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. 115, 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 2. Autocovariance function, (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 2. Autocovariance function, (k) = E[(X t -m)(X t+k -m)], decays slowly. Autocorrelation function (ACF), (k) = (k)/ 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 ( 2 ) and Hurst parameter ( H ). LRD process Input traffic process Single Server Queue (SSQ) Link Overflow probability QoS Mean (m) Variance ( 2 ) Hurst parameter (H) SSQ with buffer Steady state Queue Size (Q) Service rate () 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 [bursts/s]. Burst durations, d, are i.i.d. Lomax random variables with parameters and . Each burst contributes work with constant rate r [B/s]. PLBP: {A t, t0}, 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; : shape parameter; : scale parameter. Advantages of Lomax: takes care of small bursts; is no more minimum value of burst duration. 22 The complementary cumulative distribution function (CCDF) of Pareto distribution Lomax distribution Mean ( m(t) ): Variance ( 2 (t) ): 1x is Number of long bursts, , is Poisson distributed with mean as E(d)P(>). E(d) : the mean number of the existing burst at t ; : the forward recurrence time of a Lomax RV. Q S : the steady state queue size of an SSQ fed by S . Assuming S 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 27 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), t0} : B(0) = 0, E[B(t)] = 0 for all t0 ; Var[B(t)] = t 2H for all t0; 0.5x) 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. 387396, Sep By Hsler and Piterbarg [6]: where C is a certain constant and the it holds in sense that [6] J. Hsler 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 Hsler and Piterbargs 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 denotes the Gamma function. It holds in the sense that 36 Our approximation vs. asymptotics by Hsler 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 t. Q n denotes the queue size at the end of n th interval, defined by Lindleys equation: where Q 0 =0, is the amount of work arriving in each interval. Discrete timeContinuous time Difficulty: t 0 38 For a given H, simulate for different t with one sequence of standard fGn,, with mean and variance v 1. A new sequence is defined by where s(t) and m(t) are chosen so that has the appropriate mean and variance. 39 40 41 42 43 The density function of Q:The density function of Amoroso Distribution: g = 0, d = , p = and a = 1/ = 44 Validation 45 Negative arrivals Appropriate for 1) large buffer size; 2) 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 * : capacity; m : mean of the input process; 2 : variance of the input process; H : Hurst parameter; : required overflow probability; q : buffer threshold; G -1 (): inverse regularised incomplete Gamma function. 48 49 50 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 53