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EE 8001. Link Dimensioning for Fractional Brownian Input. BY Chen Jiongze. Supervisor: Prof. ZUKERMAN, Moshe QP Members: Dr. KO, K T Dr. CHAN, Sammy C H. Supported by Grant [ CityU 124709 ]. Outline:. Background Analytical results of a fractional Brownian motion (fBm) Queue

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Link Dimensioning for Fractional Brownian Input

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Link dimensioning for fractional brownian input

EE 8001

Link Dimensioning for Fractional Brownian Input

BY

Chen Jiongze

Supervisor: Prof. ZUKERMAN, Moshe

QP Members:Dr. KO, K T

Dr. CHAN, Sammy C H

Supported by Grant [CityU 124709]


Outline

Outline:

  • Background

  • Analytical results of a fractional Brownian motion (fBm) Queue

    • Existing approximations

    • Our approximation

  • Simulation

    • An efficient approach to simulation fBm queue

    • Results

  • Link Dimensioning

  • Discussion & Conclusion


Outline1

Outline:

  • Background

  • Analytical results of a fractional Brownian motion (fBm) Queue

    • Existing approximations

    • Our approximation

  • Simulation

    • An efficient approach to simulation fBm queue

    • Results

  • Link Dimensioning

  • Discussion & Conclusion


How to model internet traffic

How to modelInternet Traffic?

Its statistics match those of real traffic (for example, auto-covariance function)

A small number of parameters

Amenable to analysis


Background

Background

  • Self-similar (Long Range Dependency)

    • “Aggregating streams of traffic typically intensifies the self similarity (“burstiness”) instead of smoothing it.”[1]

    • Very different from conventional telephone traffic model

      (for example, Poisson or Poisson-related models)

    • Using Hurst parameter (H) as a measure of “burstiness”


Background1

Background

  • Self-similar (Long Range Dependence)

    • “Aggregating streams of traffic typically intensifies the self similarity (“burstyiness”) instead of smoothing it.”[1]

    • Very different from conventional telephone traffic model

      (for example, Poisson or Poisson-related models)

    • Using Hurst parameter (H) as a measure of “burstiness”

  • Gaussian (normal) distribution

    • When umber of source increases

Central limit

theorem

  • process of Real traffic

Gaussian process [2]

Especially for core and metropolitan Internet links, etc.


Fractional brownian motion

Fractional Brownian Motion

  • process of parameter H,MtHare as follows:

    • Gaussian process N(0,t2H)

    • Covariance function:

  • For H > ½ the process exhibits long range dependence


How to model internet traffic1

How to modelInternet Traffic?

Does fBm meets the requirements?

Its statistics match those of real traffic (for example, auto-covariance function)

- Gaussian process & LRD

A small number of parameters

- Hurst parameter (H), variance

Amenable to analysis


Outline2

Outline:

  • Background

  • Analytical results of an fractional Brownian motion (fBm) Queue

    • Existing approximations

    • Our approximation

  • Simulation

    • An efficient approach to simulation fBm queue

    • Results

  • Link Dimensioning

  • Discussion & Conclusion


Analytical results of fbm queue

Analytical results of (fBm) Queue

  • A single server queue fed by an fBm input process with

    - Hurst parameter (H)

    - variance (σ12)

    - drift / mean rate of traffic (λ)

    - service rate (τ)

    - mean net input (μ = λ- τ)

    - steady state queue size (Q)

  • Complementary distribution of Q, denoted as P(Q>x), for H = 0.5:

[16]


Analytical results of fbm queue1

Analytical results of (fBm) Queue

No exact results for P(Q>x) for H ≠ 0.5

Existing asymptotes:

  • By Norros[9]


Analytical results of fbm queue2

Analytical results of (fBm) Queue

Existing asymptotes (cont.):

  • By HuslerandPiterbarg[14]


Analytical results of fbm queue3

Analytical results of (fBm) Queue

Approximation of [14] is more accurate for large x but with no way provided to calculate

  • Our approximation:


Analytical results of fbm queue4

Analytical results of (fBm) Queue

  • Our approximation VS asymptote of [14]:

    • Advantages:

      • a distribution

      • accurate for full range of u/x

      • provides ways to derive c

    • Disadvantages:

      • Less accurate for large x (negligible)


Outline3

Outline:

  • Background

  • Analytical results of a fractional Brownian motion (fBm) Queue

    • Existing approximations

    • Our approximation

  • Simulation

    • An efficient approach to simulation fBm queue

    • Results

  • Link Dimensioning

  • Discussion & Conclusion


Simulation

Simulation

  • Basic algorithm (Lindley’s equation):


Simulation1

Simulation

  • Basic algorithm:

    m = - 0.5, Q0 = 0

    Q1 = max (0, Q0 + U0 + m)

    = max(0, 1.234 – 0.5)

    = 0.734

    Q2 = max(Q1 + U1+ m)

    = max (0, 0.734 – 0.3551 – 0.5)

    = 0

Δt

errors

Discrete time

Continuous time

Length of Un = 222 for different Δt, it is time-consuming to generate Un for very Δt)


An efficient approach

An efficient approach

Instead of generating a new sequence of numbers, we change the “units” of work (y-axis).

variance of the fBn sequence (Un): v

While

Variance in an interval of length (Δt) =

So 1 unit = S instead of 1

where

= S

Δt


An efficient approach cont

An efficient approach (cont.)

Rescale m and P(Q>x)

  • m = μΔt/S units, so

  • P(Q>x) is changed to P(Q>x/S)

    Only need one fBn sequence


Simulation results

Simulation Results

  • Validate simulation


Simulation results1

Simulation Results


Simulation results2

Simulation Results


Simulation results3

Simulation Results


Simulation results4

Simulation Results


Simulation results5

Simulation Results


Outline4

Outline:

  • Background

  • Analytical results of a fractional Brownian motion (fBm) Queue

    • Existing approximations

    • Our approximation

  • Simulation

    • An efficient approach to simulation fBm queue

    • Results

  • Link Dimensioning

  • Discussion & Conclusion


Link dimensioning

Link Dimensioning

  • We can drive dimensioning formula by

Incomplete Gamma function:

Gamma function:


Link dimensioning1

Link Dimensioning

Finally

where C is the capacity, so .


Link dimensioning2

Link Dimensioning


Link dimensioning3

Link Dimensioning


Link dimensioning4

Link Dimensioning


Link dimensioning5

Link Dimensioning


Link dimensioning6

Link Dimensioning


Outline5

Outline:

  • Background

  • Analytical results of a fractional Brownian motion (fBm) Queue

    • Existing approximations

    • Our approximation

  • Simulation

    • An efficient approach to simulation fBm queue

    • Results

  • Link Dimensioning

  • Discussion & Conclusion


Discussion

Discussion

  • fBm model is not universally appropriate to Internet traffic

    • negative arrivals (μ= λ – τ)

  • Further work

    • re-interpret fBm model to

      • alleviate such problem

      • A wider range of parameters


Conclusion

Conclusion

In this presentation, we

  • considered a queue fed by fBm input

  • derived new results for queueing performance and link dimensioning

  • described an efficient approach for simulation

  • presented

    • agreement between the analytical and the simulation results

    • comparison between our formula and existing asymptotes

    • numerical results for link dimensioning for a range of examples


References

References:


References1

References:


References2

References:


Link dimensioning for fractional brownian input

Q & A


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