High-Fidelity Real-Time Modeling and Simulation of Network Traffic Processes

October, 1998 DARPA / B. Melamed 1

High-Fidelity Real-Time Modeling and Simulation

of Network Traffic Processes

Khosrow SohrabyComputer Science TelecommunicationsUniversity of Missouri-Kansas City5100 Rockhill Rd.Kansas City, MO 64110

Benjamin MelamedRutgers UniversityFaculty of ManagementDept. of MSIS94 Rockafeller Rd.Piscataway, NJ 08854

DARPA/ITO BAA 97-04 AON F316


• Emerging high-speed telecommunications networks increasingly carry highly bursty traffic

• compressed video• file transfer

• Network modeling and analysis technologies are urgently needed (witness Internet congestion)

• network control (admission and congestion)• network provisioning and planning

• Problem: traditional analytical/simulation models are unsuitable for emerging networks

• traffic models• queueing models

MOTIVATION


• Traffic is modeled as a time series (stochastic process)

• interarrival intervals time series (between jobs)• variable bit rate (VBR) time series (e.g., compressed

VBR video)

• Traditional analysis assumes traffic time series is iid (independent identically distributed)

• assumptions ignore dependencies to simplify analysis

• But real-life traffic processes are not independent• traffic time series tend to be heavily autocorrelated• traditional analysis produces wrong predictions• autocorrelations must be incorporated into modeling!

ENTER AUTOCORRELATED TRAFFIC...


• Correlation is a measure of linear dependence between random variables

• the correlation coefficient of random variables X and Y is Corr(X,Y) = ( E[XY] - E[X]E[Y] ) / sqrt(V[X]V[Y])

• Autocorrelation function of a stationary random process {Xk} maps time lags between its random variables to their correlation coefficients

• acf(n) = Corr(Xk,Xk+n), n = 0,1,2• n is the lag

• The autocorrelation function, acf(n), captures

temporal (time) dependence• correlation / autocorrelation is one aspect of dependence• used routinely as a good proxy for temporal dependence

WHAT ARE AUTOCORRELATIONS?


IMPACT OF AUTOCORRELATIONS!!!

6000%

4000%

2000%

Acf(1)

Source : M. Livny, B. Melamed and A.K. Tsiolis,“The Impact of Autocorrelation on Queueing Systems”, Management Science 21(3), 322--339, 1993

25000%

20000%

15000%

10000%

5000%

0%

-.55 -.4 -.25 0 .25 .5 .75 .85

% error of mean waiting timeof TES/M/1 relative to M/M/1

Utilization = 80%

Acf(1)

10000%

8000%

0%

-.55 -.4 -.25 0 .25 .5 .75 .85

% error of mean waiting timeof TES/M/1 relative to M/M/1

Utilization = 25%


• The candidate model should be selected from a versatile class of stationary stochastic processes

• general marginal distributions• wide variety of autocorrelation functions (e.g., monotone, oscillatory, alternating, etc.)• broad qualitative range of sample path behavior (e.g., cyclical, non-directional, etc.)

• The candidate model should satisfy:• the marginal distribution of the model should match the empirical distribution (histogram)• the autocorrelation function of the model should approximate the empirical autocorrelation function• Monte Carlo simulated model paths (histories) should

“resemble” the empirical data

MODEL GOODNESS-OF-FIT CRITERIA


• TES is a new modeling methodology• designed to satisfy the 3 goodness-of-fit criteria• fast generation of TES sample paths• fast computation of TES autocorrelation functions• negligible memory for these computations• however, model search is not yet real-time

• QTES (Quantized TES) modeling methodology is a new discrete version of TES modeling methodology

• reduces the continuous TES state space to a finite space• integration operators reduce to finite matrices• can be used to solve queueing models with accurate traffic (arrival) processes, directly from empirical data

records of measurements

TES / QTES MODELING METHODOLOGIES


• Inversion Method• let X be an arbitrary random variable with cumulative

distribution function (cdf) F (and inverse F -1) • Let U be a Uniform random variable (available on most computers)• then Y = F -1(U) is a random variable with distribution F

• Iterated Uniformity• let <x> be the fractional part of x (modulo-1 operator)• let U be a random variable, uniform on [0,1)• let V be any random variable, independent of U• then, <U + V > is a random variable, uniform on [0,1), regardless of the distribution of V !!! • Therefore, choosing V selects a dependence structure without changing the (uniform) distribution!!!

TES MODELING ELEMENTS


• TES terminology • let H be the empirical histogram cdf and H -1 its inverse

• let Sxi be a stitching transformation, with xi in [0,1],

where Sxi(y) = y / xi, for y in [0,xi)

Sxi(y) = (1 - y) / (1 - xi), for y in [xi,1)

• let {Vn} be an innovation sequence (iid random variables,

independent of a uniform [0,1) random variable U0 )

• let D(x) = H -1(Sxi (x)) be the corresponding distortion

• Define two TES background (auxiliary) sequences• TES+: U0

+ = U0,; Un+ = < Un-1

+ + Vn >

• TES-: Un- = Un

+ for n even; Un- = 1 - Un

+ for n odd

• Define two TES foreground (target) sequences• TES+: Xn

+ = D(Un+ ) = H -1(Sxi (Un

+ ))

• TES-: Xn- = D(Un

- ) = H -1(Sxi (Un- ))

