11D-1 A Source Model for VBR Video Traffic Based on M ... - CiteSeerX

A Source

Model

for VBR

Video

Traffic

Based

on M/G/cQ

Input

Processes Marwan University

ISR

of ECE AZ

(520)

College

85721

krunzQece.arizona.

Makowski

and Department

University

of Arizona

Tucson, Tel:

Armand

Krunz

Department

of EE

of Maryland Park,

MD

20742

[email protected]

edu

Tel:

621-8731

Abstract

(301)

tence of traffic

405-6844

correlations

at multiple

time scales has

Statistical evidence suggests that the autocorrelation function of a compressed-video sequence is better cap-

motivated some researchers to consider instead longrange dependent (LRD) models, for which the ACF drops off slowly (typically as p(k) w k-~ = e-~ 10gk,

tured by p(k) = e–~fi than by p(k) = k–fi = e–~’og k (long-range dependence) or p(k) = e-~k (Markovian). A video model with such a correlation structure is introduced based on the so-called M/G/ca input pro-

O < /3 < 1), to the extent that ~k p(k) = CO. Advocates of LRD modeling argue that the LRD phenomenon has significant impact on network performance, and thus it must be accounted for in dimen-

Though

cesses.

not

Markovian,

short-range dependence. Using mance under ‘real’ video trafic study

via simulations

the model

exhibits

sioning

the queueing perforas a reference, we

the queueing

performance

un-

der two video models: the M/G/ca model and the fractional ARIMA (F-ARIMA) model (which exhibits LRD). Our results indicate that the M/G/w model is much more accurate in predicting the actual queueing

works with

two

visioned.

F-ARIMA

1

to Q(n2]

for

a

trace.

On the other side, support-

finite

buffers

it is sufficient lag that

to incorporate is proportional

The two asymptotic forms of autocorrelation associated with Markovian and LRD models represent

M/G/co

n, compared

resources.

correlations up to some finite to the buffer size [6, 7, 21].

performance than the F-ARIMA model. Furthermore, only C7(n) computations are required to generate an trace of length

network

ers of Markovian modeling, while acknowledging the presence of such a phenomenon, argue that for net-

extremes

within

which

For example,

other

forms

one can envision

can be ena versatile

class of stochastic

processes in which the ACF has the generic form p(k) w e ‘$(k) for some monotone function $: IN + IR+ which increases no slower than log k

Introduction

Several studies have recently indicated the persistence of the autocorrelations in various types of network traffic, including Ethernet LAN [4, 14], WAN [20], and variable-bite-rate (VBR) video traffic [1, 5, 11]. Such persistence have spurred an ongoing debate

but no faster thank. The challenge in traffic modeling is to identify such a diverse class and use it to display various forms of correlations. One such class, which is considered here, is the class of M/G/co processes, which are obtained from the (correlated) busy-server process of a discrete-time M/G/m. The viability of

on its relevance to the dimensioning sources. While researchers generally

of network reagree on the im-

M/G/co processes for modeling network traffic can be attributed to several factors [18]. Firstly, they con-

portance

still

of traffic

how much of them model. ture, with

Earlier

correlations, should

traffic

they

be incorporated

models

disagree

on

in a traffic

are Markovian

in na-

an autocorrelation

function p(k) that drops off exponentially; p(k) N e–~k for large k (~ > 0). These Markovian models exhibit short-range dependence (SRD), in that the autocorrelation function

stitute play

a versatile various

of which

class of processes,

forms

is governed

of time

which

dependencies,

by the service time

G. Secondly, the M/G/aJ teletraffic as the limiting

can dis-

the extent distribution

model arises naturally case for the aggregation

in of

(ACF) is summable, i.e., ~k p(k) < CO. (Note that

on/off sources [15]. Thirdly, queueing performance for these processes is sometimes feasible as demonstrated in [18, 19]. Finally, the computational complexity for

SRD model is not necessarily

generating

Markovian.)

a The persis-

synthetic

0-7803-4386-7/98/$10.00 (c) 1998 IEEE

M/G/m

traces

is only

O(n)

