View
215
Download
0
Category
Preview:
Citation preview
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
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).
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
Outline:
• Background• A New Analytical Result of an FBM Queue• Simulation• 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
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
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.
Recommended