Source-level Replay of UNC 1 AM

Time Series of Byte Throughput

Figure 6.27: Byte throughput time series for UNC 1 AM inbound and its four types of source-level trace replay.

Figure 6.28: Byte throughput time series for UNC 1 AM outbound and its four types of source-level trace replay.

The plots of the 1-minute time series of byte throughput for the original UNC 1 AM and its replays are shown in Figure 6.27 (inbound direction) and in Figure 6.28 (outbound direction). For the inbound, we observe a moderately bursty time series with a large increase in byte throughput between minutes 15 and 32. In good agreement with observation B.1, the replays track the shape of the original time series well. They also approximate some smaller spikes, such as the one in minute 45, and miss others, such as the one in minute 17. The result is similar for the outbound direction, although we again find a slightly lower overall throughput in the replays. There is also an area of higher throughput in the original trace between minutes 35 and 43 that is not properly reproduced by any of the replays. The full lossy replay provide the closest approximation, but there is still a clear difference with respect to the original time series.

The results also support the observation of higher burstiness from lossless replays, B.2, and from collapsed-epochs replays, B.3; especially for the inbound direction. The results are also consistent with observation B.7, since the full lossy replay appears closest to the original.

Time Series of Packet Throughput

Figure 6.29: Packet throughput time series for UNC 1 AM inbound and its four types of source-level trace replay.

Figure 6.30: Packet throughput time series for UNC 1 AM outbound and its four types of source-level trace replay.

The time series of packet throughput for UNC 1 AM inbound shown in Figure 6.29 are in sharp contrast to earlier results. As stated in observation P.1, the time series from the replays of the previous traces were generally below the time series of the original trace. However, the full replays of UNC 1 AM are often above the original packet throughput, especially in the case of the lossy full replay. The same is not true for the outbound direction, as shown in Figure 6.30, where the replays are again below the original for a large fraction of the time series. While the replays provide a reasonable approximation of the overall time series, the original packet throughput in the outbound direction is substantially lower between minutes 35 and 43. The difference is most apparent for the collapsed-epochs replays.

Regarding observation P.2, we can see that collapsing epochs substantially reduced packet throughput. Paradoxically, this makes the time series of the lossy collapsed-epochs match the original quite well, although the same is not true for the lossless collapsed-epochs replay. Note also that it is difficult to argue for this trace that the lossy replays are significantly more bursty than the lossy ones at the 1-minute scale. We only observe one artificial spike in minute 27 for the lossless collapsed-epochs replay.

Marginal Distributions

Figure 6.31: Byte throughput marginals for UNC 1 AM inbound and its four types of source-level trace replay.

Figure 6.32: Byte throughput marginals for UNC 1 AM outbound and its four types of source-level trace replay.

Figures 6.31 and 6.32 study the marginal distributions of the original 10-millisecond time series UNC 1 AM and those from the source-level trace replays. The bodies of the distributions from the replays are almost identical to the original for the inbound direction, and quite close for the outbound direction, which further supports observation B.1. Observation B.2 is consistent with these results, although the bodies of the lossless replays are very close to those of the lossy ones in this case. We do however observe consistently heavier tails from lossless replays. Note the much heavier tail from the lossless collapsed-epochs replay, which reveals an extra burstiness that was not visible in Figure 6.27. In agreement with observation B.3, we do not find consistently wider bodies or heavier tails from the collapsed-epochs replays. Finally, observation B.7 remains valid, with the lossy full replay being best for the tail of the inbound direction, but clearly not for the opposite direction.

Figure 6.33: Packet throughput marginals for UNC 1 AM inbound and its four types of source-level trace replay.

Figure 6.34: Packet throughput marginals for UNC 1 AM outbound and its four types of source-level trace replay.

The lesson from the plots of the packet throughput marginals shown in Figures 6.33 and 6.34 is similar to the one discussed for the time series in Section 6.5.2. In the inbound direction, the marginal from the lossy full replay is heavier than the original, while the lossless full replay and the lossy collapsed-epochs replays are rather close to the original. In the outbound direction, the results are consistent with the somewhat lower packet throughput for the replay stated in observation P.1.

The tails of the marginals are again surprising for UNC 1 AM, and do not follow observation P.3. The inbound plot shows lossless replays that are significantly heavier than the original, which exhibits the lightest tail. Lossy replays provide far better approximations. The outbound plot appears closer to the previous observation, with the lossless replays being the closest ones to the original. Note however that they are somewhat heavier, unlike in the Leipzig-II and UNC 1 PM cases.

Long-Range Dependence

Figure 6.35: Wavelet spectra of the byte throughput time series for UNC 1 AM inbound and its four types of source-level trace replay.