(n

being

the length

of the trace),

which

allows

for fast

trace generation in network simulations. In this paper, we investigate the use of M/G/co processes in modeling VBR compressed video streams. We start by examining the empirical ACF of four VBR video sequences. Statistical evidence suggests that the empirical than

ACF

by p(k)

-

is better captured by p(k) N e–~fi k–~ = e–~ log h (long-range depen-

dence) or p(k) N e‘~k (Markovian), lag between frames. Accordingly, M/G/co-based p(k)

video model Though

w e‘DA.

exhibits

with

where

an ACF

an

of the form

non-Markovian,

SRD. To evaluate

k is the

we introduce

requiring

with

the appropriateness

the additional

fewer computations

n=o,l,

. . .. The process {bn,

as the M/G/co

input

n = O, 1,...}

process.

Foreachn = 0,1,..., we introduce the (n+ 1)-tuple bm). The fact that the process {bn, n = (be, bl,..., 0,1,. ..} exhibits some form of positive dependence is indicated by the following result [17]:

Proposition 1 For any choice of the initial condition i = 1, 2,. . .}, the rv b. and of the service times {ao,i, rvs{bn, n = 0,1, . . .} are associated in the following sense: For any n = O, 1, . . . and any pair of nondecreasing

mappings

~, g : IN”+l

+ IR,

our model

advantage

for generating

traces. The remaining of the paper is structured In Section 2 we give an overview of iV1/G/co

E [fag]

of our

M/G/co model, we study its queueing performance via simulations and contrast it with the (LRD) FARIMA video model [5]. Our results indicate that the M/G/co model is much more accurate than the FARIMA model in predicting the queueing behavior of a real video stream,

[n, n+l), is known

of

provided

as follows. input pro-

cesses. In Section 3 we present the fitting results for the ACFS of four video sequences. The M/G/co-based video model is introduced in Section 4. Issues related to generating synthetic M/G/cm traces are discussed

the expectations

The notion

(1)

exist and are finite.

of association

has been found

was introduced

useful in many

contexts

in [2], and

when formal-

izing the idea of positive correlation. Although the busy-server process is in general strictly

synthetic

> E [f(bn)] E [g(bn)]

stationary,

it does admit

a stationary

godic version [17]. This corresponds to taking rv b. to be Poisson distributed with parameter

not

and er(i) the AE [a];

j = 1, 2,...} to be i.i.d. rvs dis(ii) the rvs {oo,j, tributed according to the forward recurrence time 6 associated with a; this integer-valued rv 8 is distributed

according

to

in Section 5. In Section 6 we present simulations of the queueing performance under the M/G/co and the FARIMA

models,

The paper is concluded

in Section

7.

Several

useful

{b~, n = 0,1,.

2

M/G/co In thk

Input

section,

Processes

we summarize

Proposition some of the impor-

tant properties of M/G/ca processes as they relate to our modeling effort. Additional details can be found in [17]. Consider a discrete-time M/G/w queueing system in which customers arrive in i. i. d. Poisson batches of mean A Let ~n+l be the size of the (n+ l)th batch, i.e., the number of arrivals during time slot [n, n + 1). The arrival

process

can be thought

time version of a Poisson process.

of as a discrete-

Upon arriving

system,

1, n + 2). Let service times batch, respec-

tively.

are ii. d. with

service times

of the

2 The

stationary

obtained

stationary

and

version

[17]: ergodic

version

{b~, n = 0,1,., .} of the busy server process has the following properties:

1. For each n = 0,1,.. ., the rv b; is a Poisson rv AE [a]; with parameter 2. Its covariance structure is given by = AE [a] P [6 > k]

(3)

for all n, k = O, 1, ..., where we use the notation max(s, O) for any real x.

