughput

distribution

0.01ms0.1ms1ms10ms0.1s 1s 10s 100s1000s

10−10

10−5

100

αON

=1.2

distribution

100

105

1010

10−10

10−5

100

αSI

=0.85

E{thr.|d

ur.}

0.1s 1s 10s 100s 1000s10

4

105

106

107

β−1=0.4

ON duration size duration

Heavy-tailed ON periods, αON = 1.2

Heavy tailed flow sizes, αSI = 0.85

Flow throughput and duration are correlated:

E{thr.|dur.} ∝ (dur.)β−1, β = αON/αSI (= 1.4)

⇒ Which heavy tail index does control LRD ? (αON , αSI ) ?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 7 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Taqqu model extension

Planar Poisson process to describe arrival instant vs duration

Log-diagram, β > 1

logVar{B

(∆) }

scale ∆

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 8 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Taqqu model extension

Planar Poisson process to describe arrival instant vs duration

Proposition (LGVBP, 2009)

Model: E{through.|dur.} = M · (dur.)β−1; Var{through.|dur.} = V

CovB(∆) (τ) = CM2τ−(αON−2(β−1))+1 + C ′V τ−αON+1

Log-diagram, β > 1

logVar{B

(∆) }

scale ∆

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 8 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Taqqu model extension

Planar Poisson process to describe arrival instant vs duration

Proposition (LGVBP, 2009)

Model: E{through.|dur.} = M · (dur.)β−1; Var{through.|dur.} = V

CovB(∆) (τ) = CM2τ−(αON−2(β−1))+1 + C ′V τ−αON+1

Log-diagram, β > 1

logVar{B

(∆) }

τ ∗

HTaqqu

H=HTaqqu

+(β−1)

scale ∆

threshold τ∗ =(

C ′VCM2

)1/(2(β−1))

→ if ∆� τ∗: H = HTaqqu + (β− 1)

→ if ∆� τ∗: H = HTaqqu

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 8 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Taqqu model extension

Planar Poisson process to describe arrival instant vs duration

Proposition (LGVBP, 2009)

Model: E{through.|dur.} = M · (dur.)β−1; Var{through.|dur.} = V

CovB(∆) (τ) = CM2τ−(αON−2(β−1))+1 + C ′V τ−αON+1

Log-diagram, β > 1

logVar{B

(∆) }

τ ∗

HTaqqu

H=HTaqqu

+(β−1)

scale ∆

threshold τ∗ =(

C ′VCM2

)1/(2(β−1))

→ if ∆� τ∗: H = HTaqqu + (β− 1)

→ if ∆� τ∗: H = HTaqqu

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 8 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Taqqu model extension

Planar Poisson process to describe arrival instant vs duration

Proposition (LGVBP, 2009)

Model: E{through.|dur.} = M · (dur.)β−1; Var{through.|dur.} = V

CovB(∆) (τ) = CM2τ−(αON−2(β−1))+1 + C ′V τ−αON+1

Log-diagram, β > 1

logVar{B

(∆) }

τ ∗

HTaqqu

H=HTaqqu

+(β−1)

scale ∆

threshold τ∗ =(

C ′VCM2

)1/(2(β−1))

→ if ∆� τ∗: H = HTaqqu + (β− 1)→ if ∆� τ∗: H = HTaqqu

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 8 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

Taqqu model extension

Planar Poisson process to describe arrival instant vs duration

Proposition (LGVBP, 2009)

Model: E{through.|dur.} = M · (dur.)β−1; Var{through.|dur.} = V

CovB(∆) (τ) = CM2τ−(αON−2(β−1))+1 + C ′V τ−αON+1

Log-diagram, β > 1

logVar{B

(∆) }

τ ∗

HTaqqu

H=HTaqqu

+(β−1)

scale ∆

Correlations intensify LRD (β > 1)

Traffic evolution, future Internet:“flow-aware” control mechanisms,FTTH

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 8 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

LRD impact on QoS: a brief (experimental) outlookThe situation is complex. . .