Figure 6.36: Wavelet spectra of the byte throughput time series for UNC 1 AM outbound and its four types of source-level trace replay.

Table 6.5: Estimated Hurst parameters and their confidence intervals for the byte throughput time series of UNC 1 AM and its four types of source-level trace replay.

Trace	Inbound		Outbound
UNC 1 AM		C. I.		C. I.
Original	0.9885	[0.9479, 1.0292]	0.9990	[0.9584, 1.0397]
Lossless Replay	1.0275	[0.9868, 1.0681]	0.9705	[0.9299, 1.0111]
Lossy Replay	0.9465	[0.9058, 0.9871]	0.9546	[0.9140, 0.9953]
Lossless Coll. Epochs	1.0089	[0.9683, 1.0495]	1.2036	[1.1630, 1.2443]
Lossy Coll. Epochs	0.9136	[0.8730, 0.9542]	0.9720	[0.9313, 1.0126]

While the wavelet spectra for the inbound direction shown in Figure 6.35 are in good agreement with observation B.4, we find substantially higher energy above the original in the spectrum of the lossless full replay for outbound direction. The estimated Hurst parameters shown in Table 6.5 are again difficult to assess, as mentioned in that observation. Lossless replays are the only ones within the confidence interval of the estimate for the original inbound direction, while only the lossless collapsed-epochs replay is outside the confidence interval for the outbound direction. Incidentally, the extremely high estimate for the lossless collapsed-epochs replay is remarkable. It is 0.23 above the lossless full replay, illustrating the major difference that detailed source-level modeling can make on traffic long-range dependence.

In the scaling region, collapsed-epochs replays do show higher energy than full replays, as observed in B.5. This higher energy does not translate into higher Hurst parameter estimates. Notice for example the lower estimates for the inbound direction. For both directions, the lossy collapsed-epochs replay provides a good approximation of the original spectrum, although not as good as the lossy full replay. The results for UNC 1 AM are therefore consistent with observation B.7. At the finest scales, we find that the lossy full replay approximates the energy levels of the inbound direction most closely, while it is the lossy collapsed-epochs replay the best match for the outbound direction. This inconsistency supports observation B.6.

Figure 6.37: Wavelet spectra of the packet throughput time series for UNC 1 AM inbound and its four types of source-level trace replay.

Figure 6.38: Wavelet spectra of the packet throughput time series for UNC 1 AM outbound and its four types of source-level trace replay.

Table 6.6: Estimated Hurst parameters and their confidence intervals for the packet throughput time series of UNC 1 AM and its four types of source-level trace replay.

Trace	Inbound		Outbound
UNC 1 AM		C. I.		C. I.
Original	0.9316	[0.8871, 0.9760]	0.9309	[0.8864, 0.9753]
Lossless Replay	0.9860	[0.9416, 1.0305]	0.9830	[0.9385, 1.0274]
Lossy Replay	0.9749	[0.9304, 1.0193]	0.9759	[0.9315, 1.0204]
Lossless Coll. Epochs	1.1478	[1.1034, 1.1923]	1.2128	[1.1683, 1.2572]
Lossy Coll. Epochs	0.9504	[0.9059, 0.9948]	0.9757	[0.9313, 1.0202]

Figures 6.37 and 6.38 reveal that the wavelet spectra from lossless replays do not approximate the original spectra well. For both directions, the full lossless replay shows significantly more energy, while the full collapsed-epochs replay shows higher slope in the scaling region. This poor fit for the lossless full replay contradicts observation P.4. Lossy replays appear closer to the original spectra in the scaling region, especially in the case of the lossy full replay. Replays do not consistently match the energy in the fine-scale region, as stated in observation P.5. Lossy replays are the closest ones in this region.

The estimated Hurst parameters shown in Table 6.6 do not follow observation P.4 very clearly. The estimates from the lossless collapsed-epochs replay are far larger than the original estimates. The estimates from the lossless full replays are far lower, but they are still above the upper ends of the confidence intervals. Finally, both lossy replays are within confidence intervals, although the actual estimates are higher.

Time Series of Active Connections

Figure 6.39: Active connection time series for UNC 1 AM and its four types of source-level trace replay.

The time series of active connections shown in Figure 6.39 confirm the list of observations in Section 6.4. It is clear that observations C.1 and C.2 hold, being the lossy full replay a perfect match of the original time series. Observation C.3 is also true, although the relative gap between the number of connections in full and collapsed-epochs replays is smaller for this trace. The impact of losses is somewhat more significant for the collapsed-epochs replays, which is not completely in agreement with observation C.4. Observations C.5 and C.6 are consistent with the results for UNC 1 AM.