12 LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO D ANIEL P. H EYMAN AT&T Labs, Middletown, NJ 07748 T. V. L AKSHMAN Bell Laboratories, Holmdel, NJ 07733 12.1 INTRODUCTION In this chapter we present some of our results concerningsource models for H.261 coded VBR (variable bit rate) video. Video services have been forecasted to be a substantial portion of the traf®c on emerging broadband digital networks. Statistical source models of video traf®c are needed to design networks that delivery acceptable picture quality at minimum cost, and to control and shape the output rate of the coder. For example, one issue is decidingwhether a new video connection can be admitted to a network, and the consequent determination of the bandwidth that must be allocated to the connection to ensure adequate quality of service. A model of the bandwidth that the connection will try to consume is required for this task. In addition to providinga good description of the bandwidth requirements, the source model should be usable in the connection-acceptance decision model. Other chapters that contain related material are Chapters 9, 13, 16, and 17. 12.1.1 Special Propertiesof Video There are some physical reasons why traces from video sources are special. Video is a succession of regularly spaced still pictures, called frames. Each still picture is represented in digital form by a coding algorithm, and then compressed to save Self-Similar NetworkTraf®c and Performance Evaluation, Edited by KihongPark and Walter Willinger ISBN 0-471-31974-0 Copyright # 2000 by John Wiley & Sons, Inc. 285 Self-Similar NetworkTraf®c and Performance Evaluation, Edited by KihongPark and Walter Willinger Copyright # 2000 by John Wiley & Sons, Inc. Print ISBN 0-471-31974-0 Electronic ISBN 0-471-20644-X bandwidth. See, for example, Netravili and Haskell [24] for full information about video coding. A common way to save bandwidth is to send a reference frame, and then send the differences of successive frames. This is called interframe coding. Since the adjacent pictures cannot be too different from each other (because most motion is continuous), this generates substantial autocorrelation in the sizes of frames that are near to each other. To protect against transmission errors, a full frame is sent periodically. Furthermore, when there is a scene change the frames no longer depend on the past frames, so functional correlation ends; this may also end the statistical correlation in the frame sizes. Scene changes require that a complete new picture be transmitted, so the scene lengths have an effect on the trace. For these and several other reasons that are too dif®cult to describe here, video traf®c is different from broadband data traf®c, and so the models and conclusions described in this chapter may not apply to other types of traf®c. Video quality degrades when information is lost during transmission or when the interarrival times of frames are either large or very variable. The latter is controlled by limitingbuffer sizes; frames that arrive late might as well not arrive at all. Video engineers often describe the size of a buffer by the length of time it takes to empty it (which is the maximum delay a frame can incur). Current design objectives are for a maximum delay of between 100 and 200 ms. Since several buffers may be encountered from source to destination and there are other sources of delay (e.g., propagation time), some studies use 10 ms as the maximum buffer size. Frames are transmitted in ®xed size units that we call cells. The rate of information loss is the cell-loss rate or CLR. We are interested in situations where the cell losses occur because of buffer over¯ow. We consider models of a single station, where the buffer size and buffer drain rate are given. Under these conditions, the CLR is controlled by keepingthe traf®c intensity small enough to achieve a performance goal. A typical performance goal is to keep the CLR no larger than 10 Àk , where k is usually between three and six. These contraints on buffer size and CLR (and indirectly on traf®c intensity) give rise to a practical region of operation where the constraints are satis®ed. The notion of ``high'' traf®c intensity is related to these constraints. When the design parameters (e.g., buffer size and processing speed) are speci®ed, the traf®c intensity is high when the constraints are just barely satis®ed. 12.1.2 Source Modeling The central problem of source modelingis to choose how to represent data traces by statistical models. A source model is sought for a purpose, which is usually as an input process to a performance model. We think that a source model is acceptable if it ``adequately'' describes the trace in the performance model at hand. By adequate we mean that when the source model is used in the performance model, the values of the operatingcharacteristics of interest produced are ``close enough'' to the value produced by the trace. The de®nition of close enough may depend on the use to which the performance model will be put. For example, long-range network 286 LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO planningtypically requires less accuracy for delay statistics and loss rates than equipment engineering does. We don't regard good source models for a given trace to be unique. Different purposes may best be served by different models. For example, the DAR and GBAR models described in Sections 12.2.2 and 12.2.3 are designed for different purposes. A consequence of our emphasis on testingsource models by how well they emulate the behavior of the trace they model in a performance model is that the con®dence intervals we emphasize are on the operatingcharacteristics of the performance models. 