The first technique we consider for introducing variability in the traffic generation
process is Poisson Resampling. The starting point of every method presented in this
chapter is a connection vector trace
where is an augmented connection vector
(an a-b-t connection vector plus some network-level parameters),
and is its relative start time.
The basic version of our Poisson Resampling technique
consists of deriving a new trace
by randomly choosing connection vectors from without replacement,
and assigning them start times according to an exponential distribution.
We define the duration of
as , the length of the interval in which connections are
Given a target
, the Poisson Resampling algorithm
iteratively adds a new
Each is equal to some randomly selected , and
The resampling technique described above has the advantage of its simplicity. Furthermore, it is statistically appealing, since the exponential distribution naturally arises from the combination of independent events. The use of connection inter-arrivals sampled independently from an exponential distribution is intuitively consistent with the view of traffic as a superposition of a large number of independent connections that transmit data through the same network link. Our empirical data, presented in Figures 7.1 to 7.4, confirm the applicability of this inter-arrival model. Figure 7.1 shows two pairs of CDFs comparing real connection arrivals and their exponential fits. The first pair (white symbols) corresponds to the distribution of connection inter-arrivals for UNC 1 PM (squares), and an exponential distribution7.2 with the same mean (triangles). The second pair (black symbols) shows the distribution of connection inter-arrivals for UNC 1 AM and an exponential distribution with the same mean. Both fits are excellent, so exponentially distributed connection inter-arrivals are clearly a good starting point for a trace resampling technique. The tails of the empirical distributions, shown in Figure 7.2, are also consistent with the fitted exponentials. Their slope is slightly lower, which could motivate a fit with a more general distribution like Weibull. However, a small improvement in the fit would require an increase in the complexity of the model, since the one-parameter exponential model would have to be replaced by the two-parameter Weibull model. This additional effort would produce only a limited gain given that the exponential fit is excellent for 99.9% of the distribution.
Figures 7.3 and 7.4 consider another two traces, Abilene-I and Leipzig-II. The bodies are again very closely approximated, but the tails are heavier for the original data. Note that this effect is more pronounced as the traces get longer. The duration of the UNC traces is one hour, the duration of Abilene-I is 2 hours, and the duration of Leipzig-II is 2 hours and 45 minutes. This could suggest that the worse fit is due to non-stationarity in the connection arrival process, which becomes more likely for longer traces. Further analysis is needed to confirm this hypothesis or find an alternative explanation. We must note that these results are in sharp contrast with those in Feldmann [Fel00], where the empirical inter-arrival distributions were significantly different from the bodies7.3 of fitted exponential distributions. The reason for this difference is unclear at this point7.4.
The main problem with the basic Poisson Resampling technique is the lack of control over the load offered by the replay of . As we will demonstrate, the number of connections in is only loosely correlated to the offered load. As a consequence, it becomes difficult to predict the load that a Poisson resampled trace generates, even for resamplings of the same duration and mean inter-arrival rate. This would force researchers to create many resampled traces until they hit upon a resampling with the intended offered load. We studied the wide range of offered loads that result from basic Poisson Resampling by performing a large of number of resamplings of the same connection vector trace .
As discussed in the introduction of this chapter, average load created
by is equal to the total number of bytes in divided by its duration
Given that TCP headers and retransmitted segments usually represent only a small fraction
of , the total size of the ADUs in divided by is also
a close approximation of . We use this approximation to examine the average
loads offered by a large number of Poisson resamplings, considering the offered load
of a resampling
equal to the total size of its ADUs
divided by its duration .
It is also important to note that the traces we consider are
bidirectional, and they do not necessarily create the same average load in both
directions. The analysis in the rest of this section will focus only on one direction
of the trace, the target direction, whose average load is given the total size of
the ADUs flowing in that direction divided by the duration of the trace.
