Fractional Poisson motionand network traffic models

Ingemar Kaj

Uppsala Universityikaj@math.uu.se

Isaac Newton Institute, Cambridge, June 2010

Overview

• Background• Data characteristics• Generic models• Origin of heavy tails• Can short tails generate heavy tails?

• Randomized service• M/M/∞ with CIR-rate• Does user correlation affect system workload?

• Scaling limit results• fast, slow and intermediate growth• fractional Brownian motion vs. stable Levy• fractional Poisson motion

Background

Data characteristics of packet traffic on high-speed links:LRD, self-similarity

Recent study: Grid5000 (5000 CPUs throughout France)

Explanatory models:

• Infinite source Poisson

• On-off models

• Renewal type models

More realistic: hierarchically structured model

- Web session level (infinite source Poisson)

- Web page level (on-off)

- Object level (traffic structure during on-period)

- Packet level (Poisson or renewal)

Generic heavy-tail models

• Infinite source Poisson

Integrate M/G/∞ to get aggregated workloadG (t) ∼ t−γ , 1 < γ < 2

• On-off models

Integrate alternating renewal process,heavy-tailed on- and/or off-periods

• Renewal type models

Heavy-tailed interrenewal timesgives packet arrival model

Origin of heavy tails

• Empirically based a-priori modeling input

Measurements suggest file-sizes, download times, interarrivals,etc, show evidence of finite mean, infinite variance behavior.Alternative interpretation: non-stationary arrival structure

• Ubiquitous outcome of ’robust design of complex systems’ (?)

• Intrinsic effects of protocol mechanisms, TCP, Retransmit, etc

• Randomized service rates

Can short tails generate heavy tails?

The Retransmit Protocol

Sheahan, Lipsky, Fiorini, Asmussen; MAMA2006Jelenkovic, Tan; InfoCom2007Asmussen, Fiorini, Lipsky, Sheahan, Rolski; 2008

Can short tails generate heavy tails?

The Retransmit Protocol

Consider a task of length L ∼ Exp(µ) to be transmitted on a linksubject to Poisson(λ) arrivals of disruption events, λ < µ. Put

M = number of attempts until task carried out successfully

Then

P(M > n) = E (1− e−λL)n =

∫ 1

0(1− x)nδxγ−1 dx

=1(n+γn

) ∼ Γ(1 + γ)1

nγ, n →∞, γ =

µ

λ> 1

Gamma modulated M/M/∞ model

Consider M/M/∞ model

- Poisson arrivals intensity λ.- replace service rate by random process (ξt):

stationary solution of SDE

dxt = δ(γ − xt) +√

2δxt dWt , γ > 1, δ > 0

Gamma modulated M/M/∞ model

Consider M/M/∞ model

- Poisson arrivals intensity λ.- replace service rate by random process (ξt):

stationary solution of SDE

dxt = δ(γ − xt) +√

2δxt dWt , γ > 1, δ > 0

Known that

- ξt ∈ Γ(γ, 1) for all t- Cov(ξs , ξt) = γe−δ(t−s)

Heavy tails?

Service time of job arriving at t: Vξ ∼ Exp(ξt)

P(Vξ > v) = E (e−vξt ) =1

(1 + v)γ

Heavy tails?

Service time of job arriving at t: Vξ ∼ Exp(ξt)

P(Vξ > v) = E (e−vξt ) =1

(1 + v)γ

Take 1 < γ < 2 to obtain Pareto type, heavy-tailed service times

Infinite source Poisson, CIR-service rate

Given (ξs), consider Poisson point measure Nξ(ds, dv) on R × R+,with intensity measure

nξ(ds, dv) = λds ξse−ξsv dv

The stationary, rate-modulated, M/M/∞-model on the real line is

M(y) =

∫R×R+

1{s<y<s+v} Nξ(ds, dv) = nmb of sessions at time y

The gamma-rate workload model W (t) =∫ t0 M(y) dy , t ≥ 0, is

the infinite source Poisson process

Wδ(t) =

∫R×R+

∫ t

01{s<y<s+v} dy Nξ(ds, dv)