Negative on finite queues with UDP flows [cf. Mandjes, 2004 (infinitequeues)]

– LRD degrades QoS for large queue sizes (beyond some threshold)– but the threshold depends on the considered QoS metric (loss rate vs

mean load)

Questionable with TCP flows: [Park, 1997] against [Ben Fredj, 2001]

– LRD has contradictory effects on QoS metrics depending on:

with slow start without slow start

Delay ↘ ↗

loss rate ↘ →

mean throughput → ↗

– Heavy tailed distributions (i.e LRD) can favour QoS for large flows

But in general, QOS is a complex function of multiple variables

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 9 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

LRD impact on QoS: a brief (experimental) outlookThe situation is complex. . .

Negative on finite queues with UDP flows [cf. Mandjes, 2004 (infinitequeues)]

– LRD degrades QoS for large queue sizes (beyond some threshold)– but the threshold depends on the considered QoS metric (loss rate vs

mean load)

Questionable with TCP flows: [Park, 1997] against [Ben Fredj, 2001]

– LRD has contradictory effects on QoS metrics depending on:

with slow start without slow start

Delay ↘ ↗

loss rate ↘ →

mean throughput → ↗

– Heavy tailed distributions (i.e LRD) can favour QoS for large flows

But in general, QOS is a complex function of multiple variables

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 9 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

LRD impact on QoS: a brief (experimental) outlookThe situation is complex. . .

Negative on finite queues with UDP flows [cf. Mandjes, 2004 (infinitequeues)]

– LRD degrades QoS for large queue sizes (beyond some threshold)– but the threshold depends on the considered QoS metric (loss rate vs

mean load)

Questionable with TCP flows: [Park, 1997] against [Ben Fredj, 2001]

– LRD has contradictory effects on QoS metrics depending on:

with slow start without slow start

Delay ↘ ↗

loss rate ↘ →

mean throughput → ↗

– Heavy tailed distributions (i.e LRD) can favour QoS for large flows

But in general, QOS is a complex function of multiple variables

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 9 / 25

Scaling Properties of Traffic Heavy tailed distributions and long range dependence

LRD impact on QoS: a brief (experimental) outlookThe situation is complex. . .

Negative on finite queues with UDP flows [cf. Mandjes, 2004 (infinitequeues)]

– LRD degrades QoS for large queue sizes (beyond some threshold)– but the threshold depends on the considered QoS metric (loss rate vs

mean load)

Questionable with TCP flows: [Park, 1997] against [Ben Fredj, 2001]

– LRD has contradictory effects on QoS metrics depending on:

with slow start without slow start

Delay ↘ ↗

loss rate ↘ →

mean throughput → ↗

– Heavy tailed distributions (i.e LRD) can favour QoS for large flows

But in general, QOS is a complex function of multiple variables

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 9 / 25

Scaling Properties of Traffic TCP and large deviations principle

Second level of description : single TCP source traffic

Nsrc

Sources

1

Agrégat

τON τOFF

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 10 / 25

Scaling Properties of Traffic TCP and large deviations principle

Second level of description : single TCP source traffic

i (RTT)Sources

1

Agrégat

τON τOFF

Nsrc

Wi

single TCP source traffic detail

Long-lived flow → stationary regime

⇒ How to characterize the congestion window evolution?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 10 / 25

Scaling Properties of Traffic TCP and large deviations principle

Markov model

i (RTT)

Wi (paquets)

n

long-lived flow stationary regime: AIMD

model: (Wi )i≥1 finite Markov chain (irreducible, aperiodic), transition matrixQ : {

Qw,min(w+1,wmax) = 1− p(w),Qw,max(bw/2c,1) = p(w).

p(·) loss probability of at least one packet, only depends on the currentcongestion window (hyp.)

Example: [Padhye, 1998] Bernoulli loss: p(w) = 1− (1− ppkt)w

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 11 / 25

Scaling Properties of Traffic TCP and large deviations principle

Almost sure mean throughput

W(n)

i (RTT)

Wi (paquets)

n

mean throughput at scale n (RTT): W(n)

=∑n

i=1 Wi

n

Ergodic Birkhoff theorem (1931): almost sure mean

For almost all realisation, the mean throughput at scale n converges towards a valuecorresponding to the expectation of the invariant distribution:

W(n) p.s.−−−→

n→∞W

(∞)= E{Wi}

Example: [Padhye, 1998], W(∞) ∼

ppkt→0

√3

2ppkt(RTT=1, MSS=1)

