Statistical Multiplexing of Self-Similar Video Streams: Simulation Study and Performance Results Byron Bashforth

Carey Williamson Department of Computer Science University of Saskatchewan 57 Campus Drive Saskatoon, SK, Canada S7N 5A9 Phone: (306) 966-4886 FAX: (306) 966-4884 Email: {bnb121,carey}@cs.usask.ca

Abstract Achieving statistical gains when multiplexing video streams, as in a video-on-demand (VOD) scenario, is difficult because of the stringent QOS demands and the self-similar nature of the traffic. This paper explores, through empirical simulation, the QOS, network utilization, and statistical characteristics of the aggregate traffic resulting from multiple independent MPEG video streams. In addition, the simulation results are compared against several recently-derived theoretical results for self-similar network traffic. There are three main results that are evident from our experiments. First, there is a moderate amount of statistical multiplexing gain to be had, even when aggregating multiple self-similar traffic streams. Second, video multiplexing is extremely sensitive to traffic phasing effects, and to heavy-tailed frame size distributions, as has been noted by other researchers. Finally, the theoretical approaches that we consider (namely, the Norros effective bandwidth formulation) seem promising, but still require some fine tuning if they are to be practical for call admission and network dimensioning.

1. Introduction Several recent papers in the research literature have shown that variable bit rate (VBR) video traffic is selfsimilar [Garrett and Willinger, 1994; Beran et al, 1995]. Self-similarity implies the presence of long range dependence (LRD) in the traffic. That is, there exist non-negligible positive correlations in the burst behaviour of the traffic over many time scales, ranging from milliseconds to seconds to minutes or more. The presence of self-similarity in network traffic raises serious concerns for network traffic management, particularly for video-on-demand (VOD) service providers and backbone network providers. These concerns include the following: •

•

Traffic aggregation effects. The aggregate network traffic, even if from multiple independent sources, will still be self-similar. Bursts will exist across many time scales and positive correlations in traffic will adversely affect the quality of service provided to network users [Duffield et al, 1995; Duffield, 1996]. The cell loss ratio (CLR) in a network with self-similar traffic may be several orders of magnitude higher than that predicted by the traditional Markovian traffic models used in most network planning studies [Chen et al, 1995]. Buffer ineffectiveness. Increasing the buffer sizes used in the network will have marginal impact on the cell loss ratio. The heavy tailed nature of the burst size distribution [Garrett and Willinger, 1994; Leland et al, 1993] implies that only extremely large buffers are effective in reducing CLR [Chen et al, 1995]. These large buffers, however, adversely impact the cell delay performance of delay-sensitive traffic.

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• •

QOS degradation. The two foregoing observations suggest that either CLR or cell transfer delay must be compromised if the network is to carry self-similar network traffic at any reasonably high network utilization. Low network utilization. The only way to provide reasonable QOS to sources is to operate the network at sufficiently low utilization so that queuing effects and buffer overflows are negligible. However, this does not make for efficient use of the transmission facilities.

These problems are particularly acute for VOD service providers whose networks are intended to carry video traffic streams exclusively. There are no other traffic types (such as voice, data, and available bit rate (ABR)) with which to share network resources, resulting in limited flexibility for traffic management approaches such as priority scheduling or selective cell discard. Furthermore, the delay sensitive nature of video traffic renders feedback control and end-to-end retransmission useless. All is not lost, however. There are several recent papers that show statistical multiplexing gain is still possible with self-similar traffic sources [Erramilli et al, 1996; Knightly et al, 1995; Knightly, 1997; Norros, 1995; Patel and Williamson, 1997]. That is, even though the aggregated traffic from several independent self-similar traffic sources remains self-similar, the relative variability of the aggregated stream still decreases. The purpose of this paper is to illustrate these gains in the context of a VOD service provider. In particular, this paper addresses the following research questions: • • • •

What are the statistical characteristics of individual VBR video streams? What are the statistical characteristics of aggregated VBR video streams? What effect does the number of video streams, buffer size, and output link capacity have on the quality of service provided by a VOD multiplexer? What target network utilization can VOD providers expect to achieve while still maintaining QOS?

The approach taken is a simulation of a VOD multiplexer using empirical MPEG video traffic to drive the simulations. The experiment is set up to assess the effect of the number of sources, buffer size, and link capacity on the end-to-end QOS provided to the video streams. QOS is measured using CLR and cell transfer delay (CTD) metrics. We also identify issues regarding the phasing of video sources, the statistical characteristics of multiplexed video traffic, and the trade-offs between achievable network utilization and QOS. The remainder of this paper is organized as follows. Section 2 provides some background on network traffic self-similarity and theoretical work related to long-range dependent traffic flows. Section 3 describes the statistical characteristics of the 15 empirical MPEG video traces used in our study. Section 4 presents the simulation methodology used in our experiment and Section 5 presents the simulation results. Section 6 concludes the paper.

2. Background 2.1 Self-Similarity There is ample evidence in the research literature for the presence of self-similarity in network traffic. Intuitively, self-similarity refers to the existence of visually similar traffic profiles at time scales ranging from seconds to hours. Self-similarity has been identified in Ethernet LAN traffic [Leland et al, 1993], compressed video streams [Garrett and Willinger, 1994], wide area TCP/IP traffic [Paxson and Floyd, 1994], and World-Wide Web traffic workloads [Crovella and Bestavros, 1995]. Self-similarity is a statistically rigorous property for which a number of standard statistical tests exist. For example, the autocorrelation function for a self-similar process decays very slowly (hyperbolically rather

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than exponentially) indicating positive correlations across many time scales and resulting in a nonsummable autocorrelation function. Similarly, the variability of a self-similar process diminishes very slowly as one increases the sample size over which it is averaged. Such behaviour can be illustrated in a log-log variance time plot where the slope for a self-similar process is significantly flatter than -1. Finally, the rescaled adjusted range statistic (R/S statistic) can be used to estimate H (the Hurst parameter) for a self-similar process. A plot of the R/S statistic versus the sample size results in a slope of 0.5 < H < 1.0 for a self-similar process. Most VBR video traffic tends to have 0.7 < H < 0.9. One implication of self-similarity is a heavy-tailed burst size distribution. That is, similar lasting bursts are present at time scales ranging from milliseconds to minutes. Very large buffers may be required in a network to accommodate these bursts. Another implication is that traffic does not aggregate well. When traffic is composed of flows from many independent sources, the resulting traffic aggregation is not smooth. This property implies that statistical multiplexing must be done carefully.