Workload mean and variance

E (Wδ(t)) =λ

γ − 1t

Var(Wδ(t)) =λ

γ − 1

∫ t

0

∫ t

0dydy ′

1

(1 + y ∨ y ′ − y ∧ y ′)γ−1

+λ2

∫R2

dsdr

∫ s+t

s

∫ r+t

rdydy ′

∫ 1

1−e−δ|y−y′|du

γrs

(1 + r + s + rsu)γ+1

Workload mean and variance

E (Wδ(t)) =λ

γ − 1t

Var(Wδ(t)) =λ

γ − 1

∫ t

0

∫ t

0dydy ′

1

(1 + y ∨ y ′ − y ∧ y ′)γ−1

+λ2

∫R2

dsdr

∫ s+t

s

∫ r+t

rdydy ′

∫ 1

1−e−δ|y−y′|du

γrs

(1 + r + s + rsu)γ+1

Take δ →∞ (Corr(ξs , ξt)→ 0): as t →∞

Var(Wδ(t)) ∼ const λ(t ∨ t3−γ), γ > 1

Workload mean and variance

E (Wδ(t)) =λ

γ − 1t

Var(Wδ(t)) =λ

γ − 1

∫ t

0

∫ t

0dydy ′

1

(1 + y ∨ y ′ − y ∧ y ′)γ−1

+λ2

∫R2

dsdr

∫ s+t

s

∫ r+t

rdydy ′

∫ 1

1−e−δ|y−y′|du

γrs

(1 + r + s + rsu)γ+1

Take δ → 0 (Corr(ξs , ξt)→ 1): as t →∞

Var(Wδ(t)) ∼ const λ2t2 <∞, γ > 2

Asymptotic results for a-priori heavy tailed models

- Increase aggregation level (nmb of users, connections, flows, . . . ), bysuperposing i.i.d. copies of LRD random process- Scale time (capacity) simultaneously

Which fluctuations build up?

Asymptotic results for a-priori heavy tailed models

- Increase aggregation level (nmb of users, connections, flows, . . . ), bysuperposing i.i.d. copies of LRD random process- Scale time (capacity) simultaneously

Which fluctuations build up?

Depends on relative speed of aggregation/time

• Fast growth of aggregation relative time: Strongly dependent paths,CLT, Gaussian (fBm) or stable, self-similarity

• Slow growth relative time: Independent increments, independentscattering, stable Levy, self-similarity

• Balanced growth relative time: Fluctuations influenced by twocompeting domains of attraction, less rigid paths, non-Gaussian,non-stable, non-self-similar

More exactly

X (t), t ≥ 0, X (0) = 0; continuous time random process, finite mean,stationary increments; Xi (t), i ≥ 1, i.i.d. copies

LRD:∑∞

n=1 Cov(X (1),X (n + 1)− X (n)) =∞

Centered fluctuations, aggregation level m, time scale at:

m∑i=1

(Xi (amt)− EXi (amt)), am →∞,m →∞

Normalized fluctuations

1

bm

m∑i=1

(Xi (amt)− EXi (amt))↗ fractional Brownian motion−→ fractional Poisson motion↘ stable Levy

Heavy-Tailed Renewal Process

Renewal counting process N(t), t ≥ 0,stationary incr, interrenewal times U,

µ = E (U) <∞P(U > t) ∼ t−γL(t), t →∞

1 < γ < 2

Fluctuations, aggregation level m:

1

b

m∑i=1

(Ni (at)−at

µ)

0 10 20 30 40 50 600

2

4

6

8

10

12

14

16

18

20

γ = 1.25, m = 5

Heavy-tailed renewal, result

• If m/aγ−1 →∞,

1√ma3−γ

m∑i=1

(Ni (at)−at

µ) ⇒ const BH(t),

1

2< H =

3− γ

2< 1

• If m/aγ−1 → 0,

1

(am)1/γ

m∑i=1

(Ni (at)−at

µ)

fdd−→ const Λγ(t) (γ-stable Levy)

• If m/aγ−1 → µcγ−1,

1

a

m∑i=1

(Ni (at)−at

µ) ⇒ −µ−1const cPH(t/c), H =

3− γ

2

Gaigalas-IK, Bernoulli 2003

Limit processes

Fractional Brownian motion: BH(t), 0 < H < 1, the continuous,Gaussian process with stationary increments and VarBH(t) = t2H ;

Cov(BH(s),BH(t)) =1

2

(s2H + t2H − |t − s|2H

)Fractional Poisson motion:

PH(t) = Cγ

∫R×R+

∫ t

01{s<y<s+v} dy (Nγ(ds, dv)− dsv−γ−1dv),

where Nγ(ds, dv) Poisson measure with intensity ds v−γ−1dv ,

Cov(PH(s),PH(t)) = Cov(BH(s),BH(t))

Generic pattern

Same result holds for

• infinite source Poisson, P(V > v) ∼ v−γ , 1 < γ < 2

1

b

∫R×R+

∫ at

01{s<y<s+v} dy (N(ds, dv)− λdsFV (dv))

⇒

BH(t), λ/aγ−1 →∞

cPH(t/c), λ/aγ−1 → cγ−1

Λγ(t), λ/aγ−1 → 0

Mikosh, Resnick, Rootzen, Stegemann; AAP 2003IK; Fract Eng 2005Gaigalas; SPA 2006

IK-Taqqu; Progr in Prob 2008

Generic pattern

Same result holds for

• on-off models, 1 < γ = γon < γoff ∧ 2

1

b

m∑j=1

∫ at

0(Ij(s)−

µon

µon + µoff) ds

⇒

BH(t), m/aγ−1 →∞

cPH(t/c), m/aγ−1 → cγ−1

Λγ(t), m/aγ−1 → 0

Mikosh et al 2003

Dombry-IK 2010 (archive)

Bridging property of FPM

Recall intermediate limit: m/aγ−1 → cγ−1 as m, a →∞, impliesthe limit process is

cPH(t/c), t ≥ 0.