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 12 / 25

Scaling Properties of Traffic TCP and large deviations principle

Throughput variability: Large Deviations

W(n)

Wi

n

W(n)

i

Wi

n

i

W(n) ' α 6= W

(∞)Rare events

Large Deviations theorem (Ellis, 84)

P(W(n) ' α) ∼

n→∞exp(n · f (α))

f (α) Large Deviation spectrum

→ Scale invariant quantity

W(∞)

0

α

f (α)

⇒ Does a similar theorem exist for a single realization?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 13 / 25

Scaling Properties of Traffic TCP and large deviations principle

Throughput variability: Large Deviations

2n

Wi

n

W(n)

i

Wi

n

i

W(n)

W(n) ' α 6= W

(∞)Rare events

Large Deviations theorem (Ellis, 84)

P(W(n) ' α) ∼

n→∞exp(n · f (α))

f (α) Large Deviation spectrum

→ Scale invariant quantity

W(∞)

0

α

f (α)

⇒ Does a similar theorem exist for a single realization?

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 13 / 25

Scaling Properties of Traffic TCP and large deviations principle

Large Deviation on almost all realizations

intervalle knWi

n

W(n)1

2n iknn

W(n)kn

intervalle 1

Large Deviation theorem on almost all realisations (Loiseau et al., 2010)

For a given α, if kn ≥ enR(α), then a.s.

#{j ∈ {1, · · · , kn} : W

(n)j ' α

}kn

∼n→∞

exp(n · f (α))

“Price to pay”: exponential increase of the number of intervals

Finite realization (of size N): nkn = N

⇒ [αmin(n), αmax(n)] support of observable spectrum at scale n

Theory: p(·) → Q → f (α),R(α), αmin, αmax

Practice: (Wi )i≤N → observed distribution

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 14 / 25

Scaling Properties of Traffic TCP and large deviations principle

Large Deviation on almost all realizations

intervalle knWi

n

W(n)1

2n iknn

W(n)kn

intervalle 1

Large Deviation theorem on almost all realisations (Loiseau et al., 2010)

For a given α, if kn ≥ enR(α), then a.s.

#{j ∈ {1, · · · , kn} : W

(n)j ' α

}kn

∼n→∞

exp(n · f (α))

“Price to pay”: exponential increase of the number of intervals

Finite realization (of size N): nkn = N

⇒ [αmin(n), αmax(n)] support of observable spectrum at scale n

Theory: p(·) → Q → f (α),R(α), αmin, αmax

Practice: (Wi )i≤N → observed distribution

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 14 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results: example of Bernoulli losses (ppkt = 0.02)f

(α)

4 10 16

−0.1

−0.05

0

W(∞)

theorique

α (packets)

Apex: almost sure mean: 8.6 packets (Padhye:√

32ppkt

= 8.66)

Superimposition at different scales → scale invariance

beyond n = 100: variabilityn = 100, portion of intervals with mean ∼ 11: e−100×0.01 = 0.37n = 200, portion of intervals with mean ∼ 11: e−200×0.01 = 0.14

⇒ More accurate information than the almost sure mean

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 15 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results: example of Bernoulli losses (ppkt = 0.02)f

(α)

4 10 16

−0.1

−0.05

0

W(∞)α

min(100) α

max(100)

theoriquen=100

α (packets)

Apex: almost sure mean: 8.6 packets (Padhye:√

32ppkt

= 8.66)

Superimposition at different scales → scale invariance

beyond n = 100: variabilityn = 100, portion of intervals with mean ∼ 11: e−100×0.01 = 0.37n = 200, portion of intervals with mean ∼ 11: e−200×0.01 = 0.14

⇒ More accurate information than the almost sure mean

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 15 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results: example of Bernoulli losses (ppkt = 0.02)f

(α)

4 10 16

−0.1

−0.05

0

W(∞)α

minα

maxα

min(200) α

max(200)

theoriquen=100n=200

α (packets)

Apex: almost sure mean: 8.6 packets (Padhye:√

32ppkt

= 8.66)

Superimposition at different scales → scale invariance

beyond n = 100: variabilityn = 100, portion of intervals with mean ∼ 11: e−100×0.01 = 0.37n = 200, portion of intervals with mean ∼ 11: e−200×0.01 = 0.14

