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

DARPA/ITO BAA 97-04 AON F316. High-Fidelity Real-Time Modeling and Simulation of Network Traffic Processes. Benjamin Melamed Rutgers University Faculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854. Khosrow Sohraby Computer Science Telecommunications

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

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  1. DARPA/ITO BAA 97-04 AON F316 High-Fidelity Real-Time Modeling and Simulation of Network Traffic Processes Benjamin Melamed Rutgers UniversityFaculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854 Khosrow Sohraby Computer Science Telecommunications University of Missouri-Kansas City 5100 Rockhill Rd. Kansas City, MO 64110 DARPA / B. Melamed

  2. MOTIVATION • Emerging high-speed telecommunicationsnetworks increasingly carry highlybursty 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 DARPA / B. Melamed

  3. ENTER AUTOCORRELATED TRAFFIC... • 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 ignoredependenciesto simplify analysis • But real-life traffic processes are not independent • traffic time series tend to be heavily autocorrelated • traditional analysis produceswrong predictions • autocorrelationsmust beincorporated into modeling! DARPA / B. Melamed

  4. WHAT ARE AUTOCORRELATIONS? • Correlation is a measure of linear dependence between random variables • thecorrelation coefficientof random variables X and Y is Corr(X,Y) = ( E[XY] - E[X]E[Y] ) / sqrt(V[X]V[Y]) • Autocorrelation functionof a stationary random process {Xk} maps timelags 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), capturestemporal(time)dependence • correlation / autocorrelation is one aspect of dependence • used routinely as a goodproxyfor temporal dependence DARPA / B. Melamed

  5. IMPACT OF AUTOCORRELATIONS!!! % error of mean waiting timeof TES/M/1 relative to M/M/1 % error of mean waiting timeof TES/M/1 relative to M/M/1 10000% 25000% 8000% 20000% Utilization = 25% Utilization = 80% 15000% 10000% 5000% 0% 0% Acf(1) -.55 -.4 -.25 0 .25 .5 .75 .85 -.55 -.4 -.25 0 .25 .5 .75 .85 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 DARPA / B. Melamed

  6. MODEL GOODNESS-OF-FIT CRITERIA • The candidate model should be selected from a versatile class of stationary stochastic processes • generalmarginal distributions • wide variety of autocorrelation functions (e.g., monotone, oscillatory, alternating, etc.) • broad qualitative range ofsample pathbehavior(e.g., cyclical, non-directional, etc.) • The candidate model should satisfy: • themarginal distributionof the model should match theempirical distribution (histogram) • theautocorrelation functionof the model shouldapproximate the empirical autocorrelation function • Monte Carlo simulated modelpaths (histories)should“resemble” the empirical data DARPA / B. Melamed

  7. TES / QTES MODELING METHODOLOGIES • TES is a new modeling methodology • designed to satisfy the 3 goodness-of-fit criteria • fast generation of TESsample paths • fast computation of TES autocorrelation functions • negligible memory for these computations • however, model search isnot yetreal-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 solvequeueing modelswith accuratetraffic (arrival) processes, directly from empirical data records of measurements DARPA / B. Melamed

  8. TES MODELING ELEMENTS • Inversion Method • letXbe an arbitrary random variable with cumulativedistribution function (cdf)F (and inverseF -1) • Let U be a Uniform random variable (available on most computers) • thenY =F -1(U)is a random variable with distributionF • Iterated Uniformity • let<x>be the fractional part of x (modulo-1 operator) • letUbe a random variable, uniform on [0,1) • letVbeanyrandom variable,independentof U • then,<U + V >is a random variable, uniform on [0,1),regardless of the distribution ofV!!! • Therefore, choosingVselects a dependence structurewithoutchanging the (uniform) distribution!!! DARPA / B. Melamed

  9. TES PROCESSES • TES terminology • let Hbe the empirical histogram cdf andH -1its inverse • let Sxibe astitching transformation, withxiin [0,1],whereSxi(y)=y / xi,for yin[0,xi)Sxi(y)= (1 - y) / (1 - xi),for yin[xi,1) • let {Vn} be aninnovation sequence(iid random variables,independent of a uniform [0,1) random variableU0 ) • letD(x) =H -1(Sxi(x)) be the correspondingdistortion • Define two TES background (auxiliary) sequences • TES+:U0+=U0,;Un+= <Un-1++Vn > • TES-:Un-= Un+ forneven;Un-=1 -Un+ fornodd • Define two TES foreground (target) sequences • TES+:Xn+=D(Un+) =H -1(Sxi(Un+)) • TES-:Xn- =D(Un- ) =H -1(Sxi(Un-)) DARPA / B. Melamed

  10. TES+ BACKGROUND PROCESSES Un + Step-function Innovation density < Un + Ln> < Un + Rn> + + Unit circle • Geometric interpretation DARPA / B. Melamed

  11. THE TES MODELING PARADIGM nextforeground variate Inversehistogramcdf H -1(x) stitching transformation Sxi(y) + + + + + 1 Xn= H-1(Sxi(Un)) x 0 1 Sxi(Un) Sxi(Un) y 0 xi 1 Un stitching parameter Un = < Un-1+ Vn > + + previousbackground variate Un-1 nextbackground variate unit circle DARPA / B. Melamed

  12. TES FACTS • Basic results • everybackground TES process is a Markov sequence, uniformly distributed on [0,1) • using the inversion method, a TES foreground sequencecan be endowed withanyprescribed distribution,regardlessof 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 effectivelydecomposes the fitting of theempirical autocorrelation functionandthe fitting of theempirical distribution • experience shows that it often produceshigh-fidelitymodels,both quantitatively and qualitatively DARPA / B. Melamed

  13. QTES PROCESSES • QTES terminology • let M >1 be a positive integer, representing a partition of the unit circle into M equal slices of lengthh = 1 / M • identify each slice with a state in the setS= {0, 1 ,…, M -1} • let <n>M = n (mod M ) (smallest residual of n modulo M ) • let {Jn} be aninnovation sequence(iid random variables overS, independent of a uniform {0, 1,…, M -1} variateK0) • 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) isfinite (in fact, a finite-state Markov process) DARPA / B. Melamed

  14. QTES PROCESSES (Cont.) • Define two QTESbackground (auxiliary) sequences • QTES+:K0+=K0; Kn+= <Kn-1++Jn >M • QTES-:Kn-=Kn+ forneven;Kn- =M - 1 -Kn+ fornodd • Define two QTESforeground (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 isfinite! • nevertheless, QTES satisfies the 3 goodness-of-fit criteria DARPA / B. Melamed

  15. QTES+ BACKGROUND PROCESSES + slice/statek previousbackground variate Kn = < Kn-1 + Jn >M + + slice/state1 nextbackground variate Kn-1 slice/stateM-1 slice/state0 Sliced unit circle • Geometric interpretation DARPA / B. Melamed

  16. QTES FACTS • Basic results • everybackground 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 simultaneouslyapproximate 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 adiscretetransition structure which make QTES traffic models itamenableto fast queueing analysis DARPA / B. Melamed

  17. REFERENCES • 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 DARPA / B. Melamed

  18. EXAMPLE: H.261 COMPRESSED VIDEO DARPA / B. Melamed

  19. EXAMPLE: MPEG COMPRESSED VIDEO DARPA / B. Melamed

  20. EXAMPLE: JPEG “STAR WARS” VIDEO DARPA / B. Melamed

  21. PROJECT INFORMATION • 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) DARPA / B. Melamed

  22. PROJECT SUMMARY DARPA / B. Melamed

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