A discrete wavelet transform traffic model with application to queuing critical time scales

Transcription

1 University of Roma ÒLa SapienzaÓ Dept. INFOCOM A discrete wavelet transform traffic model with application to queuing critical time scales Andrea Baiocchi, Andrea De Vendictis, Michele Iarossi University of Roma ÒLa SapienzaÓ - INFOCOM Dept. - Roma (Italy) baiocchi@infocom.uniroma1.it

2 Summary Background and motivation Wavelet based traffic modeling Comparison of performance metrics predicted by the model and trace-driven simulations An application of wavelet based traffic modeling: traffic ÒextrapolationÓ Relevance of time scales Final remarks

3 Background 3 Key concepts Long Range Dependence (LRD) Self-similarity Heavy tailed distributions ÒStructuralÓ analysis of traffic traces points out that self-similarity is essentially tied to heavy tails at application level self-similarity is an additive property: superposition of many ON-OFF sources, corresponding to typical user behaviour and application features; it appears typically at time scales beyond a few hundreds of ms (asymptotic self-similarity) at smaller time scales the effect of trasport and network protocols may shape significantly the traffic properties (multiplicative nature of traffic multifractality)

4 Objective 4 To define a computationally feasible and parsimonious way of generating synthetic aggregated IP traffic traces To understand traffic dynamlcs and its impact on queueing performance by means of a handy yet accurate traffic model Observed traffic flow Router Buffer Output line To device Òload boxesó to be used in network test-beds Synthetic injected traffic (load box)

5 Wavelet based traffic generator 5 Why wavelets? Almost perfect decorrelation for exactly self-similar processes (FGN) Computationally attractive (O(trace size)) Parsimonious modelling possible Yield naturally a multiplicative structure which appears to be consistent with fine time scale behaviour of traffic measurements Why the Haar Discrete Wavelet Tranform (Haar DWT)? Simple to compute, yet sufficiently accurate (e.g. as for H estimates) Allows explicit expressions, hence easier insight Ease of implementation of some important constraints for the traffic generation application

6 Wavelet based traffic model (1/5) 6 The key concept of MultiResolution Analysis (MRA) is to identify of nested subspaces where a function projection represents its ÒapproximationÓ and approx jð1 (t) = approx j (t) + detail j (t) The MRA based on wavelets expands a signal f(t) as J J = J + j = ( ) Jk, Jk, + ( ) jk, jk, j = k j = k ft () Aft () Dft () f, φ φ () t f, ψ ψ () t where φ J,k (t) and ψ j,k (t), j J, form an orthonormal basis on L (R); they are obtained by scaling and translation of the scaling function φ 0,0 (t) and the mother wavelet ψ 0,0 (t)

7 Wavelet based traffic model (/5) 7 MultiResolution Analysis for a discrete time sequence: ft () Aft () = Aft () + Dft () = a φ () t + d ψ () t 0 J J j Jk, Jk, jk, jk, j = 1 k j = 1k The Discrete Wavelet Tranform (DWT) of a sequence f(t) is the sequence of approximation coefficients at the coarsest scale, a J,k, and the details for the finer scales, d J,k, j=1,é,j. The simplest DWT exploits the Haar basis (Haar DWT) a d jk, jk, = = a a + a j 1, k+ 1 j 1, k a j 1, k+ 1 j 1, k and a a j 1, k+ 1 j 1, k = J = a a + d d jk, jk, jk, jk,

8 Wavelet based traffic model (3/5) 8 For a wide-sense stationary LRD process X(t), the covariance function and the spectrum exhibit power law asymptotics X γ ( H ) 1 H X f γ ( k) ~ c k k and Γ ( ν) ~ c ν ν 0 A key property of the details of the DWT of a LRD sequence is [ d ] ~ c ν Φ0( ν) dν j E j( H H j, 1 ) 1 f H can be estimated by the slope of the linear part of the wavelet scaling plot, i.e. the plot of log ( k d j,k /N j ) vs. j (provided the number of vanishing moments of the wavelet is larger than HÐ1)

9 Wavelet scaling and detail correlation CSELT00ms trace analyzed by means of Wavelet MRA and compared with a Wavelet based synthetic trace 1 0 CSELT00ms sequence Autocorrelation of Wavelet detail coefficients Scale j= x 10 4 log (E j ) 38 Wavelet scaling plot Scale j * CSELT00ms sequence Ð synthetic sequence

