Source-level Replay of UNC 7:30 PM

Time Series of Byte Throughput

Figure 6.40: Byte throughput time series for UNC 7:30 PM inbound and its four types of source-level trace replay.

Figure 6.41: Byte throughput time series for UNC 7:30 PM outbound and its four types of source-level trace replay.

The time series of byte throughput for the inbound and outbound directions of UNC 7:30 PM are shown in Figures 6.40 and 6.41 respectively. Observation B.1 is clearly applicable to these results. For the inbound direction, note the very good approximation of the time series features between minutes 20 and 60, and the accurately reproduced spikes in minutes 46 and 53. In contrast, the replay seems out of phase for the initial spike in minute 1, and the large spike in minute 32. This could be explained by one or a few fast connections in the original trace that could not be replayed fast enough. In the outbound direction, we find replays with somewhat lower byte throughput, which was also observed in the other two UNC traces.

The possible extra burstiness in lossless replays mentioned in observation B.2 is not present in the inbound direction, and the last 40 minutes in the outbound direction. We do however observe substantially higher throughputs in the outbound direction for the first 20 minutes, especially in the case of the lossless collapsed-epochs replay. Regarding observation B.3, we do observe slightly more bursty time series from the collapsed-epochs replay in both directions, although the difference seems minor in this case. A few of the (smaller) features in the inbound direction are more closely approximated by the full replays, such as the spike in minute 22 and the ditch in minute 43.

Time Series of Packet Throughput

Figure 6.42: Packet throughput time series for UNC 7:30 PM inbound and its four types of source-level trace replay.

Figure 6.43: Packet throughput time series for UNC 7:30 PM outbound and its four types of source-level trace replay.

The analysis of the packet throughput results shows again some interesting differences with respect to earlier results and observations P.1 and P.2, but only in the outbound direction. The results for the inbound direction presented in Figure 6.42 are in good agreement with observation B.1, since we observe substantially lower packet throughput for collapsed-epochs replays. The result is also consistent with observation B.2, showing higher packet throughput in the lossy replay. However, the result for the outbound direction is more surprising. Unlike previous cases, losses have a minimal impact on the replays, as shown in Figure 6.43. We could argue that the lossy replay provides a better fit between minutes 30 and 35 and after minute 52, but the rest of the time series for this replay is very similar to the one for the lossless replay. We can also say that the lossless replay makes a better attempt to match the spike in minute 14, although the replay spike seems shifted to minute 16.

Marginal Distributions

Figure 6.44: Byte throughput marginals for UNC 7:30 PM inbound and its four types of source-level trace replay.

Figure 6.45: Byte throughput marginals for UNC 7:30 PM outbound and its four types of source-level trace replay.

The marginal distributions of the 10-millisecond time series of byte throughput for the inbound direction are shown in Figure 6.44, while the ones for the outbound direction are shown in Figure 6.45. In agreement with observation B.1, the body of the marginals are closely approximated by the replays, especially in the case of the inbound direction. For the outbound, it is interesting to note a better approximation by the lossless replays for the upper part of the body and the first half of the tail.

As observation B.2 and B.3 pointed out, it is difficult to make general a statement about the approximation of the tails. For UNC 7:30 PM inbound, collapsed epoch replays match the original as closely as the full replays that match the original tail below , but they are substantially heavier above that probability. Note that this heaviness did not manifest itself in the plots of 1-minute byte throughput. For UNC 7:30 PM outbound, we however have that the lossless replays are the ones showing an excellent match below , but a far heavier tail above that probability. Overall, the only type of replay that did not show an overly heavy tail was the lossy full replay, which supports observation B.7.

Figure 6.46: Packet throughput marginals for UNC 7:30 PM inbound and its four types of source-level trace replay.

Figure 6.47: Packet throughput marginals for UNC 7:30 PM outbound and its four types of source-level trace replay.

The body of the marginal distributions for the 10-millisecond packet throughputs shown in Figure 6.46 and 6.47 do not clearly follow observation P.1. In the inbound direction, the distributions from the collapsed-epochs replays are clearly lighter than those from the full replays and the original trace for most of the distribution. However, they provide a better approximation above 100 packets. Interestingly, the distributions from the lossless replays exhibit similar shapes, but a clear offset, and the same is true for the lossy replays. For this direction, we find that the impact of detailed source-level modeling and per-connection losses is of the same order. The lesson is similar for the outbound direction, although all of the replay distributions are lighter than the original distribution in this case.

Regarding the tails of the marginal distributions, we observe similar conditions in both directions. Lossless replays exhibit substantially heavier tails than the original, while lossy replays exhibit substantially lighter tails. This is in sharp contrast to the results for the 1-minute time series studied in Section 6.6.2, where losses had a very small impact. We can argue that lossless replays are artificially more bursty only at fine-time scales for UNC 7:30 PM. The results clearly support observation P.3, showing that the tails are far more sensitive to losses than to detailed source-level modeling for this trace.

Long-Range Dependence

Figure 6.48: Wavelet spectra of the byte throughput time series for UNC 7:30 PM inbound and its four types of source-level trace replay.

