Link Dimensioning for Fractional Brownian

Input

Chen JiongzePhD student,

ElectronicEngineering Department,City University of Hong

Kong

Supported by Grant [CityU 124709]

Moshe ZukermanElectronic

Engineering Department,,City University of Hong Kong

Ron G. AddieDepartment of Mathematics

and Computing,University of

Southern Queensland, Australia

Outline:

• Background• A New Analytical Result of an FBM Queue• Simulation• Link Dimensioning• Discussion & Conclusion

Outline:

• Background• A New Analytical Result of an FBM Queue• Simulation• Link Dimensioning• Discussion & Conclusion

Fractional Brownian Motion (fBm)• process of parameter H, Mt

H are as follows:• Gaussian process N(0,t2H)• Covariance function:

• For H > ½ the process exhibits long range dependence

…

traffic Queue

…

traffic Queue

Long Range Dependence

Gaussian By

Central limittheorem

• 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

Is fBm a good model? YES

Outline:

• Background• A New Analytical Result of an FBM Queue• Simulation• Link Dimensioning• Discussion & Conclusion

A New Analytical Result of an fBm Queue

…

traffic Queue

Queuing Model

fBm trafficHurst parameter (H)variance (σ1

2)drift / mean rate of traffic (λ)

Single server Queue with ∞ buffersservice rate (τ)steady state queue size (Q)

mean net input (μ = λ - τ)

Analytical results of (fBm) Queue

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

Existing asymptotes:•By Norros [9]

[9] I. Norros, “A storage model with self-similar input,” Queueing Syst., vol. 16, no. 3-4, pp. 387–396, Sep. 1994.

Analytical results of (fBm) Queue

Existing asymptotes (cont.):•By Husler and Piterbarg [14]

[14] J. H¨usler and V. Piterbarg, “Extremes of a certain class of Gaussian processes,” Stochastic Processes and their Applications, vol. 83, no.

2, pp. 257 – 271, Oct. 1999.

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

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)

Analytical results of (fBm) Queue

[14] J. H¨usler and V. Piterbarg, “Extremes of a certain class of Gaussian processes,” Stochastic Processes and their Applications, vol. 83, no.

2, pp. 257 – 271, Oct. 1999.

Outline:

• Background• A New Analytical Result of an FBM Queue• Simulation• Link Dimensioning• Discussion & Conclusion

Simulation

• Basic algorithm (Lindley’s equation):

n 0 1 2 …

Un (Mb) 1.234 – 0.3551 0.743 …

m(t) (Mb) -0.5 -0.5 -0.5 …

Qn (Mb) 0 max(0, 1.234 – 0.5)=0.734

max (0, 0.734 – 0.3551 – 0.5)=0

…

1 ms

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

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

1ms -> Δt ms

variance of the fBn sequence (Un): V

Rescale the Y-asix

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

1 unit = S instead of 1 where

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• Validate simulation

Simulation Results

Simulation Results

Simulation Results

Simulation Results

Outline:

• Background• A New Analytical Result of an FBM Queue• Simulation• Link Dimensioning• Discussion & Conclusion

Link Dimensioning• We can drive dimensioning formula by

Incomplete Gamma function:

Gamma function:

Finally

Link Dimensioning

where C is the capacity, so .

Link Dimensioning

Link Dimensioning

Link Dimensioning

Link Dimensioning

Link Dimensioning

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

• 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

ConclusionIn this presentation, weIn 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:

References:

Q & AQ & A

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”

[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.

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

[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.

[6] M. Zukerman, T. D. Neame, and R. G. Addie, “Internet traffic modeling and future technology implications,” in Proc. IEEE INFOCOM

2003,vol. 1, Apr. 2003, pp. 587–596.

process of Real traffic Gaussian process [2]Central limit

theorem

Especially for core and metropolitan Internet links, etc.

Analytical results of (fBm) Queue

• A single server queue fed by an fBm input process with- Hurst parameter (H)- variance (σ1

2)

- 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]

[16] J. M. Harrison, Brownian motion and stochastic flow systems. New York: John Wiley and Sons, 1985.