Fractional Poisson motion and network traffic models Ingemar Kaj Uppsala University [email protected] Isaac Newton Institute, Cambridge, June 2010
Aug 16, 2019
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