1. October, 1998DARPA / B. Melamed1 High-Fidelity Real-Time Modeling and Simulation of Network Traffic Processes Khosrow Sohraby Computer Science Telecommunications University of Missouri-Kansas City 5100 Rockhill Rd. Kansas City, MO 64110 Benjamin Melamed Rutgers University Faculty of Management Dept. of MSIS 94 Rockafeller Rd. Piscataway, NJ 08854 DARPA/ITO BAA 97-04 AON F316

2. October, 1998DARPA / B. Melamed2 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

3. October, 1998DARPA / B. Melamed3 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...

4. October, 1998DARPA / B. Melamed4 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 { X k } maps time lags between its random variables to their correlation coefficients acf(n) = Corr(X k,X k+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?

5. October, 1998DARPA / B. Melamed5 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 time of 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 time of TES/M/1 relative to M/M/1 Utilization = 25%

6. October, 1998DARPA / B. Melamed6 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

7. October, 1998DARPA / B. Melamed7 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

8. October, 1998DARPA / B. Melamed8 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 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, 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

9. October, 1998DARPA / B. Melamed9 TES terminology let H be the empirical histogram cdf and H -1 its inverse let S xi be a stitching transformation, with xi in [0,1], where S xi (y) = y / xi, for y in [0,xi) S xi (y) = (1 - y) / (1 - xi), for y in [xi,1) let {V n } be an innovation sequence (iid random variables, independent of a uniform [0,1) random variable U 0 ) let D(x) = H -1 (S xi (x)) be the corresponding distortion Define two TES background (auxiliary) sequences TES + : U 0 + = U 0, ; U n + = TES - : U n - = U n + for n even; U n - = 1 - U n + for n odd Define two TES foreground (target) sequences TES + : X n + = D(U n + ) = H -1 (S xi (U n + )) TES - : X n - = D(U n - ) = H -1 (S xi (U n - )) TES PROCESSES

10. October, 1998DARPA / B. Melamed10 Geometric interpretation TES + BACKGROUND PROCESSES Step-function Innovation density UnUn + + + Unit circle

11. October, 1998DARPA / B. Melamed11 THE TES MODELING PARADIGM stitching parameter 0 1 1xi stitching transformation y S xi (y) + S xi ( U n ) + UnUn unit circle previous background variate next background variate + U n-1 U n = ++ Inverse histogram cdf next foreground variate 0 1 + S xi ( U n ) X n = H -1 (S xi (U n )) + H - 1 (x) x

12. October, 1998DARPA / B. Melamed12 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,f V ) (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

13. October, 1998DARPA / B. Melamed13 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 M = n (mod M ) (smallest residual of n modulo M ) let {J n } be an innovation sequence (iid random variables over S, independent of a uniform {0, 1,…, M -1} variate K 0 ) let {W n (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

14. October, 1998DARPA / B. Melamed14 Define two QTES background (auxiliary) sequences QTES + : K 0 + = K 0 ; K n + = M QTES - : K n - = K n + for n even; K n - = M - 1 - K n + for n odd Define two QTES foreground (target) sequences QTES + : X n + = H -1 (S xi (W n (K n + ))) QTES - : X n - = H -1 (S xi (W n (K n - ))) 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.)

15. October, 1998DARPA / B. Melamed15 Geometric interpretation QTES + BACKGROUND PROCESSES Sliced unit circle previous background variate next background variate + K n-1 K n = M ++ slice/state 0 slice/state 1 slice/state M-1 slice/state k

16. October, 1998DARPA / B. Melamed16 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,f J ) (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

17. October, 1998DARPA / B. Melamed17 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

18. October, 1998DARPA / B. Melamed18 EXAMPLE: H.261 COMPRESSED VIDEO

19. October, 1998DARPA / B. Melamed19 EXAMPLE: MPEG COMPRESSED VIDEO

20. October, 1998DARPA / B. Melamed20 EXAMPLE: JPEG “STAR WARS” VIDEO

21. October, 1998DARPA / B. Melamed21 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) PROJECT INFORMATION

22. October, 1998DARPA / B. Melamed22 PROJECT SUMMARY