Abstract
Video represents a larger and larger portion of the traffic in Internet. H.264/AVC and its scalability extension have recently become some of the most widely accepted video coding standards. Consequently, adequate models for this traffic are very important for the performance evaluation of networks architectures and protocols. Particularly, the efficient and on-line generation of synthetic sample paths is fundamental for simulation studies. In this work, we check the suitability of the M/G/to process for modeling the spatial and quality scalability extensions of the H.264 standard.
Keywords: H.264 video traffic modeling; Scalability; M/G/to process; Whittle estimator.
Introduction
Improvements in network infrastructures, storage capacity and computing power, along with advances in video coding technology, are enabling an increasing popularity of multimedia applications.
Modern video transmission systems are typically characterized by a wide range of connection qualities and receiving devices with different characteristics: displays, processing power, … Scalability, that allows to remove parts of a video stream in order to adapt it to preferences of users, characteristics of terminals and changing network conditions, is a good solution for modern video transmissions.
H.264/AVC and the SVC extension [1] have recently become some of the most widely accepted video coding standards, because they have demonstrated significantly improved coding efficiency, substantially enhanced error robustness and increased flexibility and scope of applicability relative to prior video coding standards, such as H.263 and MPEG2.
Traffic modeling plays an important role in the performance evaluation of network architectures and protocols. In last decade, several traffic studies have convincingly show the existence of persistent correlations in several kinds of traffic, as VBR video [2-5] and that the impact of the correlation on the performance metrics may be drastic, and several works have been conducted in modeling VBR video traffic based on different stochastic processes [6-12] that display different forms of correlation. We focus on the M/G/ro-type processes [13, 7, 14] for its theoretical simplicity, its flexibility to exhibit both Short-Range Dependence (SRD) and Long-Range Dependence (LRD) in a parsimonious way and its advantages for simulation studies, such as the possibility of on-line generation and the lower computational cost [15].
In order to apply a model to the synthetic generation of traces with a correlation structure similar to that of real sequences, a fundamental problem is the estimation of the parameters of the model. Between the methods proposed in the literature [16-19], those based on the Whittle estimator are especially interesting because they permit to fit the whole spectral density and to obtain confidence intervals of the estimated parameters. Moreover, in [20] we have presented a method based on the prediction error of the Whittle estimator to choose, between several models for compressed VBR video traffic based on the M/G/ro process, the one that gives rise to a better adjustment of the spectral density, and therefore of the correlation structure, of the traffic to model.
In this work we check the suitability of the M/G/ro process for modeling the spatial (different spatial resolutions) and quality (different fidelity levels) scalability extensions of the H.264 standard.
The remainder of the paper is organized as follows. We begin reviewing the main concepts related to the M/G/ro process in Section 2 and those related to the Whittle estimator in Section 3. In Section 4 we explain the M/G/ro-based models that we consider in this work and in Section 5 we apply then to the modeling of H.264/SVC video traffic at the GoP level.
Theprocess [21] is a stationary version of the occupancy process of an queueing system. In this queueing system, customers arrive according to a Poisson process, occupy a server for a random time with a generic distribution X with finite mean, and leave the system.
Though the system operates in continuous time, it is easier to simulate it in discrete-time, so this will be the convention henceforth [14]. The number of busy servers at timeis the number of arrivals at time t — i which remain active at time t, i.e., the number of active customers with age i. For any fixed t,are a sequence of independent and identically distributed (iid) Poisson variables with parameterwhereis the rate of the arrival process. The expectation and variance of the number of servers occupied at time t is
The discrete-time process Yt,t = 0,1,… is time-reversible and wide-sense stationary, with autocovariance function
The function 7(h) determines completely the expected service time
and the distribution of X, the service time, because
By (1), the autocovariance is a non-negative convex function. Alternatively, any real-valued sequence 7(h) can be the autocovariance function of a discrete-time M/G/to occupancy process if and only if it is decreasing, non-negative and integer-convex [7]. In such a case,and the probability mass function of X is given by (1).
If Aojo (i.e., the initial number of customers in the system) follows a Poisson distribution with mean, and their service times have the same distribution as the residual life X of the random variable X
then {Yt, t = 0,1,… } is strict-sense stationary, ergodic, and enjoys the following properties:
1. The marginal distribution of Yt is Poissonian for all t, with mean value
2. The autocovariance function is
If the autocovariance function is summable the process exhibits SRD. Conversely, if the autocovariance function is not summable, the process exhibits LRD. In particular, the M/G/to process exhibits LRD when X has infinite variance, as in the case of some heavy-tailed distributions. The latter are the discrete probability distribution functions satisfyingasymptotically as
Whittle Estimator
Let) be the spectral density function of a zero-mean stationary Gaussian stochastic process, X, where 0 = (0i,…, 0M) is a vector of unknown parameters that is to be estimated from observations. Let
be the periodogram of a sample of size N of the process X. The approximate Whittle estimator [16] is the vector 0 = (01,…,0M) minimizing, for a given sample X of size N of X, the statistic
If 0° is the true value of 0, then
for any e > 0, namely, 0 converges in probability to 0° (is a weakly consistent estimator). It is also asymptotically normal, sinceconverges in distribution tois a zero-mean Gaussian vector with matrix of covariances known. Thus, from this asymptotic normality, confidence intervals of the estimated values can be computed.