⇒ More accurate information than the almost sure mean

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 15 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results: example of Bernoulli losses (ppkt = 0.02)f

(α)

4 10 16

−0.1

−0.05

0

W(∞)α

minα

maxα

minα

maxα

min(500) α

max(500)

theoriquen=100n=200n=500

α (packets)

Apex: almost sure mean: 8.6 packets (Padhye:√

32ppkt

= 8.66)

Superimposition at different scales → scale invariance

beyond n = 100: variabilityn = 100, portion of intervals with mean ∼ 11: e−100×0.01 = 0.37n = 200, portion of intervals with mean ∼ 11: e−200×0.01 = 0.14

⇒ More accurate information than the almost sure mean

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 15 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results: example of Bernoulli losses (ppkt = 0.02)f

(α)

4 10 16

−0.1

−0.05

0

W(∞)α

minα

maxα

minα

maxα

minα

max

theoriquen=100n=200n=500n=1000

α (packets)

Apex: almost sure mean: 8.6 packets (Padhye:√

32ppkt

= 8.66)

Superimposition at different scales → scale invariance

beyond n = 100: variabilityn = 100, portion of intervals with mean ∼ 11: e−100×0.01 = 0.37n = 200, portion of intervals with mean ∼ 11: e−200×0.01 = 0.14

⇒ More accurate information than the almost sure mean

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 15 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results: example of Bernoulli losses (ppkt = 0.02)f

(α)

4 10 16

−0.1

−0.05

−0.010

W(∞)α

minα

maxα

minα

maxα

minα

max11

theoriquen=100n=200n=500n=1000

α (packets)

Apex: almost sure mean: 8.6 packets (Padhye:√

32ppkt

= 8.66)

Superimposition at different scales → scale invariance

beyond n = 100: variabilityn = 100, portion of intervals with mean ∼ 11: e−100×0.01 = 0.37n = 200, portion of intervals with mean ∼ 11: e−200×0.01 = 0.14

⇒ More accurate information than the almost sure mean

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 15 / 25

Scaling Properties of Traffic TCP and large deviations principle

Results II: case of a long-lived flow

losses: not Bernoulli

empirical losses p(w

)

0 20 40 60 80 1000

0.5

1

Bernoulli (ppkt

=0.007)

empirique

w

f(α

)

0 20 40 60 80−0.1

−0.08

−0.06

−0.04

−0.02

0

theo. perte emp.theo. perte Ber.n=100n=200n=500n=1000

αmin,α

max

0 200 400 600 800 10000

20

40

60

80

theo. perte emp.theo. perte Ber.empirique

α (packets) n (RTT)

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 16 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

Two important assets for Large Deviations Utility

General result ( “Large deviations for the local fluctuations of random walks", J. Barral, P. Loiseau, Stochastic

Processes and their Applications, 2011)

A wide class of processes (stationary & mixing) verifies an empirical large deviationprinciple. In particular, this results holds true any time series that can reliably bemodelled by an irreducible, aperiodic Markov process.

Theorem ( “On the estimation of the Large Deviations spectrum", J. Barral, P. G., J. stat. Phys., 2011)

We derived a consistent estimator of the large deviation spectrum from a finite size timeseries (observation samples). We proved convergence on mathematical objects withscale invariance properties (multifractal measures and processes).

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 17 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

An epidemic based model for volatile workloadGoal – Dynamic resource allocation yielding a good compromise between capex andopex costs

Approach – Combine the three ingredients:

A sensible (epidemic) model to catch the burstiness and the dynamics of theworkload

A (Markov) model that verifies a large deviation principle

A probabilistic management policy based on the large deviationcharacterisation

Number of current VoD users

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

time

A hidden state Markov process with memory effect

i, r

i, r-1

i+1, r

i-1, r+1

β(i+r)+lβ = β1

β = β2

γiμr

a1 a2

i : current # of viewers / r : current # of infected

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 18 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

An epidemic based model for volatile workloadGoal – Dynamic resource allocation yielding a good compromise between capex andopex costs

Approach – Combine the three ingredients:

A sensible (epidemic) model to catch the burstiness and the dynamics of theworkload

