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Link Dimensioning for Fractional Brownian Input

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Link Dimensioning for Fractional Brownian Input

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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]

- 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

- 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 modelInternet Traffic?

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

A small number of parameters

Amenable to analysis

- 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”

- 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.

- 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 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

- 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

- 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]

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

Existing asymptotes:

- By Norros[9]

Existing asymptotes (cont.):

- By HuslerandPiterbarg[14]

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

- Our approximation:

- 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)

- Advantages:

- 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

- Basic algorithm (Lindley’s equation):

- 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)

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

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

- Validate simulation

- 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

- We can drive dimensioning formula by

Incomplete Gamma function:

Gamma function:

Finally

where C is the capacity, so .

- 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

- 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

- re-interpret fBm model to

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

Q & A