A simplification of (2) arises by choosing a special scale parameter 0i, such that
and
whereis the optimal one-step-ahead prediction error, that is equal to the variance of the innovations in therepresentation of the process [22],
Using this normalization, equation (2) simplifies to
which is usually evaluated numerically via integral quadrature.
Additionally [22]
We useas a measure of the suitability of a model, since smaller values of mean better adjustment to the actual correlation of the sample.
Based Models
In this work we consider as distribution for the service time the discrete-time distribution S, proposed in [14]. Its main characteristic is that of being a heavy-tailed distribution with two parameters, a and m, a feature that allows to model simultaneously the short-term correlation behavior (by means of the one-lag autocorrelation coefficient r(1)) and the long-term correlation behavior (by means of the H [23] parameter) of the occupancy process. Specifically, the autocorrelation function of the resulting M/S/to process is
with
Ifthen. Hence, in this case this correlation structure gives rise to an LRD process.
The spectral density, needed to use the Whittle estimator, is given by [19]
where /h is the spectral density of a FGN [22] process withscaled by the variance.
In order to improve the adjustment of the short-term correlation of the previous process, in [24] we have proposed to add an autoregressive filter. Specifically, we focus on the particular case of an AR(1) filter.
If Y is theoriginal process, the new one is obtained as Yn.
The mean values and covariances are related by
The spectral density results
We denote the resulting process as
Modeling H.264/SVC Video Traffic at the GoP Level
We consider, as an example, the following empirical video traces of the Group of Pictures (GoP) sizes available at [25].
— Spatial scalability:
• T-1: "Star Wars IV (layer QCIF)".
• T-2: "Star Wars IV (layer CIF)".
— Quality scalability:
• T-3: "Star Wars IV (base layer)".
• T-4: "Star Wars IV (first enhanced layer)".
• T-5: "Star Wars IV (second enhanced layer)".
In order to adjust simultaneously the marginal distribution and the autocorrelation, as the marginal distribution in all cases is approximately Lognormal, we apply a change of distribution. In each case, A denotes the process we want to generate and C the M/G/to process from which we start off, that should have a high enough mean value so as the Poissonian marginal distribution can be considered approximately Gaussian (we select oQ = pc = 104). Moreover, we consider the intermediate process B = log(A), from which we estimate the parameters.
If A has Lognormal marginal distribution, then B = log(A) has Gaussian marginal distribution, with mean, variance and autocorrelation given by [26]
The estimations of the parameters of theand theprocesses,computed via the Whittle estimator, are as follows:
Table 1. Estimations of the prediction error with each model
T-l |
T-2 |
T-3 |
T-4 |
T-5 |
1.0007 |
1.0054 |
1.0002 |
1.003 |
1.0104 |
Fig. 1. Spatial scalability. Adjustment of the autocorrelation. QCIF layer (left) and CIF layer (right).
Fig. 2. Spatial scalability. Adjustment of the marginal distribution. QCIF layer (left) and CIF layer (right).
In Table 1 we show the relationship between the estimations of the prediction error
The results show that, in all cases, increasing the number of parameters leads to smaller prediction errors.
Once we have generated a sample of the process C, to obtain Lognormal marginal distribution with the mean value and variance of the empirical traces, we apply the inverse transformation
beingthe estimation of the variance of B computed with the Whittle estimator, that is, considering the autocorrelation structure.
Fig. 3. Quality scalability. Adjustment of the autocorrelation. Base layer (top), first enhanced layer (bottom left), second enhanced layer (bottom right).
Fig. 4. Quality scalability. Adjustment of the marginal distribution. Base layer (top), first enhanced layer (bottom left), second enhanced layer (bottom right).
In Figs. 1, 2, 3, and 4 we represent the autocorrelation function and the marginal distribution of synthetic traces of theprocesses. We can observe a good math with the empirical traces in both cases.
Conclusions and Further Work
In this paper, we have checked the suitability of the M/G/to process for modeling the spatial and quality scalability extensions of the H.264 standard.
The proposed generators enjoy several interesting features: highly efficient, online generation and the possibility of capturing the whole correlation structure in a parsimonious way.
As further work we are going to include these traffic models in simulators of different systems, in order to use the synthetic traces for performance evaluation of scalable video transmission.