x+ =

r(k)

s cov[b;,

b;+k]

at the

customers are presented to an Arrivals during time slot of servers. serviced at the beginning of slot [n + (7n+l,l, cTn+l,2, . . . be the integer-valued of the (n + l)th for customers 1,2,... It is assumed that

properties

..} are readily

infinite group [n, n + 1) are

a common distribution G. We use u to indicate a generic rv for the service time of a customer. Initially, there are bo customers ing (residual) service

in the system with correspondwhich are times ao, 1, 00,2, ...,

mutually independent. servers (i.e., remaining

Let bn be the number of busy customers) at the end of slot

Henceforth, its stationary

by an M/G/co version

{b~,

input

process we mean

n = O, 1,...}which

is fully

characterized by the pair (A, G) and which we use here as the basis for traffic modeling. We note from (3) that the ACF for an M/G/co process is given by

‘!&p[&>

‘(k)= r(o) By varying

k], k=

G, the process {b;,

0,1, . . . .

n = O, 1,...}

(4)

can dis-

play various forms of positive autocorrelations, the extent of which is controlled by the tail behavior of G.

0-7803-4386-7/98/$10.00 (c) 1998 IEEE

3

Autocorrelations

in

Video

Traffic Star Wars 11

Our modeling study which were compressed

1

is based on four video traces, using three different compres-

sion schemes (see Table 1). These traces are available in the public domain, and the details of their compression can be found in the cited references. Frame sizes are given in 48-byte ATM cells (rounded up to give an integer

number

of cells per frame).

y 20.4

J, \

Compression

Trace

Star Beauty

DCT

Wars [5]

Wizard

Dundee

0.2 -

~.

“..., ---:,

-

-. ;----“’. . . .. .

---------““’ .,...

JPEG MPEG-2

~ -

0.1 -

JPEG

[3]

of Oz [13]

“.., ‘\

(intra-coding)

and the Beast [3]

Crocodile

\

Scheme

100

00

200

300

(1 sequence)

____

_____

-_

. . . . . . . . . ,,, ,,, ,,

400 500 600 k (lag in frames)

700

800

800

1000

(i) Table

1: VBR

traces used in the study.

Beauty and the Seast

In teletrafllc studies, the usefulness of a traffic model lies in its ability to predict network performance for the purpose of dimensioning network resources. Since traffic correlations are known to have a profound impact on queueing behavior, preliminary indications

of the goodness of a model can be obtained

by examining

its correlation

structure.

The ACFS for

the four traces are shown in Figure

1. Each empirical

ACF

(1) e-~~

is fitted

by three

functions:

vian), (2) k–~ (LRD), and (3) e–~fi. chosen because its drop-off behavior

0.8 : \ ‘\.

~8 30.4 -

from

the estimated

.. ...

—

Real

------

a4w6 @j

“,’ .,.,,,

(Marko-

The last fit was is similar to that

0.2

fWG1-)

t

I e4,C01

0.1

k (Ma~k@@

t

of the empirical ACF (other forms are also possible). For fits (1) and (3), /3 is obtained by least-square fitting. For the LRD fit of Star Wars trace, B = 0.4 was obtained

-

,= o.6 z ~ - 0.5

i

01 o

100

200

300

400 500 600 k (lag in frames)

700

800

900

I

1000

(ii)

value of the Hurst

Crocodile Dundee

parameter (Ill = 1 – @/2 x 0.8), which was reported in [5]. For the other traces, the Hurst parameter was estimated using several tools, including variance-time plots, R/S analysis, and Whittle’s approximation. For brevity, we only display the estimated values for the various parameters in Figure 1. Clearly, the Markovian fit drops off much faster than the real ACF, so it only captures the short-term correlations. The LRD fit is not adequate either since it underestimates the correlations Only

In contrast,

and large

e–@

~ - 0.5 g) 20.4

-.~-

-~-

.=---

-----

-----

_

1000, and even beyond.

at very large lags does the LRD

ceptable. small

at lags 1 through

\ \

fit become

ac-

gives a very good fit at

lags, particularly

for the first

traces. Using a larger value for H would not the LRD fit, since k–~ always drops off fast maintains almost a flat appearance. Hence, underestimates the correlations up to some overestimates them beyond that lag.

three

improve and then it always lag, and

0.3 -. ...,,,

,,

900

I 1000

0.2 0.1

I

OL o

100

200

0-7803-4386-7/98/$10.00 (c) 1998 IEEE

300

400 500 eoo k (lag in framea)

(iii)

700

800

Wizard of Oz

fork

1

=0,1,

. . .. Specializing

this last relation

for k =

O and using the fact P [a > O] = 1, we obtain

E[a]-l

= p(0) – p(l)

= 1 – p(l)

= 1 – e-e.