2.2 Effective Bandwidth Despite the foregoing observations, there is still statistical gain to be had with self-similar traffic streams (see, for example, Section V of [Erramilli et al, 1996]). That is, even though the resulting aggregate stream is self-similar, the relative variability of the aggregate stream is still reduced. Norros [Norros, 1995] provides a mathematical characterization of the effective bandwidth or capacity demanded by the resulting traffic. Norros states that the effective bandwidth (CE) for a self-similar traffic source is:

(

CE = m + κ ( H ) − 2 ln(ε )

)

1 H

a

1 2H

x

− (1− H ) H

m

1 2H

where m is the mean bit rate of the traffic stream (in bps), a is the variance coefficient of the traffic stream (in bit-s), H is the Hurst parameter of the stream (a dimensionless measure of long range dependence, with 0.5 < H < 1), k(H) = HH(1-H)(1-H), x is the buffer size (in bits), and ε is the target cell loss ratio (CLR) for the traffic stream. Note that the formula does not explicitly address the issue of cell transfer delay. The two new parameters in the Norros formula (compared to earlier work on the effective bandwidth problem [Guerin, 1991; Chang and Thomas, 1995; Elwalid et al, 1995; Choudhury et al, 1996]) are the variance coefficient (a) and the Hurst parameter (H). These are actually two orthogonal parameters which, combined with the mean bit rate (m), provide a statistical characterization of a traffic source. A similar characterization is used in [Fan and Mars, 1996]. Two assumptions in the theoretical derivation of the formula must be noted. First the formula depends on the traffic being sufficiently Gaussian, which is more likely to be the case when traffic is aggregated from a large number of independent sources. Second, the derivation of the formula uses the Weibull distribution to approximate the tail of the queue length distribution. This approximation is logarithmically accurate for large buffers, so the buffer must be sufficiently large for the formula to apply. The Norros formula provides a means to estimate the statistical gain across multiple self-similar traffic streams. In particular, since the mean and variance of independent sources sum, the variance coefficient of the aggregate stream is the weighted sum of the variance coefficients of the input streams. Furthermore, the Hurst parameter of the aggregate traffic stream is determined by the largest Hurst parameter in the input streams. Table 1 summarizes several theoretical results regarding the aggregation of self-similar traffic sources. We will use these theoretical results later to assess effective bandwidths in comparison to simulation results.

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Table 1: Statistical Characteristics of Aggregated Self-Similar Traffic Incoming Traffic Stream 1 Incoming Traffic Stream 2 Outgoing Traffic Stream H1 H2 max(H1,H2) m, a, H m, a, H 2m, a, H m 1, a 1, H m 2, a 2, H m1a1 + m2 a2 m 1 + m 2, ,H

m1 + m2

3. Video Traffic Analysis 3.1 Video Traces The video traces used to drive the simulation were obtained from a previous study of self-similar network traffic [Deng, 1996]. In that study, Deng analyzed 20 MPEG-1 encoded video streams obtained from various sources on the Internet [Rose, 1995]. Each trace contains 40,000 frames, representing 26.7 minutes of video at 25 frames per second (fps). The size of each encoded frame is listed in the trace data. For the experiments in this paper, fifteen of these video traces were selected, covering a wide variety of video content (i.e., movies, sitcoms, cartoons, news broadcasts, sports events, music videos, and talk shows). MPEG encoded video has three different frame types: • I frames are intra-frame coded and have complete information for one frame. I frames typically require the most bandwidth of the frame types. • P frames are “predictively” coded using motion compensation and represent relative differences from a I or P frame. • B frames are “bidirectionally” coded with motion predication and compensation and encode the relative difference to the previous I or P frame, next I or P frame, or an interpolation between them. B-frames are usually the smallest of the frame types (ie, require the least bandwidth). MPEG video streams are broken into segments, each of which is known as a group of pictures (GOP). Each GOP in the stream is encoded with the same pattern of I, P, and B frames. The video traces in this simulation use the pattern “IBBPBBPBBPBB” [LeGall, 1991; Pancha and Zarki, 1994]. Table 2 summarizes the traces, their content, and their mean bit rates. The average of the mean bit rates for all of the video streams is 4.17 Mbps. Not enough traces are available to make any statistically valid statement about the content types but a number of interesting observations can be made. As a group, the sporting events demand the most bandwidth, requiring an average of 5.97 Mbps and ranging up to 6.79 Mbps for Sports 1. The average bandwidth requirement for the movie samples is 2.52 Mbps which is even less than the average bandwidth for the two cartoons (4.53 Mbps). Unfortunately, Deng provides little information about the source of the traces [Deng, 1996]. Intuitively, it is difficult to see why an actionoriented movie like "Terminator" (Movie 1) requires less bandwidth than a cartoon such as "The Simpsons" (Cartoon 1). Since the movie trace is only 27 minutes long, it may have been taken from a calmer (ie, less action and movement) or darker (ie, less colour variation) segment of the movie. Another explanation is that many movies have dark scenes, while cartoons rarely do.