As c →∞,cPH(t/c)

c1−H⇒ BH(t)

as c → 0c1/γ−1 cPH(t/c)

fdd→ Λγ(t)

Gaigalas-IK, 2003

Gaigalas, SPA 2006

Aggregate-similarity property of FPM

Recall intermediate limit: m/aγ−1 → cγ−1 as m, a →∞, impliesthe limit process is

cPH(t/c), t ≥ 0.

There is cn →∞ s.t.

cnPH(t/cn)fdd=

n∑j=1

P jH(t),

there is cn → 0 s.t.

n∑j=1

cnPjH(t/cn)

fdd= PH(t),

Workload approximation

Generic workload: Wm(at), aggregation level m, time scale aConsider m ≈ (ca)γ−1, then

1

amWm(at) ≈ νt +

1

mcPH(t/c)

Workload approximation

Generic workload: Wm(at), aggregation level m, time scale aConsider m ≈ (ca)γ−1, then

1

amWm(at) ≈ νt +

1

mcPH(t/c)

≈ νt +

c1−H

m BH(t), c →∞

c1−1/γ

m Λγ(t), c → 0

Workload approximation

Generic workload: Wm(at), aggregation level m, time scale aConsider m ≈ (ca)γ−1, then

1

amWm(at) ≈ νt +

1

mcPH(t/c)

≈ νt +

c1−H

m BH(t), c →∞

c1−1/γ

m Λγ(t), c → 0

≈ νt +

1√

ma1−H BH(t), c →∞

1(ma)1−1/γ Λγ(t), c → 0

Taqqu’s Theorem

Taqqu, Sherman, Willinger 1995, . . .

Center and normalize Workload for heavy-tailed on-off process:

Sequential limits,Take m →∞, then a →∞: fractional Brownian motionTake a →∞, then m →∞: stable Levy process

Scaling limit for CIR-service rate model?

As λ, a →∞ such that λ/aγ−1 →∞,

Wδ(at)− λ atγ−1√

λa3−γ→ const BH(t)

Fractional Poisson motion, 0 < H < 1/2

Given H ∈ (0, 1/2) take γH = 1− 2H ∈ (0, 1), put

PH(t) =

∫R

∫ ∞

0

(1{|t−s|<v} − 1{|s|<v}

)N(ds, dv)

where N(ds, dv) is Poisson measure with intensity ds v−γH−1dv .

The marginal distribution is “symmetrized Poisson”

PH(t) ∼ Po(|t|2H)− Po′(|t|2H),

covariance is

Cov(BH(s),BH(t)) = const(|t|2H + |s|2H − |t − s|2H)

Bierme, Estrade, IK; JTP09

Fractional Poisson motion in Rd

PH(t), t ∈ Rd

1/2 < H < 1: βH = d + 2(1− H).

PH(t) =

cH

∫R×R+

∫B(x,r)

(1[0,t](y)− 1[t,0](y)

)dy N(dx , dr), d = 1

cH

∫Rd×R+

∫B(x,r)

(1

|t−y |d−1 − 1|y |d−1

)dy N(dx , dr), d ≥ 2

0 < H < 1/2: βH = d − 2H.

PH(t) = cH

∫Rd

∫ ∞

0

(1B(x,r)(t)− 1B(x,r)(0)

)N(dx , dr),

where N(dx , dr) = N(dx , dr)− dxr−βH−1dr ,

“Telecom process”

For d = 1, 1 < γ < δ < 2, put

Zγ,δ(t) =

∫R×R+

∫ t

0

1{|x−y |<r} dy Mδ(dx , dr)

for d ≥ 2, d < γ < δ < d + 1, put

Zγ,δ(t) =

∫Rd×R+

{∫B(x,r)

(1

|t − y |d−1− 1

|y |d−1

)dy

}Mδ(dx , dr)

where Mδ is δ-stable random measure with control measure dx , r−γ−1dr .The resulting field is δ/d-stable and self-similar with index

H ′ =δ + d − γ

δ∈ (d/δ, 1)

Pipiras, Taqqu 2004, IK-prep