10 Wavelet based traffic model (4/5) 10 Positivity through multiplication: for the Haar DWT, requiring that X(k)=a 0,k 0 for any k turns out to be equivalent to d a d = c a, c ~ C [ 11, ] jk, jk, jk, jk, jk, jk, ( j) A Haar DWT of a LRD sequence can be so generated the coarsest approximation is a J,0 = ( ) J E [ X ] Multipliers at scale j are independent samples of the zero-mean symmetric random variable C (j) with variance given by E[ C ] E[ H ( j + 1) C( j )]= 1 E C E[ X ] E X J = + [ ] ( j + 1) [ ] j = 1 1+ E[ C ] ( j )

11 The traffic model as a cascade 11 a 3,0 a,0 = a 3,0 (1Ð c 3,0 )/ () a,1 = a 3,0 (1+ c 3,0 )/ () a 1,0 = a,0 (1Ð c,0 )/ () a 1,1 = a,0 (1+ c,0 )/ () a 1, = a,1 (1Ð c,1 )/ () a 1,3 = a,1 (1+ c,1 )/ ()

12 Wavelet based traffic model (5/5) 1 The wavelet based generator requires three input parameters to produce a traffic trace mean variance and Hurst parameter - or wavelet scaling plot Traffic is generated by constructing a DWT sequence, then inverting it, so as to obtain a sequence {X k, k=1,é,n } representing the number of bytes per time quantum Computational complexity is O(N): a few tens of seconds are required for a synthetic trace of length in the order of 100,000

13 Performance comparison (1/3) 13 DEC10ms LBL10ms Loss probability load = 0.8 Loss probability for real trace Loss probability for synthetic trace Loss probability Loss probability for real trace Loss probability for synthetic trace load = Buffer size (byte) Buffer size (byte)

14 Performance comparison (/3) 14 CSELT00ms Bellcore10ms Loss probability 10 0 load = 0.8 Loss probability for real trace Loss probability for synthetic trace Loss probability load = 0.8 Loss probability for synthetic trace Loss probability for real trace Buffer size (byte) Buffer size (byte)

15 Performance comparison (3/3) 15 CSELT10ms trace 10 0 load = buffer size = 10 6 bytes Loss probability Real trace Synthetic trace (real marginal) Synthetic trace Loss probability Real trace Synthetic trace (real marginal) Synthetic trace Buffer size (byte) Output capacity load

16 From large to small scales 16 Loss probability As long as we can assume linear wavelet scaling (self-similarity), the traffic trace can be ÒextrapolatedÓ to fine scale from knowledge of some ÒcoarseÓ scale actual trace extrapolated trace approx. to order 11 ( 6 min 50 s) CSELT00ms ρ= Buffer size (byte) Loss probability 10 0 LBL10 ms ρ= actual trace extrapolated trace approx. to order 13 ( 8 s) Buffer size (byte)

17 Information through time scales 17 % error on loss probability buffer size (byte) B/C B/(CÐE[X]) Bellcore 0 trace rho= Wavelet time scale, j (time = 10 ms x j ) % error on loss probability buffer size (byte) Shuffled Bellcore trace rho= Wavelet time scale, j (time = 10 ms x j )

18 Information through time scales 18 % error on loss probability buffer size (byte) B/C B/(CÐE[X]) Bellcore 0 trace rho= Wavelet time scale, j (time = 10 ms x j ) % error on loss probability B/C B/(CÐE[X]) Bellcore trace rho=0.6 buffer size (byte) Wavelet time scale, j (time = 10 ms x j )

19 Time scale relevance 19 % error on loss probability ms B/C=930 ms DEC10ms rho=0.5 buffer size=100,000 byte Wavelet time scale, j (time = 10 ms x j ) % error on loss probability ms B/C=67 ms 30 CSELT10ms 0 180ms rho= buffer size=100,000 byte Wavelet time scale, j (time = 10 ms x j )

20 Summary and a look ahead 0 The Haar DWT traffic generator produces ÒpessimisticÓ traces for TCP aggregated traffic (e.g. DEC, LBL), ÒoptimisticÓ traces for overall aggregated traffic (e.g. CSELT, Bellcore), overall achieveing a satisfactory accuracy correctly captures the qualitative behaviour of the real traces (large lag autocovariance decay, marginal distribution, scaling behaviour) Research directions Analysis of individual TCP connection traffic Effect of end-to-end feedback congestion control on traffic modeling and resource dimensioning: what about Òopen-loopÓ models? Impact of UDP traffic supporting real time traffic flows