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Video Stream Cartoon 1 Cartoon 2 Movie 1 Movie 2 Movie 3 Movie 4 Music 1 Music 2 News 1 Sitcom 1 Sports 1 Sports 2 Sports 3 Talk 1 Talk 2 Average

Table 2: General Description of Video Streams Content Description Cartoon Cartoon Movie Movie Movie Movie Music Videos Music Videos News Sitcom Sports Sports Sports Talk Show Talk Show

“The Simpsons” episode “Asterix” episode “Terminator” “Silence of the Lambs” “Jurassic Park” Unknown MTV program MTV program BBC News broadcast “Mr. Bean” episode Car race “Super Bowl” Soccer game Unknown Unknown

Mean Bit Rate (Mbps) 4.11 4.94 2.41 1.62 2.89 3.16 5.44 4.37 4.57 3.90 6.79 5.20 6.00 3.21 3.96 4.17

3.2 GOP-Level Analysis Since the video traces are MPEG encoded, there is a high degree of correlation structure between frames within a GOP [Deng, 1996]. For example, the smaller B or P frames follow the larger I frames in a set pattern. Figure 1 shows the frame sizes of Movie 1 while Figure 2 displays the autocorrelation function of those frame sizes. The correlations between the frame sizes in each GOP is clearly evident. Long range dependence in the traffic may be obscured by analyzing the video traces on a frame-by-frame basis (though it is still evident if, for example, one looks at the correlation coefficients for the I frames). Each GOP, however, contains the same number and arrangement of I, P, and B frames, and thus, analyzing GOP sizes can illustrate LRD more clearly. Figure 3 shows the GOP sizes of Movie 1 and shares the same bursty appearance as Figure 1. However, the autocorrelation graph of GOP sizes [Figure 4] clearly demonstrates the presence of LRD in the trace.

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Figure 1: Movie 1 Frame Sizes.

Figure 2: Movie 1 Frame Size Autocorrelation.

Figure 3: Movie 1 GOP Sizes.

Figure 4: Movie 1 GOP Size Autocorrelation.

The following table [Table 3] analyzes the distribution of GOP sizes of the video traces. The Hurst parameter, used to indicate self-similarity, was estimated by using the graphical R/S method [Leland et al., 1994] (with a sample size of 10 for the R/S statistic).

Video Stream

Cartoon 1 Cartoon 2 Movie 1 Movie 2 Movie 3 Movie 4 Music 1 Music 2 News 1 Sitcom 1 Sports 1 Sports 2 Sports 3 Talk 1 Talk 2 Average

Total GOPs

3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333 3333

Max (KB)

837.89 1079.43 407.51 462.05 627.78 682.36 1280.79 1457.94 1114.77 863.27 1332.63 850.11 1281.24 470.64 669.07 894.5

Table 3: GOP Statistics Min Mean Std Dev (KB) (KB) (KB)

25.02 30.93 24.84 26.63 49.66 19.29 39.61 27.12 20.47 58.82 112.26 27.05 66.30 72.85 92.40 46.22

222.91 268.19 130.86 87.73 156.93 171.46 295.26 237.37 184.27 211.70 368.99 282.07 325.54 174.44 214.98 222.18

95.04 124.82 45.17 52.99 62.98 86.32 137.56 167.24 86.13 105.95 139.09 103.92 123.78 56.53 57.89 96.36

COV

0.43 0.47 0.35 0.60 0.40 0.50 0.47 0.70 0.47 0.50 0.38 0.37 0.38 0.32 0.27 0.44

Peak to Mean Ratio 3.76 4.02 3.11 5.27 4.00 3.98 4.34 6.14 6.05 4.08 3.61 3.01 3.94 2.70 3.11 4.07

Hurst Param

0.83 0.84 0.79 0.91 0.83 0.84 0.84 0.85 0.83 0.93 0.75 0.77 0.74 0.83 0.85 0.83

Across all of the video streams, GOPs are highly variable, ranging from 19.29 KB to 1457.94 KB, with an average of 222.18 KB. The coefficient of variation is noticeably large (0.44 on average and ranging up to 0.70). Large bursts exist in the video streams as indicated by the high peak-to-mean ratios. The average video stream experiences GOPs which are 4.07 times as large as the mean GOP. Music 2 sees bursts up to 6.14 times as large as the mean while the smallest peak to mean GOP size ratio of 2.70 (Talk 1) is still substantial.

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All of the video streams have Hurst parameters between 0.5 and 1.0 demonstrating that self-similarity is present in the traces. Furthermore, a high-degree of self-similarity is indicated by an average Hurst parameter of 0.83 and a range from 0.74 (Sports 3) to 0.91 (Movie 2). As a supplement to Figures 3 and 4, Figures 5 and 6 illustrate properties of self-similarity. As already mentioned, Figure 3 demonstrates the bursty nature of the trace while Figure 4 depicts the presence of LRD. Figure 5 shows a slowly decaying variance for GOP sizes, which is another indicator of self-similarity. Finally, self-similarity is demonstrated by the majority of the R/S samples lying in a band with slope 0.8 [Figure 6].

Figure 5: Movie 1 Variance-Time Plot.

Figure 6: Movie 1 R/S Plot.

3.3 Frame-Level Analysis Even though analyzing the video streams on a frame-by-frame basis will introduce unwanted correlations, it is of interest to study the I, P, and B frames separately. The results from such an analysis are presented in Table 4. Note that the results from only one of each content type is presented here. The results from the other traces are similar.

