Link Dimensioning for Fractional Brownian Input
Supervisor: Prof. ZUKERMAN, MosheQP Members: Dr. KO, K T
Dr. CHAN, Sammy C H
EE 8001
BYChen Jiongze
Supported by Grant [CityU 124709]
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
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?• Its statistics match those of real traffic (for example,
auto-covariance function)• A small number of parameters• Amenable to analysis
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.
Fractional Brownian Motion
• process of parameter H, MtH are as follows:
• Gaussian process N(0,t2H)• Covariance function:
• For H > ½ the process exhibits long range dependence
How to model
Internet Traffic?• 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
Does fBm meets the requirements?
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
• 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.
Analytical results of (fBm) Queue
No exact results for P(Q>x) for H ≠ 0.5Existing 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• 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
• Basic algorithm (Lindley’s equation):
Simulation
• Basic algorithm:m = - 0.5, Q0 = 0Q1 = max (0, Q0 + U0 + m) = max(0, 1.234 – 0.5) = 0.734Q2 = max(Q1 + U1+ m) = max (0, 0.734 – 0.3551 – 0.5) = 0…
Length of Un = 222 for different Δt, it is time-consuming to generate Un for very Δt)
Δt
Discrete time Continuous timeerrors
An efficient approachInstead of generating a new sequence of numbers, we change the “units” of work (y-axis).
Δt
variance of the fBn sequence (Un): v
WhileVariance in an interval of length (Δt) =
So 1 unit = S instead of 1where
= S
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
An efficient approach (cont.)
Simulation Results• Validate simulation
Simulation Results
Simulation Results
Simulation Results
Simulation Results
Simulation Results
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• 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, 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 & A