Performance Evaluation of Long Range Dependent Queues Performance Evaluation of Long Range Dependent Queues Chen Jiongze Supervisor: Moshe ZukermanCo-Supervisor: Ronald G. Addie Department of Electronic Engineering (EE) City University of Hong Kong Supported by Grants [CityU ] and [CityU 8/CRF/13G]

 Introduction  Poisson Lomax Burst Process (PLBP) Queue  Fractional Brownian motion (fBm) Queue  Conclusions 2

 Introduction  Poisson Lomax Burst Process (PLBP) Queue  Fractional Brownian motion (fBm) Queue  Conclusions 3

Not enough capacity Angry customer 4 Credit:

Too much capacity Lose Money 5 Credit:

CapacityQoS Traffic Engineering Network Engineering Network Planning 6 Credit:

… Data … A Link Traffic Data 7 Credit: Data …

… … … A Link Traffic Data It is unrealistic to replicate the entire traffic on an Internet link! Quality of Service (QoS) ? 8

A Link Traffic Internet Link Input process Queue Queueing System Modelling 9 Data …

… … Traffic Sampling Input process Traffic model Fitting the parameters 10 A good traffic model capture the nature of the traffic: Long Range Dependence (LRD); A small number of parameters; Amenable to analysis.

[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

k LRD MMPP IID Poisson 0 IID – Independent and Identically Distributed MMPP – Markov Modulated Poisson Process  A process, X={X t, t=1,2,…}, with mean m and variance σ 2.  Autocovariance function, γ(k) = E[(X t -m)(X t+k -m)], decays slower than exponential. 12

 A process, X={X t, t=1,2,…}, with mean m and variance σ 2.  Autocovariance function, γ(k) = E[(X t -m)(X t+k -m)], decays slowly.  Autocorrelation function (ACF), ρ(k) = γ(k)/σ 2, follows  Hurst parameter H : the measure of the degree of the LRD. 0.5 < H < 1  the process is LRD  The aggregate process of X with interval t, X (t) follows 13

… Data … Traffic Sampling Input process Traffic model Fitting the parameters 14 Important statistics of traffic: mean ( m ), variance ( σ 2 ) and Hurst parameter ( H ).

 LRD process  Input traffic process  Single Server Queue (SSQ)  Link  Overflow probability  QoS Mean (m) Variance (σ 2 ) Hurst parameter (H) SSQ with ∞ buffer Steady state Queue Size (Q) Service rate (μ) LRD process Output SSQ P(Q>x)? LRD Queue 15

Mean Variance Hurst parameter Capacity Buffer size Blocking probability Mean Variance Hurst parameter Blocking probability Buffer size Capacity Helpful for Traffic Engineering Network Engineering & Network Planning 16

 Introduction  Poisson Lomax Burst Process (PLBP) Queue  Fractional Brownian motion (fBm) Queue  Conclusions 17

 The process is based on a stream of bursts.  Burst arrivals following a Poisson process with rate λ [bursts/s].  Burst durations, d, are i.i.d. Lomax random variables with parameters γ and δ.  Each burst contributes work with constant rate r [B/s].  PLBP: {A t, t≥0}, where A t is the work contributed by all bursts during the interval (0,t]. 18

t Work t Exponential ( ) Lomax ( ,  ) … … PLBP 4r 3r 2r r … 19

t BtBt t Exponential ( ) Common probability mass function (PMF) G … … M/G/ ∞ process … 20

t Work t Exponential ( ) Pareto ( ,  ) … … Poisson Pareto Burst Process (PPBP) 4r 3r 2r r … 21

d : burst duration; γ : shape parameter; δ : scale parameter. Advantages of Lomax: takes care of small bursts; δ is no more minimum value of burst duration. 22 The complementary cumulative distribution function (CCDF) of Pareto distribution Lomax distribution

 Mean ( m(t) ):  Variance ( σ 2 (t) ): 1<γ<2  LRD 23

InputOutput PLBP with parameters: λ, γ, δ and r. SSQ with ∞ buffer and service rate, μ. SSQ Obtain the overflow probability, P(Q>x), by: Analytical result: the Quasi-stationary (QS) approximation, Simulation: the fast simulation method. 24

Long burst process ( L τ ) Short burst process ( S τ ) t+τ t t PLBP Process τ For a certain period τ, the probability of Q>x is Number of long bursts, η, is Poisson distributed with mean as λE(d)P(ω>τ).  λE(d) : the mean number of the existing burst at t ; ω : the forward recurrence time of a Lomax RV. Q S : the steady state queue size of an SSQ fed by S τ.  Assuming S τ is Gaussian, we can obtain P(Q S >x) by [2]. QS approximation: [2] R. G. Addie, P. Mannersalo, and I. Norros, Performance formulae for queues with Gaussian input, ser. Teletraffic Engineering in a Competitive World. Elsevier Science, Jun. 1999, pp. 1169–