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Video Stream

Cartoon 1 (I) Cartoon 1 (P) Cartoon 1 (B) Movie 1 (I) Movie 1 (P) Movie 1 (B) Music 1 (I) Music 1 (P) Music 1 (B) News 1 (I) News 1 (P) News 1 (B) Sitcom 1 (I) Sitcom 1 (P) Sitcom 1 (B) Sports 1 (I) Sports 1 (P) Sports 1 (B) Talk 1 (I) Talk 1 (P) Talk 1 (B)

Total Frames 3334 10000 26666 3334 10000 26666 3334 10000 26666 3334 10000 26666 3334 10000 26666 3334 10000 26666 3334 10000 26666

Max (KB) 148.50 136.06 240.38 79.56 61.93 50.66 197.69 229.20 113.05 189.89 166.34 78.69 150.18 119.21 229.07 186.05 202.42 165.45 106.77 69.55 36.00

Table 4: Frame Statistics Min Mean Std Dev (KB) (KB) (KB) 15.30 1.06 0.34 16.46 1.01 0.32 18.52 1.86 0.37 13.83 0.744 0.27 14.27 1.72 0.34 36.79 6.04 4.19 36.19 3.24 2.08

74.05 21.53 10.53 37.39 14.12 6.39 69.86 39.31 13.43 15.60 15.44 8.42 75.16 18.28 10.21 79.24 38.20 21.89 64.73 14.81 8.16

19.27 15.62 7.06 8.34 7.21 3.82 24.17 21.79 9.20 20.99 12.67 5.04 19.45 14.32 6.79 20.82 18.23 10.00 10.18 8.71 3.48

COV

0.26 0.73 0.67 0.22 0.51 0.60 0.35 0.55 0.69 0.30 0.82 0.60 0.26 0.78 0.66 0.26 0.48 0.46 0.16 0.59 0.43

Peak to Mean Ratio 2.01 6.32 22.82 2.13 4.39 7.93 2.83 5.83 8.42 2.69 10.77 9.35 2.00 6.52 22.42 2.35 5.3 7.56 1.65 4.69 4.41

Hurst Param 0.87 0.83 0.85 0.76 0.80 0.81 0.84 0.84 0.86 0.84 0.83 0.91 0.92 0.93 0.96 0.79 0.79 0.85 0.83 0.85 0.89

The results in Table 4 are consistent with those in Table 3 in that they indicate a high degree of variability and a high degree of self-similarity in the video traffic streams. The peak-to-mean ratios are again large, ranging from 2.01 (Cartoon 1 (I)) to 22.82 (Cartoon 1 (B)). While I frames are generally larger than P or B frames, it is interesting to note that the largest frame sizes seen are a B frame in Cartoon 1 and Sitcom 1, and a P frame in Sports 1. Our speculation is that the motion prediction and compensation algorithms of MPEG work less well for these types of video. These "rogue frames" are indicative of a heavy-tailed frame size distribution, and will likely present a significant challenge to a video multiplexer. As a final observation, the Hurst parameters indicate a high degree of self-similarity between frames of the same type. As with the GOP analysis, Figure 7 through Figure 10 illustrate the different manifestations of self-similarity, using only the I frames from Movie 1.

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Figure 7: Movie 1 I Frame Sizes.

Figure 8: Movie 1 I Frame Autocorrelation.

Figure 9: Movie 1 I Frame Variance-Time Plot.

Figure 10: Movie 1 I Frame R/S Plot.

4. Experimental Methodology This experiment focuses on multiplexing video streams onto a single ATM link as in a typical VOD scenario [Figure 11].

Video Source V1

Video Source V2

C1

Multiplexer

C2 Buffer B

. . . Video Source Vn

C

Output

Cn

Figure 11: System under study. Each video source (V1...Vn) produces a stream of video data. Video data is generated a frame at a time every 40 ms (25 fps). Once generated, the frame is broken into cells and sent over an ATM link. The capacity of this link (C1...Cn) must be sufficient to deliver the entire frame to its destination before the next frame is generated. The multiplexer accepts the cells from each source and places them into a buffer (B) until the output link (C) has space to send them. The capacity of the output link determines how quickly the buffer is emptied. If the buffer is full when a cell arrives, the cell is lost.

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The metrics used to describe the performance of the system are: • Cell loss ratio. This is the number of cells lost due to a full buffer compared to the number of cells sent to the multiplexer. For VOD providers, a CLR below 10-5 is acceptable. • Cell delay (seconds). Cells experience a delay waiting in the multiplexer's buffer. Long or variable delays can be detrimental to video quality. It is assumed that delays longer than 10 ms are unacceptable. • Output link utilization (%). VOD providers will be concerned with how efficiently their resources are being utilized. This metric will measure the number of data cells delivered compared to the number of empty cells. A number of parameters will affect system performance. In the system under study [Figure 11], the capacity of incoming links (C1...Cn), output capacity (C), buffer size (B), and the number of video sources (n) will influence the QOS. Additionally, the choice to use multiple buffers (one for each input) or a single buffer in the multiplexer will affect performance. The following system parameters will not be studied and will be given fixed values: • Capacity of incoming links (C1...Cn). The capacity of each link is assigned to be 100,000 cells/second (42 Mbps). This capacity was chosen because it is not unrealistically fast as to overwhelm the multiplexer’s buffer and still fast enough to deliver all of the cells of a frame with ample time before the next frame is generated. It is also very close to the capacity of a standard DS-3 link. • Multiple buffers or a single buffer. The effect of multiple buffers will not be explored. It should also be noted that time required for a cell to traverse the electrical paths in the buffer from the input to the output is assumed to be 0 s. The following system parameters will be studied: • Buffer size (B). B will be assigned values of 1,000, 2,000, 4,000, and 8,000 cells. Buffer sizes larger than this are undesirable because of the potential delays that could arise. • Number of video sources (n). n will be varied through 5, 10, and 15 sources. These three groups of five streams were selected to represent a mix of programming being delivered at the same time. The first group of five consists of Cartoon 1, Movie 1, Sports 1, Talk 1, and News 1 (19.9 Mbps total). The second group adds Sitcom 1, Movie 2, Sports 2, Talk 2, and Music 2 (39.0 Mbps total) while the final group adds Cartoon 2, Movie 3, Sports 3, Movie 4, and Music 1 (61.4 Mbps total). • Output capacity (C). C will be assigned levels of 60,000 (25.4 Mbps) to 230,000 cells/s (97.5 Mbps) in increments of 10,000 cells/s (4.24 Mbps). Only 7 of the total 17 levels will be used in each experiment. Depending on the number of sources involved (5, 10, or 15), certain output capacities are uninteresting. For example, with 15 video sources, the lower output capacities result in unrealistically high cell loss ratios. A number of parameters in the workload will also affect performance, such as mean bit rate, burstiness, selfsimilarity, and encoding scheme. However, this experiment will not explore the effects of those properties explicitly. Instead, the workload model of chosen empirical traces is designed to reflect a realistic VOD scenario. Before any data is collected, the simulation is run for 1.0 (simulated) min. This startup phase allows the multiplexer's buffer to fill with some amount of cells. Since the length of the traces is relatively short (less than 30 minutes), collecting statistics immediately while the buffers are filling may have an effect on the results. After the start up phase, the simulation runs for 25 (simulated) min. All of the traces are longer than 26 min so the simulation ends before any of the video sources stop transmitting cells. This removes end-effects from the statistics. Phasing presents a serious concern for VOD providers and this experiment [Krunz and Tripathi, 1996]. Phasing occurs when two or more video sources are sending data synchronously. That is, they are bursting or silent at the same time. Phasing can occur on two levels in this experiment. First, since I frames are typically much larger than P or B frames, a number of video sources sending synchronized I frames will