12.1.3 Outine We divide VBR video into two classes, video conferences and entertainment video. Section 12.2 contains two models for video conferences, and Section 12.3 contains a model for entertainment video. The models are vetted by comparingthe performance measures they induce in a simulation to the performance measures induced by data traces. All of these models are Markov chains, so they are short-range dependent (SRD). Hurst parameter estimates for the time series these models describe indicate the presence of long-range dependence. The reasons that short-range dependent models can provide good models for time series that exhibit long-range dependence are given in detail in Section 1.4. Our results are summarized in the last section. 12.2 VIDEO CONFERENCES Video conferences show talkingheads and may be the easiest type of video to model. The models developed for them will be expanded to describe entertainment video in Section 1.3. 12.2.1 Source Data We have data from three different coders and four video teleconferences of about one-half hour in length. The data consists of the size of each still picture, that is, of each frame. All of the teleconferences show a head-and-shoulders scene with moderate motion and scene changes, and with little camera zoom or pan. All of the coders use a version of the H.261 video codingstandard. The key differences in the sequences are that sequence A was recorded by a coder that uses neither discrete- cosine transform (DCT) nor motion compensation, sequence B was recorded by a coder that uses both DCT and motion compensation, and sequences C and D were recorded by a coder that used DCT but not motion compensation. The graphs in Figs. 12.1 and 12.2 show that the details (presence or absence of DCT or motion compensation) do not have a signi®cant effect on the statistics of interest to us here. The summary statistics of these sequences are given in Table 12.1. All of these sequences are adequately described by negative-binomial marginal distributions and geometric autocorrelation functions. Figure 12.1 shows Q-Q plots of the marginal distributions, which have been divided by their means; the ®t is 12.2 VIDEO CONFERENCES 287 D ATA gamma Fig. 12.1 Q-Q plots for four sequences. Autocorrelation function Lag (frames) Fig. 12.2 Autocorrelation functions. 288 LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO excellent for sequences C and D, good for sequence A, and adequate for sequence B. The negative-binomial distribution is the discrete analog of the gamma distribution, and a discretized version of the latter can be used when it is more convenient to do so. Figure 12.2 shows the autocorrelation functions. The ordinate has a log scale, so geometric functions will appear as straight lines. The geometric property holds for at least 100 lags (2.5 seconds) for sequences B, C, and D, and for 50 lags for sequence A. For lags larger than 250, the geometric function underestimates the autocorrela- tion function. We examined sequences A, B, and C and concluded they possess long- range dependence. Since the autocorrelation functions shown in Fig. 12.2 are so large for small lags, it seem intuitive (to us, at least) that the short-range correlations should be the important ones to capture in a source model. We propose usingthe geometric function r k for the autocorrelation function. Since the negative-binomial and gamma distributions are speci®ed by two parameters, these parameters can easily be esimated from the mean and the variance of the number of cells per frame by the method of moments. Only those two moments and the correlation coef®cient (r) are needed to specify the key properties of VBR teleconference traf®c. The correlation coef®cient can be estimated from the geometric portion of the autocorrelation function by taking logarithms and doing a linear regression. 12.2.2 The DAR Model Our ®rst investigations of these sequences with Tabatabai [16] and Heeke [18] focused on multiplexingissues. First, we established that the time series were stationary. This was done by examiningplots of smoothed versions of the time series and boxplots of many partitions of the time series. Next, we showed a Markov chain provided a good description of the time series. This was done via simulations as described in Section 12.2.2.1. This means that the marginal distributions of the time series can be viewed as the steady-state distributions of the Markov chain. A Markov chain that has a geometric autocorrelation function and whose steady-state distribu- tion can be speci®ed is the DAR(1) process introduced by Jacobs and Lewis [20]. The only member of the DARk family that is used here is the DAR(1), so the (1) will be deleted. The transition matrix is given by P  rI 1 À rQ; 12:1 TABLE 12.1 Summary Statistics of Data Sequences Sequence Bytes per Cell Mean (cells) Standard Deviation (cells) r A 14 1506.4 512.7 0.981 B 48 104.9 29.7 0.984 C 64 130.3 74.4 0.985 D 64 170.6 107.6 0.970 12.2 VIDEO CONFERENCES 289 where r is the lag-one autocorrelation coef®cient. I is the identity matrix, and each row of Q consists of the steady-state