(7)

!

0.7 f

‘\

Combining

~ 1 f! - 0.5 -’1 8 ~ !, 20.4

(6) and (7), we conclude

P[a>kl=

p(k)–p(k+l),

that ~=12

1 – p(1)

-.

\ ‘,

0.3 -

,.

Hence,

. . ...,. ,

0.2 -

... .

P[a=k]

.. . . . ““. -...,,

.,, . . . . .

0.1 -

. . . . . . . . . ... ,,, ,,,

01 o

100

=

P[L7>k-1]-P[f7>k]

=

p(k – 1) – 2p(k) + p(k + 1) 1 – p(1)

I 200

300

(8)

> ,...

400 500 600 k (lag in framas)

700

800

900

.

(9)

1000

Substituting

(5) into

(9) we find

(iv)

Figure

1: ACFS

various

fits.

of four

sequences

along

with With a correlation model is SRD since

M/G/m-Based

4

video

As indicated

Video

in Figure

quence is adequately

A model

with

ing M/G/co

of a video

k = 0,1,...

such an ACF input

se-

(5)

can be constructed

processes.

In teletraffic

studies, it is common practice to try first two moments, the autocorrelation the general shape of the marginal

us-

modeling

to capture structure,

distribution

the and

(partic-

ularly, the tail of this distribution [5, 6, 16]). Of the parameters G and A of the M/G/co process that can be used in the fitting, G can be chosen to provide a given autocorrelation structure via (4), but A can only be fitted

to one moment

capture

both the complete

cluding

mean and variance)

ture,

we proceed

(mean or variance). marginal

and the correlation

in two steps.

Thus, to

distribution

First,

(instruc-

we choose G in

the M/G/co model that provides the target ACF (5). Then, we transform the Poisson marginal distribution into a more appropriate distribution.

4.1

Modeling

the

(5), the

by

Modeling

4.2 p(k) = e-6fi,

of the form

Model

1, the ACF

captured

structure

Correlation

M/G/co

Marginal

model

produces

a Poisson marginal

faster than that video

the frame-size

correlated

distribution,

of the empirical

sequence.

Distribution

Several

variates

whose tail drops

distribution

of a real

fits have been suggested

distribution,

including

Gamma

for

[9], log-

normal [8, 12], and hybrid Gamma/Pareto distributions [5]. Of these fits, the last is particularly appropriate for the tail of the frame-size distribution, and is thus used in our modeling study. As explained in [5], the Gamma part fit is used to capture pirical

distribution,

its tail. functions

in the hybrid Gamma/Pareto the general shape of the em-

whereas

the Pareto

part

Let Fr and FP be the cumulative for Gamma

and Pareto

captures

probability

distributions,

tively. Although no closed-form expression Fr, its density ~r has the following form:

respecexists for

Structure

An ACF of the form (5) is generated

for some parameters

by an M/G/co

process whose service distribution G is determined follows: From (4) and (2) we have p(k) – p(k + 1) = E[a]-l

The with

the

P[u

> k]

as

(6)

the Gamma function. explicit form

Fp(z)

w >0

= 1 – min

0-7803-4386-7/98/$10.00 (c) 1998 IEEE

and s >0,

The Pareto

()

1, ~

with

I’(.)

being

distribution

has the

X>o

(12)

‘“,

with parameters ting. Thehybrid given by

a > 0 and a > 1 determined Gamma/Pareto distribution

Star Wars

by fitFr/P is

‘~ 0.9 *4.074T

0.8

%/~(~)=

{

ifx

Fr (x) ~P(Z)



(13)

x”.

of the Gamma

to deviate from the tail of the Gamma the continuity condition Fr (x*) = FP(x*)

The A

of the Pareto

=

tail,

1 is

process

we can obtain

{Xn,

n

into

a new

transformed

Gamma/Pareto

variates

{Y~,

=

O, 1,...}

F;;p(d

ically.