A (Markov) model that verifies a large deviation principle

A probabilistic management policy based on the large deviationcharacterisation

Number of current VoD users

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

time

A hidden state Markov process with memory effect

i, r

i, r-1

i+1, r

i-1, r+1

β(i+r)+lβ = β1

β = β2

γiμr

a1 a2

i : current # of viewers / r : current # of infected

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 18 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

An epidemic based model for volatile workloadGoal – Dynamic resource allocation yielding a good compromise between capex andopex costs

Approach – Combine the three ingredients:

A sensible (epidemic) model to catch the burstiness and the dynamics of theworkload

A (Markov) model that verifies a large deviation principle

A probabilistic management policy based on the large deviationcharacterisation

Number of current VoD users

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

time

A hidden state Markov process with memory effect

i, r

i, r-1

i+1, r

i-1, r+1

β(i+r)+lβ = β1

β = β2

γiμr

a1 a2

i : current # of viewers / r : current # of infected

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 18 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

An epidemic based model for volatile workloadGoal – Dynamic resource allocation yielding a good compromise between capex andopex costs

Approach – Combine the three ingredients:

A sensible (epidemic) model to catch the burstiness and the dynamics of theworkload

A (Markov) model that verifies a large deviation principle

A probabilistic management policy based on the large deviationcharacterisation

Number of current VoD users

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

time

A hidden state Markov process with memory effect

i, r

i, r-1

i+1, r

i-1, r+1

β(i+r)+lβ = β1

β = β2

γiμr

a1 a2

i : current # of viewers / r : current # of infected

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 18 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

An epidemic based model for volatile workloadCalibration and evaluation

VoD workload trace Memory Markov model Modul. Markov Poisson

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140 160 180 2000

20

40

60

80

100

120

140

0 20 40 60 80 100 120 140 160 180 2000

50

100

150

Steady state distribution Autocorrelation function Param. estimation precision

0 50 100 15010

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

101

Modelled trace I

Real trace I

ctmc−modelled Trace I

MMPP−modelled trace I

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 210

−8

10−7

10−6

10−5

10−4

10−3

10−2

10−1

100

Modelled trace I

Real trace I

ctmc−modelled Trace I

MMPP−modelled trace I

−0.2

−0.1

0

0.1

0.2

0.3

β2

µ lγ a2

a1

β1

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 19 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

Markov processesUnder mild conditions, a Markov processes It verifies a large deviation principle:

P{〈It〉τ ≈ α} ≡ exp (τ · f (α)) , τ →∞

f (α) : theoretically (from the transition matrix) or empirically (from a finite trace)identifiable

"Dynamic" implies time scale: a notion that is explicit in large deviation principle

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 20 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

Markov processesUnder mild conditions, a Markov processes It verifies a large deviation principle:

P{〈It〉τ ≈ α} ≡ exp (τ · f (α)) , τ →∞

f (α) : theoretically (from the transition matrix) or empirically (from a finite trace)identifiable

"Dynamic" implies time scale: a notion that is explicit in large deviation principle

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 20 / 25

Scaling Properties of Traffic Large Deviations applied to dynamic resource management

Markov processesUnder mild conditions, a Markov processes It verifies a large deviation principle:

P{〈It〉τ ≈ α} ≡ exp (τ · f (α)) , τ →∞

f (α) : theoretically (from the transition matrix) or empirically (from a finite trace)identifiable

"Dynamic" implies time scale: a notion that is explicit in large deviation principle

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 20 / 25

Semi-supervised machine learning

Parametric generalisation of semi-supervised learning

Standard classification

Training set

(X (t),Y (t)) −→ classifier

Validation set

X (v) classifier−→ Y (v′) : |Y (v) − Y (v′)| ' 0

Real data

Xclassifier−→ Answer

Semi-supervised classification

Validation set

(X (v),Y (L))classifier−→ Y (v′)

such that |Y (v) − Y (v′)| ' 0

Real data

(X ,Y (L))classifier−→ Answer

Allow to constantly update the classifierto match data evolution

Dataset X = X1,X2, . . . ,Xp︸ ︷︷ ︸labeled points

,Xp+1, . . . ,XN

Similarity matrix W and D (reap. D∗) the row-sum (reap. column) of WLabel matrix Y = {Yi,k ∈ (0, 1) for i = 1, . . .N and k = 1, . . .K}Objective (classification) matrix FN×K : element i belongs to class k∗ = argmax