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overwhelm the multiplexer's buffer much more quickly than if a mixture of I, P, and B frames are being sent. Second, phasing occurs when any two sources begin sending a frame at the same time (regardless of type). It can be assumed that VOD providers are aware of the encoding scheme used and can take steps to avoid delivering video streams in-phase (ie, sources sending synchronized frames). In this experiment, the start times are equally spaced within the time required to send a GOP. This ensures that no two frames are sent at the same time (and, therefore, no two I frames are delivered synchronously). The following experiments will be run: • The effect of the number of video sources, buffer size, and output capacity on cell loss, output utilization, and bandwidth demands will be explored by running a full-factorial experiment with the simulator [Jain, 1991]. This is reasonable since the number of factors is small (three) and the number of levels of each factor is also small (seven or fewer). • Cell delay will be explored by holding the number of sources at 10 and varying the buffer sizes and output capacities. With 10 video streams, a variety of performance regions are represented. Collecting cell delay information involves recording the time when a cell enters the multiplexer and when it exits. Due to disk space limitations, cell delay information could only be collected for 30 s (A 30 s trace created a 50 MB uncompressed text file). Collection occurred between 5.0 min and 5.5 min of the simulation. • The effect of phasing will be presented by re-running the cell loss simulation with the number of video sources fixed at 10 and starting all of the sources simultaneously.

5. Simulation Results 5.1 Cell Loss The following figures (Figures 12, 13, and 14) illustrate the effect of the number of video streams, buffer size, and output capacity on cell loss.

1

1

0.1

0.1 0.01 CLR

0.0001

1000 Buffer Size (cells)

1

CLR

0.01 0.001 0.0001

1000 2000 Buffer Size (cells)

8000 230000

210000

220000

200000

180000

190000

170000

4000

8000 170000

150000

Buffer Size (cells)

Figure 13: Cell Loss Ratio with 10 Video Sources.

0.1

0.00001

160000

Output Capacity (cells/s)

Figure 12: Cell Loss Ratio with 5 Video Sources.

Output Capacity (cells/s)

4000 130000

110000

8000

0.000001

2000

0.000001

120000

100000

Output Capacity (cells/s)

110000

90000

70000

4000 80000

60000

0.000001

1000

0.00001

2000

140000

0.0001 0.00001

0.001

120000

CLR

0.01 0.001

Figure 14: Cell Loss Ratio with 15 Video Sources.

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(Note in the above figures that 0 values cannot be plotted on the logarithmic axis. The surface that lies along 0.000001 represents areas where the cell loss ratio is 0.000001 or less.) With 5 video streams (19.9 Mbps), 2.8% of the cells are lost when the output capacity is 60,000 cells/s (25.4 Mbps) and the buffer size is 1000 cells. Increasing the buffer size or the output capacity decreases the CLR, as expected. At lower output capacities, the effect of increasing the buffer size on the CLR is not as pronounced as at higher output capacities. At 60,000 cells/s, enlarging the buffer from 1,000 cells to 8,000 cells decreases the CLR from 2.8% to 0.5% (approximately 5.6 times better). At 120,000 cells/s, doubling the buffer size from 1,000 to 2,000 cells improves the CLR from 0.00635% to 0.000287% (approximately 22 times better). With 5 video sources, an acceptable CLR is not obtained until the output capacity is faster than 90,000 cells/s (38.2 Mbps) and the buffer size is larger than 4,000 cells, or the buffer is larger than 2,000 cells and the output capacity exceeds 110,000 cells/s (46.6 Mbps). With 10 video streams (39.0 Mbps), cell loss is more pronounced. In the worst case examined (a 1,000 cell buffer and a 110,000 cell/s output link), 3% of the incoming cells are dropped. The desired CLR is not achieved until the output link is faster than 160,000 cells/s (67.8 Mbps) and the buffer holds more than 4000 cells. The scenario is very similar when 15 video sources (61.4 Mbps) are multiplexed. Cell loss of 2% is experienced when the output link is 170,000 cells/s (72.1 Mbps) and the buffer size is 1,000 cells. Acceptable loss is accomplished only when the output capacity is increased to 220,000 cells/s (93.3 Mbps) or more and the buffer size exceeds 4,000 cells. Statistical gain is evident in the above three figures. With the 5 source scenario, a CLR of 0 is achieved when the output capacity is 110,000 cells/s (46.6 Mbps). At this speed, each source is using approximately 9.3 Mbps of the total output. An output capacity of 170,000 cells/s (72.1 Mbps) is required for 0% cell loss with 10 sources and 230,000 cells/s (97.5 Mbps) with 15 sources. In these two scenarios, each video stream is using 7.2 Mbps and 6.5 Mbps, respectively. As the number of video streams increase, the apparent bandwidth demanded by each individual source decreases.