Figure 6.49: Wavelet spectra of the byte throughput time series for UNC 7:30 PM outbound and its four types of source-level trace replay.

Table 6.7: Estimated Hurst parameters and their confidence intervals for the byte throughput time series of UNC 7:30 PM and its four types of source-level trace replay.

Trace	Inbound		Outbound
UNC 7:30 PM		C. I.		C. I.
Original	0.8927	[0.8520, 0.9333]	0.9424	[0.9018, 0.9830]
Lossless Replay	0.8490	[0.8083, 0.8896]	1.0191	[0.9784, 1.0597]
Lossy Replay	0.8449	[0.8043, 0.8856]	1.0044	[0.9637, 1.0450]
Lossless Coll. Epochs	0.8392	[0.7985, 0.8798]	0.9984	[0.9578, 1.0390]
Lossy Coll. Epochs	0.8655	[0.8249, 0.9062]	1.0971	[1.0564, 1.1377]

The spectra of the byte throughput in the full replays are close to the original spectrum, as shown in Figure 6.48. However, the spectra of the outbound byte throughput shown in Figure 6.49 reveals a lossless replay with substantially more energy in the scaling region (which starts at octave 6). As in the case of UNC 1 AM outbound, this finding does not support observation B.4 regarding lossless full replays. The lossy full replay provides however a closer approximation to the original spectrum. Estimated Hurst parameters in Table 6.7 show similar results. Estimates from the replays are within the confidence interval of the inbound estimate, but somewhat lower. However, they are above the upper end of the confidence interval of the outbound estimate. The estimate from the lossy collapsed-epochs replay is specially high in this case, probably driven by the spike in octave 11. It is again difficult to draw any strong conclusion other than the general finding of inconsistent results already stated in observation B.4.

Collapsed-epochs replays show substantially more energy in the lossless case, but the difference in not so substantial for the lossy replay, especially in the inbound direction. This is in agreement with observation B.5.

The lossy full replay provides the best approximation again, as pointed out in observation B.7, However, some regions, such as the one between octaves 9 to 12 in the inbound direction, are more closely reproduced by the lossy collapsed-epochs replay. Interestingly, the four replays track the fine-scale energy profile of the original spectrum for the inbound direction, but only the lossless ones do so for the outbound direction. Observation B.6 already reflected this type of inconsistency in the results.

Figure 6.50: Wavelet spectra of the packet throughput time series for UNC 7:30 PM inbound and its four types of source-level trace replay.

Figure 6.51: Wavelet spectra of the packet throughput time series for UNC 7:30 PM outbound and its four types of source-level trace replay.

Table 6.8: Estimated Hurst parameters and their confidence intervals for the packet throughput time series of UNC 7:30 PM and its four types of source-level trace replay.

Trace	Inbound		Outbound
UNC 7:30 PM		C. I.		C. I.
Original	0.9560	[0.9116, 1.0005]	1.0061	[0.9617, 1.0506]
Lossless Replay	0.9655	[0.9210, 1.0099]	1.0043	[0.9599, 1.0488]
Lossy Replay	0.9186	[0.8742, 0.9631]	0.9524	[0.9080, 0.9969]
Lossless Coll. Epochs	0.9491	[0.9047, 0.9936]	0.9931	[0.9487, 1.0375]
Lossy Coll. Epochs	0.9967	[0.9523, 1.0411]	1.0508	[1.0064, 1.0953]

The lessons from the wavelet spectra in Figures 6.50 and 6.51 are surprisingly similar to those from the analysis of the packet throughput spectra for UNC 1 AM, discussed in Section 6.5.4. The full lossless replay shows higher energy, and the full collapsed-epochs replay shows substantially higher slope in the scaling region. Lossy replays provide far better approximations of the original spectra. As we observed for UNC 1 AM, these findings are inconsistent with observation P.4. Regarding fine scale energy levels in the outbound direction, lossless replays show higher energy than lossy ones, which are still above the original. In the outbound direction, lossless replays are very close to the original, while lossy ones are below it. Once again, the difficulties for matching fine scale energies mentioned in observation B.6 are present in this trace.

The estimates of the Hurst parameters are not so consistent with the results for UNC 1 AM, and are in better agreement with observation P.4. Here lossless replays approximate the original spectra closely, while lossy replays appear lower (full case) or higher (collapsed-epochs case) than the original.

Time Series of Active Connections

Figure 6.52: Active connection time series for UNC 7:30 PM and its four types of source-level trace replay.

The time series of active connections shown in Figure 6.52 are in good agreement with earlier traces. Every observation listed in Section 6.4.3 is confirmed by the UNC 7:30 PM results. Unlike the two previous UNC traces, the lossy full replay is not a perfect fit of the original time series, but it still provides a very close approximation, well within the 1% bound mentioned in observation C.2. The large spike around minute 23 does not appear in the collapsed-epochs replay, providing another clear illustration of observation C.6. Note that whatever the cause of this spike, it is not due to a difference in the number of connections started, since they are identical in the four replays. The spike is necessarily explained by a set of connections with substantially longer lifespans in the full replays than in the collapsed-epochs replays.