Department of Electronic Engineering (EE)
City University of Hong Kong
Performance Evaluation of
Long Range Dependent Queues
Chen Jiongze
Supervisor: Moshe Zukerman
Co-Supervisor: Ronald G. Addie
Supported by Grants [CityU 124709] and [CityU 8/CRF/13G]
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Introduction
Poisson Lomax Burst Process (PLBP) Queue
Fractional Brownian motion (fBm) Queue
Conclusions
2
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Introduction
Poisson Lomax Burst Process (PLBP) Queue
Fractional Brownian motion (fBm) Queue
Conclusions
3
Not enough capacity
Angry customer
Credit: http://inwritefield.com/2012/06/22/are-you-being-served/
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Too much capacity
Lose Money
Credit: http://www.neverpaintagain.co.uk/blog/how-to-lose-money-with-bad-home-improvement-choices/
5
Capacity
QoS
Traffic Engineering Network Engineering Network Planning
Credit: http://ineed.coffee/
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Data
Data
Data
A Link
Traffic
Data
Data
…
Data
Data
Data
…
…
Credit: http://www.fastcodesign.com/1662881/infographic-of-the-day-the-facebook-map-of-the-world
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Data
Data
Data
A Link
Traffic
Data
Data
…
Data
Quality of
Service (QoS) ?
Data
Data
…
…
It is unrealistic to replicate the entire traffic
on an Internet link!
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A Link
Traffic
Internet Link
Data
…
Data
Modelling
Queueing System
Input process
Queue
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Input process
Traffic
Data
Data
Sampling
Data
…
…
A good traffic model
• capture the nature of the traffic:
Long Range Dependence (LRD);
• A small number of parameters;
• Amenable to analysis.
Fitting the
parameters
Traffic model
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[1] W. E. Leland, M. S. Taqqu, W. Willinger, and D. V. Wilson, “On the self-similar nature of ethernet traffic (extended version),” IEEE/ACM
Trans. Networking, vol. 2, no. 1, pp. 1–15, Feb. 1994.
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A process, X={Xt, t=1,2,…}, with mean m and variance σ2.
Autocovariance function, γ(k) = E[(Xt-m)(Xt+k-m)], decays
slower than exponential.
IID – Independent and Identically Distributed
MMPP – Markov Modulated Poisson Process
IID
Poisson
0
LRD
MMPP
k
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A process, X={Xt, t=1,2,…}, with mean m and variance σ2.
Autocovariance function, γ(k) = E[(Xt-m)(Xt+k-m)], decays
slowly.
Autocorrelation function (ACF), ρ(k) = γ(k)/σ2, follows
Hurst parameter H : the measure of the degree of the LRD.
0.5 < H < 1  the process is LRD
The aggregate process of X with interval t, X(t) follows
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Input process
Traffic
Data
Data
Sampling
Data
…
…
Important statistics of traffic:
mean (m), variance (σ2) and Hurst parameter (H).
Fitting the
parameters
Traffic model
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

LRD process  Input traffic process
Single Server Queue (SSQ)  Link
Overflow probability  QoS
LRD Queue
LRD process
Mean (m)
Variance (σ2)
Hurst parameter (H)
SSQ
Output
P(Q>x)?
SSQ with ∞ buffer
Steady state Queue Size (Q)
Service rate (μ)
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Hurst
parameter
Capacity
Variance
Hurst
parameter
Variance
Blocking
probability
Buffer
size
Buffer
size
Mean
Blocking
probability
Helpful for
Traffic Engineering
Mean
Capacity
Network Engineering
& Network Planning
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
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
Introduction
Poisson Lomax Burst Process (PLBP) Queue
Fractional Brownian motion (fBm) Queue
Conclusions
17
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The process is based on a stream of bursts.
Burst arrivals following a Poisson process with rate λ
[bursts/s].
Burst durations, d, are i.i.d. Lomax random variables
with parameters γ and δ.
Each burst contributes work with constant rate r [B/s].
PLBP: {At, t≥0}, where At is the work contributed by
all bursts during the interval (0,t].
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Exponential ()
Lomax (, )
…
…
t
Work
4r
3r
2r
r
…
t
PLBP
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Exponential ()
Common probability mass
function (PMF) G
…
…
t
Bt
4
3
2
1
…
t
M/G/∞ process
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Exponential ()
Pareto (, )
…
…
t
Work
4r
3r
2r
r
…
t
Poisson Pareto Burst Process (PPBP)
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The complementary cumulative distribution
function (CCDF) of
• Pareto distribution
• Lomax distribution
d: burst duration; γ: shape parameter;
δ: scale parameter.
Advantages of Lomax:
• takes care of small bursts;
• δ is no more minimum value of burst
duration.
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
Mean (m(t)):