kFi,k

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 21 / 25

Semi-supervised machine learning

Parametric generalisation of semi-supervised learning

Standard classification

Training set

(X (t),Y (t)) −→ classifier

Validation set

X (v) classifier−→ Y (v′) : |Y (v) − Y (v′)| ' 0

Real data

Xclassifier−→ Answer

Semi-supervised classification

Validation set

(X (v),Y (L))classifier−→ Y (v′)

such that |Y (v) − Y (v′)| ' 0

Real data

(X ,Y (L))classifier−→ Answer

Allow to constantly update the classifierto match data evolution

Dataset X = X1,X2, . . . ,Xp︸ ︷︷ ︸labeled points

,Xp+1, . . . ,XN

Similarity matrix W and D (reap. D∗) the row-sum (reap. column) of WLabel matrix Y = {Yi,k ∈ (0, 1) for i = 1, . . .N and k = 1, . . .K}Objective (classification) matrix FN×K : element i belongs to class k∗ = argmax

kFi,k

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 21 / 25

Semi-supervised machine learning

Parametric generalisation of semi-supervised learning

Standard Laplacian solution

argmaxF

{N∑i=1

N∑j=1

wij || Fi. − Fj. ||2 +µN∑i=1

di || Fi. − Yi. ||2}

Generalised semi-supervised classification [M. Sokol, 2012]

argmaxF

{N∑i=1

N∑j=1

wij || dσ−1i Fi. − dσ−1j Fj. ||2 +µN∑i=1

d2σ−1i || Fi. − Yi. ||2

}

σ = 1 Standard Laplacian (Random walk from unlabelled to labelled points)σ = 1/2 Normalised Laplacianσ = 0 PageRank method (Random walk from labelled to unlabelled points)

F.k =µ

2 + µ

(I − 2

2 + µD−σWDσ−1

)−1Y.k , for k = 1, . . . ,K

Tune the value of parameter σ to match the dataset

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 22 / 25

Semi-supervised machine learning

Parametric generalisation of semi-supervised learning

Standard Laplacian solution

argmaxF

{N∑i=1

N∑j=1

wij || Fi. − Fj. ||2 +µN∑i=1

di || Fi. − Yi. ||2}

Generalised semi-supervised classification [M. Sokol, 2012]

argmaxF

{N∑i=1

N∑j=1

wij || dσ−1i Fi. − dσ−1j Fj. ||2 +µN∑i=1

d2σ−1i || Fi. − Yi. ||2

}

σ = 1 Standard Laplacian (Random walk from unlabelled to labelled points)σ = 1/2 Normalised Laplacianσ = 0 PageRank method (Random walk from labelled to unlabelled points)

F.k =µ

2 + µ

(I − 2

2 + µD−σWDσ−1

)−1Y.k , for k = 1, . . . ,K

Tune the value of parameter σ to match the dataset

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 22 / 25

Semi-supervised machine learning

Parametric generalisation of semi-supervised learning

Standard Laplacian solution

argmaxF

{N∑i=1

N∑j=1

wij || Fi. − Fj. ||2 +µN∑i=1

di || Fi. − Yi. ||2}

Generalised semi-supervised classification [M. Sokol, 2012]

argmaxF

{N∑i=1

N∑j=1

wij || dσ−1i Fi. − dσ−1j Fj. ||2 +µN∑i=1

d2σ−1i || Fi. − Yi. ||2

}

σ = 1 Standard Laplacian (Random walk from unlabelled to labelled points)σ = 1/2 Normalised Laplacianσ = 0 PageRank method (Random walk from labelled to unlabelled points)

F.k =µ

2 + µ

(I − 2

2 + µD−σWDσ−1

)−1Y.k , for k = 1, . . . ,K

Tune the value of parameter σ to match the dataset

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 22 / 25

Semi-supervised machine learning

Parametric generalisation of semi-supervised learning

Standard Laplacian solution

argmaxF

{N∑i=1

N∑j=1

wij || Fi. − Fj. ||2 +µN∑i=1

di || Fi. − Yi. ||2}

Generalised semi-supervised classification [M. Sokol, 2012]

argmaxF

{N∑i=1

N∑j=1

wij || dσ−1i Fi. − dσ−1j Fj. ||2 +µN∑i=1

d2σ−1i || Fi. − Yi. ||2

}

σ = 1 Standard Laplacian (Random walk from unlabelled to labelled points)σ = 1/2 Normalised Laplacianσ = 0 PageRank method (Random walk from labelled to unlabelled points)