5.2 Cell Delay Figure 15 and 16 show the average cell delay and the standard deviation of cell delay, respectively, for the 10 video source simulation.

0.025

0.012

Std Dev of Delay (s)

0.010 0.008 0.006 1000 2000 Buffer Size (cells)

2000

0.000

Buffer Size (cells)

8000 170000

Output Capacity (cells/s)

160000

4000 150000

8000

Figure 15: Average Cell Delay with 10 Video Sources.

1000 0.005

130000

150000

170000

Output Capacity (cell/s)

160000

140000

120000

4000 130000

110000

0.000

0.010

110000

0.002

0.015

140000

0.004

0.020

120000

Average Delay (s)

0.014

Figure 16: Standard Deviation of Cell Delay with 10 Video Sources.

Increasing the buffer size in order to reduce cell loss is not always the best alternative. As can be seen in Figure 17, increasing the buffer size also increases the average cell delay. For example, with an output capacity of 110,000 cells/s, the average cell delay increases from 3.6 ms to 13.4 ms as the buffer size moves from 1,000 to 8,000 cells. However, the time each cell is delayed is also affected by the output capacity.

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With a faster output link, fewer cells wait in the buffer, reducing the amount of time a cell spends in the multiplexer. Variance in cell delay follows the same pattern as the average cell delay. The variance (or, equivalently, standard deviation) is highest when the output capacity is slower (110,000 cells/s) and the buffer is large (8,000 cells). Cells entering the buffer during periods of lower activity spend very little time in the buffer. However, cells entering the multiplexer during bursts will end up near the end of the queue and spend much more time waiting to be delivered. This can be avoided by increasing the output capacity. As the speed increases, fewer cells are in the buffer (on average) and the delay variance is reduced. Figure 16 also suggests that large variances in cell delay can be eliminated by decreasing the buffer size. Although this is true, the overall effect is undesirable. Variance is reduced because the buffer is always or nearly always full when it is small. Cells that are not dropped face a relatively consistent delay.

5.3 Utilization VOD providers are also interested in the utilization of their hardware. The following figures [Figures 17, 18, and 19] illustrate the usage.

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Figure 19: Output Link Utilization with 15 Video Sources. Regardless of the number of video sources, increasing output capacity decreases utilization. This is obvious since approximately the same amount of information is being delivered over a faster link. With all three figures, buffer size appears to make little difference. There is a slight (1% to 2%) improvement in output link utilization as the buffer size is increased from 1,000 cells to 2,000 cells. With a small buffer of 1,000 cells, the CLR is relatively high. When the buffer is increased, more of the cells are saved and delivered on the output link. In order to guarantee an acceptable CLR (10-5) a high percentage of the available outgoing capacity is wasted. With 5 video sources (19.9 Mbps) and an output capacity of 120,000 cells/s (50.9 Mbps), there is

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very little contention for the outgoing link and the entire system is only 40% utilized. However, with 10 and 15 sources, the buffer size and output link capacity must be chosen carefully to keep the CLR at an acceptable level. With 10 video sources (39.0 Mbps), this can be achieved with a buffer size of 8,000 cells and an output capacity of 170,000 cells/s (72.1 Mbps). At this point, 54% of available bandwidth is being used. With 15 sources (61.4 Mbps), an acceptable CLR can be obtained by having a buffer 8,000 cells large and an output link capable of 230,000 cells/s (97.5 Mbps). At this point, 63% of the output link is being utilized, indicating that the statistical gain across sources can improve network utlization. However, a substantial amount of bandwidth is still being wasted.

5.4 Multiplexed Video Stream Network providers are also interested in the statistical characteristics of the aggregate network traffic from many sources. The obvious concern is that self-similar traffic does not get smoother when it is aggregated [Leland et al, 1993]. Although not a particular concern to VOD providers, it does present additional problems for switches and other devices in the network. This brief analysis focuses on the multiplexer's output produced with 10 video sources. In particular, we sample the outgoing traffic in 40 ms intervals so the samples can be treated as being weakly analogous to video frames. The samples are then aggregated over larger intervals to study traffic burstiness and self-similarity. Table 5 summarizes the traffic as it left the multiplexer on the output link. As described above, the cells have been collected into 40 ms groups to represent video "frames". There is still a significant degree of variability in the "frame" sizes observed, though not as much as in the individual input streams. As might be expected, the outgoing traffic has a consistently high Hurst Parameter and is, therefore, still self-similar. As an example, the usual tests for self-similarity are applied to the output stream generated by 10 video sources (39.0 Mbps), an output capacity of 140,000 cells/s (59.4 Mbps), and a buffer size of 4,000 cells and are presented in Figures 20 through 23.

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Table 5: Multiplexer “Frame” Output with 10 Sources Max Min (KB) Mean Std Dev COV (KB) (KB) (KB) 233.20 254.40 275.6 296.80 318.00 339.20 351.97 233.20 254.40 275.60 296.80 318.00 339.20 360.40 233.20 254.40 275.60 296.80 318.00 339.20 360.40 233.20 254.40 275.60 296.80 318.00 339.20 360.40

109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82 109.82

194.11 197.08 199.00 200.25 200.94 201.30 201.49 198.82 200.23 201.30 201.53 201.59 201.60 201.60 198.80 200.84 201.60 201.60 201.60 201.60 201.60 199.52 201.15 201.60 201.60 201.60 201.60 201.60

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27.68 31.30 33.96 38.86 37.09 37.85 38.13 29.60 33.99 36.70 37.81 38.30 38.46 38.37 30.00 34.50 37.06 37.91 38.33 38.46 38.37 30.27 34.72 37.06 37.90 38.33 38.46 38.37

0.14 0.16 0.17 0.18 0.18 0.19 0.19 0.15 0.17 0.18 0.19 0.19 0.19 0.19 0.15 0.17 0.18 0.19 0.19 0.19 0.19 0.15 0.17 0.18 0.19 0.19 0.19 0.19

Peak to Mean Ratio 1.20 1.29 1.38 1.48 1.58 1.69 1.75 1.18 1.27 1.37 1.47 1.58 1.68 1.79 1.17 1.27 1.37 1.47 1.58 1.68 1.79 1.17 1.26 1.37 1.47 1.58 1.68 1.79

Hurst Param 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90 0.90

Figure 20: Multiplexer Output "Frame" Sizes with 10 video sources, a 4,000 cell buffer, and a 140,000 cell/s output link.