Fast simulation method Conventional simulation method T Initial short bursts Initial long burst … 26

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 Introduction  Poisson Lomax Burst Process (PLBP) Queue  Fractional Brownian motion (fBm) Queue  Conclusions 28

 Three characteristic features:  A continuous Gaussian process;  Self-similar with parameter H ;  Stationary increment.  A normalized fBm process, B={B(t), t≥0} :  B(0) = 0, E[B(t)] = 0 for all t≥0 ;  Var[B(t)] = t 2H for all t≥0;  0.5<H<1  LRD;  Covariance function: 29

 The increment process – fractional Gaussian Noise (fGn):  Cumulative work arrival process: X = {X(t), t≥0}, where X(t) = mt+σB(t), thus  E[X(t)] = mt, Var[X(t)] = σ 2 t 2H ;  the mean and variance of its increment process are m and σ 2. InputOutput fBm process with parameters: m, σ and H. SSQ with ∞ buffer and service rate, μ. SSQ 30

 The mean net input ι = m – μ, so an fBm queue has three parameters, ι, σ and H.  By Reich’s formula [3], the queue size ( Q ) is  Exact solution for P(Q>x) for H = 0.5 by Harrison [4] ： [3] E. Reich, “On the integrodifferential equation of takács I,” Annals of Mathematical Statistics, vol. 29, pp. 563–570, [4] J. M. Harrison, Brownian motion and stochastic flow systems. New York: John Wiley and Sons, No exact results for P(Q>x) for H ≠

By Norros [5]: It holds in sense that [5] I. Norros, “A storage model with self-similar input,” Queueing Syst., vol. 16, no. 3-4, pp. 387–396, Sep

By Hüsler and Piterbarg [6]: where C is a certain constant and the it holds in sense that [6] J. Hüsler and V. Piterbarg, “Extremes of a certain class of Gaussian processes,” Stochastic Processes and their Applications, vol. 83, no. 2, pp. 257 – 271, Oct  No method to determine C.  Since for RHS  ∞ as x  0.

Revise Hüsler and Piterbarg’s approach by supposing that it is not the CCDF, but rather the density whose character remains stable for x near ∞. where the density cf(x) is characterized according to 34

Dominant For x near ∞ : We have 35

where Let and, we have and Γ denotes the Gamma function. It holds in the sense that 36

 Our approximation vs. asymptotics by Hüsler and Piterbarg  Advantages: ▪ a distribution ▪ accurate for full range of buffer size ▪ provides ways to derive c  Disadvantages: ▪ Slightly less accurate for very large x 37

 Discrete-time simulation.  Divide time into N intervals of equal length Δt.  Q n denotes the queue size at the end of n th interval, defined by Lindley’s equation: where Q 0 =0, is the amount of work arriving in each interval. Discrete timeContinuous time Difficulty: Δt  0 38

For a given H, simulate for different Δt with one sequence of standard fGn,, with mean ι and variance v 1.  A new sequence is defined by where s(Δt) and m(Δt) are chosen so that has the appropriate mean and variance.  39

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The density function of Q:The density function of Amoroso Distribution: g = 0, d = β, p = ν and a = α−1/ν = 44

Validation 45

 Negative arrivals  Appropriate for 1) large buffer size; 2) σ is large relative to m.  Gaussian  Appropriate for high multiplexing. To illustrate the weaknesses, we compared it with the PPBP model. 46

Small buffer size Large buffer size CISCO routers 47

Deriving the inverse function of our approximation, we have the dimensioning formula as where  μ * : capacity;  m : mean of the input process;  σ 2 : variance of the input process;  H : Hurst parameter;  ε : required overflow probability;  q : buffer threshold;  G -1 (): inverse regularised incomplete Gamma function. 48

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 Introduction  Poisson Lomax Burst Process (PLBP) Queue  Fractional Brownian motion (fBm) Queue  Conclusions 51

Main contributions:  PLBP queues  The PLBP model (a variant of PPBP) is proposed.  An approximation based on the QS algorithm is provided.  A fast simulation method is applied.  fBm queues  New results for queueing performance and link dimensioning are derived.  Important statistics of fBm queues are provided.  An efficient approach for simulation is proposed.  The weaknesses of the fBm model are discussed. 52

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