F.k =µ

2 + µ

(I − 2

2 + µD−σWDσ−1

)−1Y.k , for k = 1, . . . ,K

Tune the value of parameter σ to match the dataset

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 22 / 25

Follow-up Dynamic graphs analysis

Duality and semi-supervised learning

graph (similarity)ordination←→ process (metric)

formulation (multidimensional scaling) : argmaxF

{N∑i=1

N∑j=1

(|| Fi. − Fj. || −wij)2

}

bridge ordination (MDS) and generalised semi-supervised learning

B leverage σ flexibility to vary duality principle

data adaptivity of semi-supervised learning

B use to update dynamic graph ↔ non-stationary time series

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 23 / 25

Follow-up Dynamic graphs analysis

Duality and semi-supervised learning

graph (similarity)ordination←→ process (metric)

formulation (multidimensional scaling) : argmaxF

{N∑i=1

N∑j=1

(|| Fi. − Fj. || −wij)2

}

bridge ordination (MDS) and generalised semi-supervised learning

B leverage σ flexibility to vary duality principle

data adaptivity of semi-supervised learning

B use to update dynamic graph ↔ non-stationary time series

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 23 / 25

Follow-up Dynamic graphs analysis

Duality and semi-supervised learning

graph (similarity)ordination←→ process (metric)

formulation (multidimensional scaling) : argmaxF

{N∑i=1

N∑j=1

(|| Fi. − Fj. || −wij)2

}

bridge ordination (MDS) and generalised semi-supervised learning

B leverage σ flexibility to vary duality principle

data adaptivity of semi-supervised learning

B use to update dynamic graph ↔ non-stationary time series

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 23 / 25

Follow-up Dynamic graphs analysis

Graph diffusionEpidemic diffusion (MOSAR): Apply standard tools. . .

B Relationship between virus spreading and graph structure: Can diffusionwavelets help?

Coarse Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

Fine Scaling Function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0.05

B How to take into account / reflect dynamicity of graphsP. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 24 / 25

Follow-up Dynamic graphs analysis

Context and collaborations

Dante (B. Girault, E. Fleury,. . . )

Institut des Systèmes Complexes

Sisyphe (ENS Lyon, P. Borgnat)

Other teams (e.g. Geodyn, Maestro. . . )

International cooperations (e.g. EPFL)

. . .

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 25 / 25

Follow-up Dynamic graphs analysis

Context and collaborations

Dante (B. Girault, E. Fleury,. . . )

Institut des Systèmes Complexes

Sisyphe (ENS Lyon, P. Borgnat)

Other teams (e.g. Geodyn, Maestro. . . )

International cooperations (e.g. EPFL)

. . .

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 25 / 25

Follow-up Dynamic graphs analysis

Context and collaborations

Dante (B. Girault, E. Fleury,. . . )

Institut des Systèmes Complexes

Sisyphe (ENS Lyon, P. Borgnat)

Other teams (e.g. Geodyn, Maestro. . . )

International cooperations (e.g. EPFL)

. . .

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 25 / 25

Follow-up Dynamic graphs analysis

Context and collaborations

Dante (B. Girault, E. Fleury,. . . )

Institut des Systèmes Complexes

Sisyphe (ENS Lyon, P. Borgnat)

Other teams (e.g. Geodyn, Maestro. . . )

International cooperations (e.g. EPFL)

. . .

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 25 / 25

Follow-up Dynamic graphs analysis

Context and collaborations

Dante (B. Girault, E. Fleury,. . . )

Institut des Systèmes Complexes

Sisyphe (ENS Lyon, P. Borgnat)

Other teams (e.g. Geodyn, Maestro. . . )

International cooperations (e.g. EPFL)

. . .

P. Gonçalves (Inria) Scaling properties of traffic Complex Networks (Lip6) 25 / 25