F~l

30.4 0.3 0.2 0.1 -

I

01 o

100

200

=

F~l

{

derived

(y)

if y >1

F~l (y)

of

(14)

of parameter

– (k/z*)a

from

In principle,

(12) ,and F;l obtained numerthe nonlinear ~ransformation (14)

of the non-transformed

M/G/co

Figure

2: Impact

Synthetic

Trace realizations

process.

is a realization

are often

used in trace-driven simulations of the queueing performance. To investigate the queueing performance under our model and contrast it with the (LRD) FARIMA model [5], we generated many synthetic realizations from the two models with the same hybrid

Gamma/Pareto

marginal

distribution

(in the F-

ARIMA model, the marginal distribution formed from a normal distribution). Each

M/G/co

realization

consists

is trans1,000,000

points (i.e., frame sizes), and each F-ARIMA realization consists of 500,000 points. Due to the correlated nature of cell losses, such long traces are needed to obtain meaningful results under small cell loss probabilities.

Intuitively,

correlations

on the ACF.

of n identically

make it more likely

distributed

is 33, trace Poisson

we have

p.g::nxi > 1 x

[

Taking

is more than

long realization

S 1-

(15)

FPOim(X)n,

n = 100,000

in (15), we get

(32) )100000 = 0.4745,

5070 chance that

an 100,000-

will never reach the maximum

frame-

size of the real trace. The F-ARIMA traces are not as long as the M/G/ca traces since generating F-ARIMA traces of length 1,000,000 is computationally prohibitive. Hosking’s algorithm [10] requires 0(n2 ) computations to generate

a F-ARIMA

formation). trace requires

of

1000

rvs Xl,..., Xn which are associated (Proposition 1). Thus, by well-known properties of associated rvs [2],

i.e., there

models

900

to display the extreme tail of the frame-size distribution. For example, the maximum frame size in the real Star Wars trace is 894 cells. In order to display this value in a transformed M/G/co trace,

P [max Xi > 32] < 1 – (Fp.iss.n

Generation of traffic

of transformation

800

for large frames to follow each other, causing extended periods of buffer overflow. Long traces are also needed

for each $ in Ill.

Synthetic

700

the corresponding value before transformation i.e., Fj\(FpOi.,0n(33)) = 894. The M/G/co

otherwise

could affect the original correlation structure. However, in all our experiments the effect of transformation was barely noticeable. This is illustrated in Figure 2, which depicts the average ACF of ten transformed M/G/oo traces along with the analytical ACF

5

400 500 600 Lag (In frames)

300

using

n = 1,2,...

where FpoaSSO~is a Poisson distribution E [a], and

with

sequence

n = O, 1, ...}

Yn = Fr~~(FPOi,,Om(Xn)),

with

fit. Using along with

- 0.5 g

of a and a. &f/G/co

realizations

z ~

rv. Once the Gamma part is fitted, x* can be estimated graphically by inspecting the tail of the empirical distribution, and determining where it starts

estimates

ACF of transformed

.=0.6

distribution

are obtained by matching the first and second moments of the empirical sequence to those of a Gamma

fitting

---i

As in [5], the parameters

least-square

0.7

trace of length

In contrast, only O(n)

two days of execution

n (before

trans-

the generation

of an M/G/m

computations.

It takes about

to generate

a 500,000-long

F-

ARIMA trace using the S-Plus package, (running on a Spare-10 workstation), compared to less than a minute for a 1,000,000-long M/G/co trace (the M/G/cm traces were generated using a simple C program that simulates an M/G/co queueing system).

0-7803-4386-7/98/$10.00 (c) 1998 IEEE

Queueing

6

Performance

model.

In fact, when U = 40% and B is small,

ARIMA To verify model,

the appropriateness

we investigate

compare

it

to

the

of the M/G/co

its queueing performance

video

performance of

the

and

model

eventually

overestimates

underestimates

the CLR

the F-

and FER,

and

them as B increases to 1000

cells.

