Fractional Poisson motion and network traffic models fileFractional Poisson motion and network...
Transcript of Fractional Poisson motion and network traffic models fileFractional Poisson motion and network...
Fractional Poisson motionand network traffic models
Ingemar Kaj
Uppsala [email protected]
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