Figure 21: Multiplexer Output "Frame" Autocorrelation with 10 video sources, a 4,000 cell buffer, and a 140,000 cell/s output link.

Figure 22: Multiplexer Output "Frame" VarianceTime Plot with 10 video sources, a 4,000 cell buffer, and a 140,000 cell/s output link.

Figure 23: Multiplexer Output "Frame" R/S Statistic with 10 video sources, a 4,000 cell buffer, and a 140,000 cell/s output link.

Even though the outgoing traffic is self-similar, the coefficient of variation and the peak-to-mean ratios are lower with the multiplexed output as compared to Tables 3 and 4. With the scenario of an 8,000 cell buffer and a 170,000 cell/s output link, the reduced COV and peak-to-mean ratio could indicate that some statistical gain is being made. In the other cases (with smaller buffers and slower output links), cell loss exists and the output stream is "clipped" (as evident in Figure 20).

5.5 Comparison to Theoretical Results As mentioned in Section 2.2, there has been significant theoretical work done on long range dependent network traffic. We have compared our simulation results with one such theoretical formulation, namely the Norros effective bandwidth formula. Table 6 summarizes the statistical characterizations of the aggregate network traffic for the 5 source, 10 source, and 15 source scenarios, for both theoretical results and simulation results. The characterizations are in terms of the three Norros parameters, namely the mean bit rate m (in Mbps), the variance coefficient a (in bit-sec), and the Hurst parameter H. These simulation results are all from scenarios with CLR < 10-5. Table 6: Norros Predictions vs Simulation Results 5 Sources 10 Sources Value Theory Sim Theory Sim Mean Bit Rate (m) 19.9 19.9 39.0 39.0 Variance Coefficient (a) 640866 661876 777689 795436 Hurst Parameter (H) 0.86 0.85 0.93 0.90

15 Sources Theory Sim 61.4 61.4 795931 849396 0.93 0.90

The results in Table 6 indicate very close agreement between the simulation results and the expected traffic characteristics. These results suggest that the Norros traffic parameters provide a good means to assess aggregate traffic characteristics, assuming that the input traffic characteristics and traffic mix are fairly precisely known (which may not be true in general, but should be true for a VOD provider). Furthermore,

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note that this characterization is only accurate when the CLR is very low, since "clipping effects" reduce the traffic variability at higher CLR values. For example, the 5 source scenario with 2.8% cell loss has a Norros variance coefficient of 431438 bit-sec, and a mean bit rate of 19.3 Mbps. The next question is how adequate are the Norros traffic parameters for determining effective bandwidth and CLR values, given the link capacity and buffer size. This question is answered, in part, by Figure 24, which plots the CLR as a function of output link capacity, for both the Norros formula and the simulation results. Four different buffer sizes are shown, ranging from 1000 cells to 8000 cells.

Figure 24: Simulation Results vs Norros Predictions for 10 Sources. Figure 24 indicates that the Norros theoretical results are definitely "in the right ballpark" for effective bandwidth determination, but not perfect. For example, the Norros bandwidth predictions for small buffer systems tend to be overly optimistic. The fact that the Norros prediction is closer to the simulation results for larger buffer sizes is not surprising because of the large buffer size assumption in the derivation of the Norros formula. Further experiments (not shown here) show that the Norros effective bandwidth is fairly tight for selected individual video streams, provided that the buffer size is 4000 cells or more. A more serious concern is the qualitative difference in the shape (i.e., flatness) of the curves. Our speculation is that the differences are due to several reasons. First, there is significant short range dependence (SRD) in the video traffic streams, which is not explicitly accounted for in the Norros formula. SRD can be very important in determining the queueing behaviour and cell loss in a network [Grossglauser and Bolot, 1996; Ryu and Elwalid, 1996]. Second, the empirical video traces have heavy-tailed frame size distributions [Krunz et al, 1995; Krunz and Tripathi, 1997]. As a result, the overall bit rate of the traffic stream may not be Gaussian (an assumption in the Norros derivation), which make the Norros formula overly optimistic as well. In particular, the CLR values in the simulation are very sensitive to the "rogue frames" seen in the empirical traces. Only by substantially increasing the buffer size or the link capacity can all such video frames be accommodated with zero cell loss. For CLR values around 10-4, the Norros prediction is fairly reasonable. Krunz [Krunz and Tripathi, 1997] cautions that basing effective bandwidth determination on CLR values less than 10-4 is a bit dangerous, since the small number of lost cells may be from only one frame or scene. Increasing network resources (buffer size and/or link capacity) to accommodate these "rogue frames" may not be worthwhile, since substantial increases are required in order to achieve a small improvement in CLR. Our results are consistent with this observation. Finally, CLR values are extremely sensitive to traffic phasing effects (see the next section), which are not captured in the

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higher time scales of the Norros traffic characterization. Nevertheless, we find the Norros formulation interesting, and of potential use for call admission control and network dimensioning activities.

5.6 Phasing The final part of our simulation experiment addressed the phasing issue for video traffic sources. As discussed earlier, phasing can dramatically increase the size of the bursts seen at the multiplexer and place additional strains on resources. Figure 13 and Figure 25 illustrate the dramatic performance degradation that phasing can cause.