F-ARIMA

model. For brevity, we show the results based for one real trace (the Star Wars) and its corresponding M/G/co and F-ARIMA models. The queueing system consists of a single-server FIFO queue with capacity B (in cells) and constant service rate C (in cells per time slot). Two types of simulations are conducted: (1) single-stream performance (i.e., no multiplexing), and

While many video models do not lend themselves to queueing analysis, they still provide the means to generate independent, homogeneous synthetic streams, which are ideal for statistical multiplexing studies. As a proof-of-concept study, we evaluate the multiplexing

(2) multiplexing

performance

assume that

performance.

In all experiments,

cells in each frame

we

6.2

Multiplexed

models,

are evenly distributed

Streams

under

both

For simplicity,

(e.g., 1/30 see). The two per-

aries of the multiplexed

formance measures considered here are the cell loss rate (CLR), and the frame error rate (FER). A frame is in error when one or more of its cells are lost. FER

the time axis is slotted

over the frame duration

is a useful measure for applications that do not implement error concealment mechanisms for recovery from partial frame losses.

6.1

Single

A summary nificant

digits)

of the simulation

results

is given in Table

(to two sig-

2 for three

different

loads: U = 80%, 60%, and 40%. For the M/G/w and F-ARIMA models, the depicted results represent the averages of ten independent runs, At U = 80% and under ‘real’ traffic, both the CLR and the FER are expectedly high. Adding extra buffer barely provides any improvement in performance. In contrast, reducing the load from 8070 to 6070 (i.e., increasing

bandwidth

by 3370) improves

about an order of magnitude. have a bigger impact both

streams in frame

are aligned, periods.

so that

This restric-

sum of the K traces. the jth

Thus, if X~k) indicates

frame in the kth trace, then ~j (slot) in which

not occur,

the aggregate

the buffer

occupancy

trace

= ~~=1

buffer overflow

X~~). can-

can be used to update

at the end of that

erwise, the individual traces performance on a cell-by-cell

the size of

period.

are used to simulate basis.

Oththe

Fortunately, buffer overflow occurs in a small fraction of frame periods. Let Qk denote the queue length at the beginning of the kth slot. It can be shown that either of the following two conditions guarantees no buffer overflow during the kth slot:

by

The buffer size seems to

on the FER than on the CLR. For

U = 80% and U = 60%, the FER

decreases by about

the CLR

and F-ARIMA

tion allows us to significantly reduce the simulation time. To multiplex K streams, we first obtain an aggregate trace {~j : j = 1,. ... n} from the pointwise

For a frame period

Stream

the M/G/co

we assume that frames bound-

for real traffic

50% when B is increased

from

100

under

In the first condition,

the total

arrival

rate over a time

the

slot is less than the service rate. However, this alone is not sufficient to prevent buffer overflow, which could

two models to that under the real stream, it is clear that the M/G/ccJ model provides much accurate pre-

be caused by simultaneous cell arrivals from several streams. For this reason, we also require that Q~ s

dictions

of both CLR

model,

the performance

B – K. In the second condition, the total arrival rate is greater than the service rate over a time slot, but

to 2500 cells.

Comparing

the performance

and FER.

Under

the F-ARIMA

is much more sensitive

to the

buffer size. When B = 2500 and U = 80% or 60%, the F-ARIMA model underestimates the CLR and FER by several orders of magnitude, underestimates the performance

The M/G/co slightly by no more than an

the difference is not large enough to overflow during that slot. Based on these facts,

the number

the buffer

of computations

order of magnitude, and often less than that. The M/G/w model is quite accurate at extremely

needed to simulate the queueing performance for K multiplexed streams is O(n + amWK), where a is the fraction of slots for which neither of the above two con-

small CLRS, as is the case when U = 40Y0. In this regime, errors are mainly due to very large frames. The F-ARIMA predictions improves as well at this

ditions is satisfied and W is the average number of cells per frame during buffer overflow. Typically, alt’