TES PROCESSES


• Geometric interpretation

TES+ BACKGROUND PROCESSES

Step-function Innovation density

Un+< Un + Ln>+ < Un + Rn>+

Unit circle


THE TES MODELING PARADIGM

stitching parameter

0

1

1xi

stitching transformation

y

Sxi(y)

+Sxi(Un )

+Un

unit circle

previousbackground

variatenext

background variate

+Un-1

Un = < Un-1 + Vn >+ +

Inversehistogram

cdf

nextforeground

variate

0 1+Sxi(Un )

Xn = H-1(Sxi(Un ))+

H -1(x)

x


• Basic results • every background TES process is a Markov sequence, uniformly distributed on [0,1)• using the inversion method, a TES foreground sequence can be endowed with any prescribed distribution, regardless of its autocorrelation structure !!!• the TES modeling methodology searches for pairs (xi,fV) (stitching parameter and innovation density) that approximate the empirical autocorrelation function

• Conclusion• TES modeling effectively decomposes the fitting of the empirical autocorrelation function and the fitting of the empirical distribution• experience shows that it often produces high-fidelity models, both quantitatively and qualitatively

TES FACTS


• QTES terminology • let M >1 be a positive integer, representing a partition of the unit circle into M equal slices of length h = 1 / M • identify each slice with a state in the set S = {0, 1 ,…, M -1}• let <n>M = n (mod M ) (smallest residual of n modulo M )

• let {Jn} be an innovation sequence (iid random variables

over S, independent of a uniform {0, 1,…, M -1} variate K0 )

• let {Wn(j)

} be an iid sequence uniform on slice [hj, h(j+1) )

• Interpretation• each slice is “collapsed” into a single state, resulting in a finite state space• values within a slice are “indistinguishable”, since as slices get small, these values lie “near” each other• the underlying transition structure (among slices) is finite (in fact, a finite-state Markov process)

QTES PROCESSES


• Define two QTES background (auxiliary) sequences• QTES+: K0

+ = K0; Kn+ = < Kn-1

+ + Jn >M

• QTES-: Kn- = Kn

+ for n even; Kn- = M - 1 - Kn

+ for n odd

• Define two QTES foreground (target) sequences• QTES+: Xn

+ = H -1(Sxi (Wn(Kn+ )))

• QTES-: Xn- = H -1(Sxi (Wn(Kn

- )))

• Interpretation• QTES background processes are random walks on a “circular lattice”, S, of integers (residuals)• QTES foreground sequences “randomize” the discrete state (slice index) to obtain a continuous state space• however, the underlying transition structure is finite! • nevertheless, QTES satisfies the 3 goodness-of-fit criteria

QTES PROCESSES (Cont.)


• Geometric interpretation

QTES+ BACKGROUND PROCESSES

Sliced unit circle

previousbackground

variate

nextbackground

variate

+Kn-1

Kn = < Kn-1 + Jn >M+ +

slice/state0

slice/state1

slice/stateM-1

slice/statek


• Basic results • every background QTES process is a Markov sequence, uniformly distributed on the integers {0, 1, … , M -1}• the randomization step results in a process which is distributed uniformly on [0,1) • thus, a QTES process can match to any prescribed distribution, and simultaneously approximate a large variety of autocorrelation functions !!!• the TES modeling methodology searches for pairs (xi,fJ)

(stitching parameter and innovation density) that approximate the empirical autocorrelation function

• Conclusion• QTES modeling enjoys all the benefits of TES modeling• however, it has a discrete transition structure which make QTES traffic models it amenable to fast queueing analysis

QTES FACTS


• TES • B. Melamed, "An Overview of TES Processes and Modeling Methodology", in Performance Evaluation of Computer and Communications Systems, (L. Donatiello and R. Nelson, Eds.), 359--393, Lecture Notes in Computer Science, Springer-Verlag, 1993• D.L. Jagerman and B. Melamed, "The Transition and Autocorrelation Structure of TES Processes Part I: General Theory", Stochastic Models 8(2), 193--219, 1992• D.L. Jagerman and B. Melamed, "The Transition and Autocorrelation Structure of TES Processes Part II: Special Cases", Stochastic Models 8(3), 499--527, 1992

• QTES• P. Jelenkovic and B. Melamed, "Algorithmic Modeling of TES Processes", IEEE Trans. on Automatic Control 40(7), 1305--1312, 1995

REFERENCES


EXAMPLE: H.261 COMPRESSED VIDEO


EXAMPLE: MPEG COMPRESSED VIDEO


EXAMPLE: JPEG “STAR WARS” VIDEO


• Numbers • empirical data set size: 500-1000 observations and up• modeling time: 5-10 minutes• analysis time: seconds• Monte Carlo traffic generation: can support 1000-10,000 traffic streams per second of CPU

• Goals• speed up modeling search to seconds (new algorithms and representations, parallelize algorithms)• real time / near real time procedure from traffic measurements to performance predictions

• Status (as of 10/97)• design and implementation of serial version for modeling testbed has begun• serial version of analysis engine is complete

• available in public domain as TELPACK (TELetraffic PACKage) at http://www.cstp.umkc.edu/org/tn/telpack/home.html (information) ftp://ftp.cstp.umkc.edu/telpack/software/ (anonymous FTP)

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High-Fidelity Real-Time Modeling and Simulation of Network Traffic Processes