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HUGHES Research Labs The Importance of Long-Range Dependence of VBR Video Traffic in ATM Traffic Engineering: Myths and Realities Bo Ryu Anwar Elwalid (Bell Labs) ACM SIGCOMM 96 Stanford University, CA 1 of 22 HUGHES Research Labs

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The Importance of Long-Range Dependence of VBR Video Traffic in ATM Traffic Engineering: Myths and Realities Bo Ryu Anwar Elwalid (Bell Labs)

ACM SIGCOMM ‘96 Stanford University, CA

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Background

  • 1. VBR video traffic exhibits long-range dependence (LRD).
  • 2. Wide interest & general concern.
  • 3. Debate on the relevance of LRD.
  • “Fatter-than-exponential” tail of ATM buffer overflow

probability.

  • Prior work on video modeling with simple Markovian model

produces good results.

[Elwalid, Heyman, Lakshman, Mitra, and Weiss; IEEE JSAC, Aug. 1995]

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200 400 600 Lag (k) 0.0 0.2 0.4 0.6 0.8 1.0 Autocorrelation r(k)

Star Wars Movie

Hurst =~ 0.8 [Garrett and Willinger 1994]

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LRD SRD B (buffer size) Log[cell loss prob.] ~ A1B ~ C1B2-2H

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Outline How important is LRD of real-time video applications in ATM traffic engineering? Buffer Size (max delay): < 20 ~ 30 msec Cell Loss Prob. (PLoss): < 10-6 Ι: Effect of long-term and short-term correlations on PLoss ΙΙ: Efficacy of Markov models in predicting PLoss ΙΙΙ: Relevant range of dependence (critical time scale)

Note: (i) video model rather than trace (ii) same marginal distribution of frame size (Gaussian)

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Definitions X = {X1, X2,...} WSS process with ACF r(k).

  • X is asymptotic LRD process if

r(k) ≈ Ak-(2-2H), (k large)

  • H: Hurst parameter (1/2 < H < 1)

(Note: Short-term correlations are arbitrary)

  • X is exact LRD if

r(k) = 1/2 δ2(k2H), k = 1, 2,...

ex) Fractional Gaussian Noise, Fractal modulated Poisson processes

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  • I. Effect of short- and long-term correlations on PLoss
  • Construct two asymptotic LRD processes Za & Vv by

Za, Vv = DAR(1) + FMPP DAR(p): Discrete Auto-Regressive model with order p. FMPP: Fractal Modulated Poisson Process

  • Za: same long-term, varying short-term correlations.
  • Vv: same short-term, varying long-term correlations.

rv k ( ) rz k ( ) v v 1 +

  • ak

1 v 1 +

  • 1

2

  • δ k2H

( ) ⋅ ⋅ + ⋅ = =

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DAR(p) Process

{εn}: i.i.d. R.V. with distribution π (εn ∈ Ζ) {Vn}: Bernoulli R.V. (Vn ∈{0,1}) {An}: i.i.d. R.V with Pr(An = i) = ai, i = 1, 2,..., p. (An ∈ {1, 2,..., p})

  • Correlations independent of marginal distribution π.
  • Correlations matching up to p lags.
  • Computationally efficient.

Xn VnXn An – 1 Vn – ( )εn + = rX k ( ) bizi k – i 1 = p ∑ =

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  • Completely characterized by R, M, and pdf of on/off sojourn times.
  • , marginal distribution of {Yn} controlled by M.
  • Computationally efficient

Fractal Modulated Poisson Process (ex: FBNDP)

  • Poisson

Generator N(t)

R FBN

  • R

I(t) M On/Off Processes Yn = N[nT] -N[(n-1)T] Heavy-tailed rY k ( ) 1 2

  • δ2 k2H

( ) =

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10 of 22 Buffer Size (msec) Log10[ Buffer Overflow Probability ] 5 10 15

  • 8
  • 6
  • 4
  • 2

Z with a = 0.7 Z with a = 0.9 Z with a = 0.975 Z with a = 0.99 Lag k (time unit Ts = 40 msec) Autocorrelation r(k) 20 40 60 80 100 0.0 0.2 0.4 0.6 0.8

v = 0.67 v = 1.0 v = 1.5

Lag k (time unit Ts = 40 msec) Autocorrelation 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8

L with H = 0.86 Z with a = 0.7 Z with a = 0.9 Z with a = 0.975 Z with a = 0.99

Buffer Size (msec) Log10[ Buffer Overflow Probability ] 5 10 15

  • 8
  • 6
  • 4
  • 2

v = 0.67 v = 1.0 v = 1.5

Varying Short-term Correlations (Za)

Effect on Cell Loss Prob.