probabilities. In our case the steady-state probabilities are the negative-binomial probabilities described above, truncated at some convenient value at least as large as the peak rate (the missing probability is added to the last probability kept). Equation (12.1) is convenient for analytical work, but it masks the simplicity of the DAR model. Let X n be the size (in bits, bytes, or cells as appropriate) of the nth frame and r be as above; the DAR model is X n  X nÀ1 with probability r; X H with probability 1 À r;  12:2 where X H is a sample from the marginal distribution (negative-binomial in our case). From Eq. (12.2) we see that the X -process maintains a constant value (cell rate) for a geometrically distributed number of steps (frames) with mean 1=1 À r, and then another value (possibly the same as the old value) is chosen. When r is close to one (it is about 0.98 in our examples, see Table 12.1), the mean time between cell rate changes is large (about 50 frames in our examples). This means that the sample paths are constant for longintervals. The data trace doesn't have this property, which is the reason the GBAR model described in Section 12.2.3 was introduced. This difference between the sample paths of the model and the data trace is mitigated when several sources are multplexed. The probability that X H n  X nÀ1 is small enough to be ignored in the following calculation. When k sources are multiplexed, X n  X n À 1 with probability r k , so the mean time between potential cell rate changes with r  0:98 and k  16 is 3.6. Consequently, sample paths of the multiplexed cell streams from 16 sources are not constant for longintervals. 12.2.2.1 Validating the DAR Model We validate the DAR model by lookingat performance models for multplexinggain and connection admission control. To estimate statistical multiplexinggain, we use cell-loss probabilities [16] from a simple model of a switch. The source model is a FIFO buffer that is drained at 45 Mb=s. The length of the buffer is expressed as the time to drain a full buffer; this is the maximum possible delay. The results of ten simulations of the DAR model for sequence C are given by 95% con®dence intervals and are shown in Table 12.2. The TABLE 12.2 Cell-Loss Rates for Trace and 95% Con®dence Intervals for DAR Model of Sequence C Buffer Size (ms) Source 1 2 3 4 5 Probability of Loss Â10 À6 for Various Buffer Sizes Trace 2070.0 527.0 141.0 33.3 2.88 DAR model (1738, 2762) (433, 775) (107.4, 212.6) (15.1, 54.1) (2.26, 9.34) 290 LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO results of these simulations show that the DAR model does a good job of estimating the cell-loss rate when 16 sources are multiplexed. Similar results were obtained for the other sequences [18]. Now we consider connection admission control (CAC). Since the DAR model is a Markov chain model of the source, it conforms to one of the sets of conditions a source model must have for the effective bandwidth (EBW) theory of Elwalid and Mitra [8]. Moreover, the DAR model is a reversible Markov chain, and so it inspired a powerful extension of the EBW method, called the Chernoff-dominated eigenvalue (CCE) method [7]. Suppose we have a switch that can process at rate C (Mb=s) and has a buffer of size B (ms). We want to ®nd the maximum number of statistically homogeneous sources that can be admitted while keeping the cell-loss rate no larger than 10 À6 . The CDE method gives an approximate analytic solution with known error bounds; this solution is denoted by K CDE . Another way to obtain the solution is to test candidate values by evaluatingthe cell-loss rate by simulation; we treat this as the exact solution and denote it by K sim . Table 12.3 compares the results of the CDE method to the CAC found from simulations. The number admitted by the CDE method is a very close approximation to the ``true'' value obtained by simulation. This implies that the DAR model captures enough of the statistical properties of the trace to produce good admission decisions. 12.2.3 The GBAR Model The DAR model may not be suitable for a single source (by a single source we mean a source that does not interact with other sources) as described above. Lucantoni et al. [22] give three areas where single source models are useful: studying what types of traf®c descriptors make sense for parameter negotiation with the network at call setup, testing rate control algorithms, and predicting the quality of service degrada- tion caused by congestion on an access link. For this reason, Heyman [11] proposed the GBAR model. Lakshman et al. [21] use the GBAR model to predict frame sizes in a rate control algorithm. Lucantoni et al. [22] propose a Markov-renewal process model to describe a single source. This model has the advantage of being very general, and the disadvantage that it is not parameterized by some simple summary statistics of the data trace. The GBAR model exploits the properties enjoyed by teleconferencing traf®c described in Section 12.2.2, the geometrically decaying autocorrelation TABLE 12.3 CAC Performance for Video Conference A and Video Conference C Video Conference A Video Conference B B 57 9 50 44 5 0.5 9 23 7 1 8.5 9.5 10 11 C 45 67 81 103 125 145 195 245 270 310 110 185 280 375 K sim 20 30 40 50 60 70 98 128 139 156 16 30 49 66 K CDE 16 25 33 44 53 63 90 120 130 150 15 30 50 70 12.2 VIDEO CONFERENCES 291 function and the negative-binomial (or gamma) marginal distributions, to produce a simple model based on the three parameters that describe these features. The GBAR(1) process was introduced by McKenzie [23], alongwith some other interestingautoregressive processes. (As with the DAR model we will drop the argument (1).) Two inherent features of this process are the marginal distribution is gamma and the autocorrelation function is geometric. Toward de®ningthe GBAR model, let Gab; l denote a random variable with a gamma distribution with shape parameter b and scale parameter l; that is, the density function is f G t llt b Gb  1 e Àlt ; t > 0: 12:3 Similarly, let Bep; q denote a random variable with a beta distribution with parameters p and q; that is, with density function f B t G p  q G p  1Gq  1 t p 1 À t q ; 0 < t < 1; 12:4 where p and q are both larger than À1. The GBAR model is based on two well- known results: the sum of independent Gaa; l and Gab; l random variables is a Gaa  b; l random variable, and the product of independent Bea; b À a and Gab; l random variables is a Gaa; l random variable. Thus, if X nÀ1 is Gab; l, A n is Bea; b À a, and B n is Gab À a; l, and these three are mutually independent, then X n  A n X nÀ1  B n 12:5 de®nes a stationary stochastic process X n with a marginal Gab; l distribution. Furthermore, the autocorrelation function of this process is given by rk a b  k ; k  0; 1; 2; . 12:6 The process de®ned by Eq. (12.5) is called the GBAR processes. The G and B denote gamma and beta, respectively, and the AR stands for autoregressive. Since the current value is determined by only one previous value, this is an autoregressive process of order one. A possible physical interpretation of Eq. (12.5) is the following. Interpret A n as the fraction of frame n À 1 that is used in the predictor of frame n, so the ®rst term on the left of Eq. (12.5) is the contribution of interframe prediction. In hybrid PCM=DPCM coding[24], for example, resistance to transmission error is accom- plished by periodically settingsome differential predictor coef®cients to zero and sendinga PCM value. We can think of B n as the number of cells to do that. If the 292 LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO distributional and independence assumptions listed above Eq. (12.5) are valid, then the GBAR process will be formed. Simulatingthe GBAR process only requires the ability to simulate independent and identically distributed gamma and beta random variables. This is easily done; for example, algorithms and Fortran programs are presented in Bratley et al. [3]. The GBAR process is used as a source model by generating noninteger values from Eq. (12.5) and then roundingto the nearest integer. It would be cleaner if a discrete process with negative-binomial marginals could be generated in the ®rst place. McKenzie describes such a process (his Eq. (3.6)). Unfortunately, that process requires much more computation to simulate, and the extra effort does not appear to be worthwhile. 12.2.3.1 Validating the GBAR Model Ten sample paths of the GBAR process were generated and used as the arrival process (number of cells per frame with a ®xed interframe time) in a simulation of a service system with a ®nite buffer and a constant-rate server. The cell-loss rates from these paths were averaged to obtain a point estimate for the GBAR model. The traf®c intensity is varied by changing the service rate. The points produced by the simulations are denoted by an asterisk. In Fig. 12.3, we see that cell-loss rates computed from the GBAR model are close to the cell-loss rates computed from the data. Note that for each traf®c intensity, the decrease in the cell-loss rate as the buffer size increases is very slight, for both the model and the data. This con®rms the prediction of Hwangand Li [19] of buffer ineffectiveness. Since the GBAR model has only short-range dependence, this effect is not caused by long-range dependence here. B() Fig. 12.3 Cell-loss rates for sequence A. 12.2 VIDEO CONFERENCES 293 Figure 12.4 shows that the mean queue lengths in an in®nite buffer computed from ®ve GBAR paths are similar to the mean queue lengths computed using the data. In Fig. 12.4 the vertical axis on the left shows the mean queue length in cells. Video quality is poor when the cell delays are large; 100 ms is an upper bound on the acceptable delay at a node in a network that provides video services. Two buffer drain times are also shown; the practical region for the maximum is below and to the left of the 100 ms line. The range of the mean queue lengths shown exceeds the practical region for maximum queue length. In the practical region, the model and the trace give very similar mean delays. The differences between the mean queue length from the GBAR model and the mean queue lengths from the data would be even smaller if a ®nite buffer were imposed. (This is the truncatingeffects of ®nite buffers that is described in Section 12.4.2). The comparisons for sequences B, C, and D are qualitatively the same as for sequence A. 12.3 BROADCAST VIDEO Now we turn to more dynamic sequences, such as ®lms, news, sports, and entertainment television. Since the main purpose of the models is to aid in network performance evaluations, we are particularly interested in usingthe models to predict cell-loss rates. T M () Fig. 12.4 Mean queue lengths for sequence A. 294 LONG-RANGE DEPENDENCE AND QUEUEING EFFECTS FOR VBR VIDEO