Variance (σ2(t)):
1<γ<2LRD
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Input
PLBP with parameters:
λ, γ, δ and r.
SSQ
Output
SSQ with ∞ buffer
and service rate, μ.
Obtain the overflow probability, P(Q>x), by:
• Analytical result: the Quasi-stationary (QS) approximation,
• Simulation: the fast simulation method.
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τ
PLBP Process
Short burst process (Sτ)
Long burst process (Lτ)
t
t+τ
t
t+τ
For a certain period τ, the probability of Q>x is
Number of long bursts, η, is Poisson distributed with mean as λE(d)P(ω>τ).
 λE(d): the mean number of the existing burst at t; ω: the forward recurrence time of a Lomax RV.
QS: the steady state queue size of an SSQ fed by Sτ .
 Assuming Sτ is Gaussian, we can obtain P(QS>x) by [2].
QS approximation:
[2] R. G. Addie, P. Mannersalo, and I. Norros, Performance formulae for queues with Gaussian input, ser. Teletraffic Engineering in a
Competitive World. Elsevier Science, Jun. 1999, pp. 1169–1178.
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Conventional simulation method
…
Fast simulation method
Initial short bursts
Initial long burst
T
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
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
Introduction
Poisson Lomax Burst Process (PLBP) Queue
Fractional Brownian motion (fBm) Queue
Conclusions
28

Three characteristic features:
 A continuous Gaussian process;
 Self-similar with parameter H;
 Stationary increment.

A normalized fBm process, B={B(t), t≥0}:
 B(0) = 0, E[B(t)] = 0 for all t≥0;
 Var[B(t)] = t2H for all t≥0;
 0.5<H<1  LRD;
 Covariance function:
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The increment process – fractional Gaussian Noise (fGn):

Cumulative work arrival process: X = {X(t), t≥0}, where
X(t) = mt+σB(t), thus
 E[X(t)] = mt, Var[X(t)] = σ2t2H;
 the mean and variance of its increment process are m and σ2.
Input
fBm process with
parameters: m, σ and H.
SSQ
Output
SSQ with ∞ buffer
and service rate, μ.
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The mean net input ι = m – μ, so an fBm queue has
three parameters, ι, σ and H.
 By Reich’s formula [3], the queue size (Q) is
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Exact solution for P(Q>x) for H = 0.5 by Harrison
[4]：
No exact results for P(Q>x) for H ≠ 0.5.
[3] E. Reich, “On the integrodifferential equation of takács I,” Annals of Mathematical Statistics, vol. 29, pp. 563–570, 1958.
[4] J. M. Harrison, Brownian motion and stochastic flow systems. New York: John Wiley and Sons, 1985.
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By Norros [5]:
It holds in sense that
[5] I. Norros, “A storage model with self-similar input,” Queueing Syst., vol. 16, no. 3-4, pp. 387–396, Sep. 1994.
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By Hüsler and Piterbarg [6]:
where C is a certain constant and the it holds in
sense that
 No method to determine C.
 Since for
RHS ∞ as x 0.
[6] J. Hüsler and V. Piterbarg, “Extremes of a certain class of Gaussian processes,” Stochastic Processes and their Applications, vol. 83, no. 2,
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pp. 257 – 271, Oct. 1999.
Revise Hüsler and Piterbarg’s approach by supposing
that it is not the CCDF, but rather the density whose
character remains stable for x near ∞.
where the density cf(x) is characterized according to
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For x near ∞:
Dominant
We have
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Let
and
, we have
where
and
Γ denotes the Gamma function.
It holds in the sense that
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Our approximation vs. asymptotics by Hüsler and
Piterbarg
Advantages:
▪ a distribution
▪ accurate for full range of buffer size
▪ provides ways to derive c
Disadvantages:
▪ Slightly less accurate for very large x
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Discrete-time simulation.
 Divide time into N intervals of equal length Δt.
 Qn denotes the queue size at the end of nth interval,
defined by Lindley’s equation:

where Q0=0,
each interval.
is the amount of work arriving in
Difficulty: Discrete time
Δt  0
Continuous time
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For a given H, simulate for different Δt with one sequence
of standard fGn,
, with mean ι and variance
v1.

A new sequence is defined by
where s(Δt) and m(Δt) are chosen so that
has the appropriate mean and variance.

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The density function of Q:
The density function of
Amoroso Distribution:
=
g = 0, d = β, p = ν and a = α−1/ν
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Negative arrivals
 Appropriate for 1) large buffer size; 2) σ is large
relative to m.

Gaussian
 Appropriate for high multiplexing.
To illustrate the weaknesses, we compared it
with the PPBP model.
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Small buffer size
Large buffer
size
CISCO routers
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Deriving the inverse function of our approximation, we
have the dimensioning formula as
where
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μ*: capacity;
m: mean of the input process;
σ2: variance of the input process;
H: Hurst parameter;
ε: required overflow probability;
q: buffer threshold;
G-1(): inverse regularised incomplete Gamma function.
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
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

Introduction
Poisson Lomax Burst Process (PLBP) Queue
Fractional Brownian motion (fBm) Queue
Conclusions
51
Main contributions:

PLBP queues
 The PLBP model (a variant of PPBP) is proposed.
 An approximation based on the QS algorithm is provided.
 A fast simulation method is applied.
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fBm queues
 New results for queueing performance and link dimensioning are
derived.
 Important statistics of fBm queues are provided.
 An efficient approach for simulation is proposed.
 The weaknesses of the fBm model are discussed.
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