Varying Long-term Correlations (Vv)

Effect on Cell Loss Prob.

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  • II. Efficacy of Markov models in predicting PLoss of LRD traffic
  • Target (asymptotic) LRD process Za
  • DAR(p): matches the first p (p small) correlations
  • Exact LRD model L based on FMPP: matches only the long-term

correlations (Hurst parameter) of Za.

  • Marginal distribution is same for all the models.
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Buffer Size (msec) Log10[ Buffer Overflow Probability ] 5 10 15 20

  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2

Z with a = 0.7 DAR(1) DAR(3) DAR(6)

N = 30 Sources, mean = 500 (cells/frame), variance = 50000, capacity = 608 (cells/frame), util = 82%

Bahadur-Rao Asymptotic (Gaussian marginal distribution)

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  • L underestimating Za
  • Larger p, better prediction

Buffer Size (msec) Log10[ Buffer Overflow Probability ] 5 10 15

  • 8
  • 6
  • 4
  • 2

Z with a = 0.975 DAR(1) DAR(2) DAR(3) L with H = 0.86

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  • L eventually outperforms DAR(p), but only over the range of no

interest.

Buffer Size (msec) Log10[ Buffer Overflow Probability ] 20 40 60 80 100 120

  • 20
  • 15
  • 10
  • 5

Z with a = 0.975 DAR(1) DAR(2) DAR(3) L with H = 0.86

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Analysis of Buffer Overflow Probability [Courcoubetis & Weber, Duffield, De Veciana, etc.]

  • For N Gaussian sources, each with mean µ, variance σ2, and ACF r(k),
  • b = amount of buffer space per source (B = Nb)
  • c = amount of bandwidth per source (C = Nc)

P W B > ( ) NI c b , ( ) – g c b N , , ( ) + ( ) exp =

g c b N , , ( ) N ⁄ N ∞ → lim = I c b , ( ) infm 1 ≥ b m c µ – ( ) + [ ]2 2V m ( )

  • ≡

V m ( ) Var Xi i 1 = m

∑

      ≡ σ2 m 2 m i – ( )r i ( ) i 1 = m

∑

+ =

src 1 src N B C

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Relevant Range of Dependence (Correlation): Critical Time Scale (CTS)

  • For given buffer size b and link capacity c,

(in units of frame) ☞ Only the first (m*

b-1) correlations are needed to evaluate P(W>B).

☞ Correlations beyond time scales ≥ m*

b are irrelevant to P(W>B).

☞ CTS ≡ m*

b

m∗b inf arg m 1 ≥ b m c µ – ( ) + [ ]2 2V m ( )

  • =
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Facts on CTS

  • m*

0 = 1

⇒ No buffer, no effect of correlation on cell loss rate!

  • m*

b < ∞ as long as b < ∞.

  • m*

b is linear with b for large b.

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Related Work on CTS

  • Frequency domain analysis [Li and Hwang]

⇒ cutoff frequency. ⇒ Low frequency behavior (long-term correlations) dominant impact on queueing performance. (???)

  • Direct relation between cutoff frequency and CTS

[Montgomery and DeVaciana].

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Simulation Study with Star Wars Movie

Simulation Setting: Trace: Star Wars (intra-frame coding only) [Garrett 1993 PhD thesis] Hurst parameter: about 0.8 cell size: 44 bytes/cell mean rate = 632 cells/frame min rate = 196 cells/frame max rate = 1784 cells/frame capacity = 725 cells/frame link utilization = 0.89 number of sources = 20 cell loss curve of the trace averaged over 5 different sets of starting points Empirical marginal distribution used for DAR(p), p = 1,2,3. Length of each replication = 10,000 sec number of replications per model = 60

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N B = Nb C = Nc Cell Loss (finite buffer)

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100 200 300 400 500 600 Lag (k) 0.0 0.2 0.4 0.6 0.8 1.0 Autocorrelation r(k) Trace DAR(1) DAR(2) DAR(3)

Short-term Correlations Matching for Star Wars trace with DAR(p)

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20 40 60 80 Buffer Size (msec) −7 −6 −5 −4 Log10[ CLR ] Trace DAR(1) DAR(2) DAR(3)

Comparison of Cell Loss Probability