Link Dimensioning for Fractional Brownian Input

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 • 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

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”

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 • 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 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

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 (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) Queue No exact results for P(Q>x) for H ≠ 0.5 Existing asymptotes: • By Norros [9]

Analytical results of (fBm) Queue Existing asymptotes (cont.): • By Husler and Piterbarg [14]

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) 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)

Simulation • Basic algorithm (Lindley’s equation):

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 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.) 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

Link Dimensioning • We can drive dimensioning formula by Incomplete Gamma function: Gamma function:

Link Dimensioning Finally where C is the capacity, so .

Link Dimensioning

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 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:

Q & A

Link Dimensioning for Fractional Brownian Input