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Figure 13 shows the cell loss ratios with 10 video sources starting at different times (equally spaced within the time required to deliver a GOP). Figure 25 shows a worst-case scenario where all 10 sources start sending frames simultaneously. This means that not only are frame boundaries synchronized, I frames are also sent at the same time by all of the sources. With out-of-phase sources, the worst case CLR is 3.0%. 0% cell loss can be achieved by having an output capacity of 170,000 cells/s and a buffer size of 8000 cells. With the phased sources, cell loss is much worse. With a buffer size of 1,000 cells and an output capacity of 110,000 cells/s, the CLR is 48.5%. Even when the buffer is increased to 8,000 cells and the output capacity to 170,000 cells/s, the CLR is still 4.5%.

6. Conclusions This paper addresses the statistical multiplexing of multiple self-similar video traffic streams. In particular, we focus on the characterization of the input streams, the characterization of the output streams, and the dimensions of buffer size and link capacity to meet target QOS requirements for a VOD service provider. Simulation is used as the vehicle to explore statistical multiplexing gain, and for comparison with theoretical results for long range dependent traffic. There are three main conclusions that follow from this work. First, there is statistical multiplexing gain to be had, even when aggregating multiple self-similar traffic streams. The keys to the statistical gain are the independence of the video sources, and the relative reduction in the variability of the aggregate traffic, even though the aggregate traffic remains self-similar. The statistical gain is not as large as for Markovian traffic flows, obviously, but it is still significant enough to exploit for VOD service providers. Network utilizations of 50-70% should be achievable with reasonable buffer sizes, while still maintaining acceptable cell loss, cell delay, and cell delay variation QOS requirements. Second, video multiplexing is extremely sensitive to traffic phasing effects, and to heavy-tailed frame size distributions. Proper phasing of video sources can result in immense savings in network resource

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requirements for a given QOS target. Furthermore, dimensioning for a low (but non-zero) CLR is more economical than dimensioning for a CLR of zero. Finally, theoretical work such as the Norros effective bandwidth formulation seem promising, but still require some fine tuning if they are to serve as tight bounds for call admission or network dimensioning functions in a real network. Our simulation experiment shows reasonably good qualitative and quantitative agreement with the Norros results, but still room for improvement. At the very least, our simulation results provide an indication of the conditions under which the Norros formula should and should not be used. Further research effort on this topic is clearly warranted.

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[Garrett and Willinger, 1994] M. Garrett and W. Willinger, “Analysis, Modeling and Generation of SelfSimilar VBR Video Traffic”, Proceedings of ACM SIGCOMM '94, London, UK, pp. 269-280, August 1994. [Grossglauser and Bolot, 1996] M. Grossglauser and J. Bolot, “On the Relevance of Long Range Dependence in Network Traffic”, Proceedings of ACM SIGCOMM '96, Stanford, CA, pp. 15-24, August 1996. [Guerin, 1991] R. Guerin, “Equivalent Capacity and Its Application to Bandwidth Allocation in HighSpeed Networks”, IEEE Journal on Selected Areas in Communications, Vol. 9, No. 7, pp. 968-981, September 1991. [Jain, 1991] R. Jain, The Art of Computer Systems Performance Analysis, John Wiley & Sons, Incorporated, New York, 1991. [Knightly, 1997] E. Knightly, “Second Moment Resource Allocation in Multi-Service Networks”, Proceedings of the 1997 ACM SIGMETRICS Conference, Seattle, WA, pp. 181-191, June 1997. [Knightly et al, 1995] E. Knightly, D. Wrege, J. Liebeherr, and H. Zhang, “Fundamental Limits and Tradeoffs of Providing Deterministic Guarantees to VBR Video Traffic”, Proceedings of the 1995 ACM SIGMETRICS Conference, Ottawa, ON, pp. 98-107, May 1995. [Krunz et al, 1995] M. Krunz, R. Sass, and H. Hughes, “Statistical Characteristics and Multiplexing of MPEG Streams”, Proceedings of the IEEE INFOCOM'95 Conference, Boston, MA, pp. 455-462, April 1995. [Krunz and Tripathi, 1996] M. Krunz and S. Tripathi, “Impact of Video Scheduling on Bandwidth Allocation for Multiplexed MPEG Streams”, Multimedia Systems Journal, 1996. [Krunz and Tripathi, 1997] M. Krunz and S. Tripathi, “On the Characterization of VBR MPEG Streams”, Proceedings of the 1997 ACM SIGMETRICS Conference, Seattle, WA, pp. 192-202, June 1997. [LeGall, 1991] D. LeGall, “MPEG: A Video Compression Standard for Multimedia Applications”, Communications of the ACM, Vol. 34, No. 4, pp. 46-58, April 1991. [Leland et al, 1994] W. Leland, M. Taqqu, W. Willinger, and D. Wilson, “On the Self-Similar Nature of Ethernet Traffic (Extended Version)”, IEEE/ACM Transactions on Networking, Vol. 2, No. 1, pp. 1-15, February 1994. [Norros, 1995] I. Norros, “On the Use of Fractional Brownian Motion in the Theory of Connectionless Networks”, IEEE Journal on Selected Areas in Communications, Vol. 13, No. 6, pp. 953-962, August 1995. [Pancha and Zarki, 1994] P. Pancha and M. Zarki, “MPEG Coding for Variable Bit Rate Video Transmission”, IEEE Communications Magazine, Vol. 32, No. 5, pp. 54-66, May 1994. [Patel and Williamson, 1997] A. Patel and C. Williamson, “Effective Bandwidth of Self-Similar Traffic Sources: Theoretical and Simulation Results”, Proceedings of the IASTED Conference on Applied Modeling and Simulation, Banff, AB, July 1997 (to appear). [Rose, 1995] O. Rose, “Statistical Properties of MPEG Video Traffic and Their Impact on Traffic Modeling in ATM Systems”, Proceedings of the IEEE 20th Conference on Local Computer Networks, Minneapolis, MN, October 1995.

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