More formally, the total size of the ADUs in the target direction is
Figure 7.5 shows a scatterplot of the results of 1,000 resamplings of UNC 1 PM. The duration of the resamplings and their mean rate of connection inter-arrival were equal to the ones in UNC 1 PM. For each resampling, the total number of connections is shown on the x-axis, while the resulting offered load is shown on the y-axis. This plot demonstrates that basic Poisson Resampling results in traces with very small variability in the number of connection vectors, between 1,409,727 and 1,417,664 (the standard deviation was equal to 1,191.71). On the contrary, the range of offered loads is very wide, between 143.55 and 183.44 Mbps ( Mbps), centered around the offered load of , 161.89 Mbps. The distribution of offered loads and its spread is further illustrated by the histogram in Figure 7.6.
The wide range of offered loads that can result from Poisson Resampling is due to
the heavy-tailed nature of the distribution of the total number of bytes contributed
by each connection vector.
The tail of this distribution for the UNC 1 PM trace is shown
in Figure 7.7.
The values in the plot correspond to the target direction, i.e.,
for each connection in and
for each connection in .
The tails show non-negligible probabilities for very large sizes, and a linear decay of
the probability over six orders of magnitude in the log-log CCDF.
As a consequence, the contribution to the offered load made by each connection is highly
variable and thus the number of connections in a trace is a poor predictor of its offered
This makes basic Poisson Resampling inadequate for controlling load.
Its only parameter is the mean inter-arrival rate of connections. This
rate controls the same number of connections in the resampling, but not the total size of these connections,
which varies greatly due to the heavy-tailed connection sizes.
Figure 7.8 further illustrates this point using six sets of 1,000
trace resamplings, each set with a different target offered load. The plot shows
a cross marking the mean of the load achieved by each of the sets of 1,000 experiments.
The variability in the offered loads is illustrated using error bars for each set of
experiments. The lower and upper endpoints of the error bars correspond to
the 5th and 95th percentiles of the loads offered by each set of trace resamplings.
Each set of trace resamplings had a fixed mean inter-arrival rate . Under the
assumption that the mean offered load is proportional to
the number of connections , we would expect the load to be inversely
proportional to the mean arrival rate , since .
Therefore, if the resamplings have the same duration , we would
expect to achieve an offered load of
As the previous section demonstrated, the number of connection vectors in a trace is a poor predictor of the mean offered load achieved during the replay of a resampled trace . Therefore, controlling the number of connections in a resampling does not provide a good way of achieving a target offered load, and an alternative method is needed. In the idealized model of offered load in the previous section, the offered load was said to be exactly equal to . If so, we need to control the total number of bytes in the resampled trace to precisely match a target offered load. In Byte-driven Poisson Resampling, the mean inter-arrival rate of is not computed a priori using Equation 7.2. Instead, the target load is used to compute a target size for the payloads in the scaled direction.
Byte-driven Poisson Resampling has two steps:
Figure 7.11 summarizes the results of 4 sets of 1,000 byte-drive Poisson resamplings. The plot uses the same type of visualization found in Figure 7.8. The error bars, barely visible in this case, illustrate the accurate load scaling that can be achieved with Byte-driven Poisson Resampling.
The previous analysis demonstrated that accurate load scaling requires control of the total number of bytes in rather than of the total number of connections. This demonstration was based on the computation of the offered load using equation 7.1. It is important to verify that the actual load generated during a testbed replay of is similar to the computed load. To show that this is indeed the case, we replayed a number of resampled traces with four different target loads. Each resampled trace was then constructed using byte-driven Poisson Resampling, with a duration of 1 hour. To eliminate startup and shutdown effects, we only considered the middle 40 minutes for computing the achieved load. Figure 7.12 summarizes the results of the experiments for the resamplings of the UNC 1 PM trace. Each point corresponds to a separate replay experiment, showing the target load on the x-axis and the achieved load on the y-axis. We ran three experiments for each target load, and the results show a good approximation of the intended scaling line. Several experiments achieved loads a few Megabytes above the target. In general, we expected the experiments to have slightly higher achieved loads, since the scaling method focuses on the offered payload, ignoring packetization overhead (i.e., extra load from bytes in the packet headers). A more precise tuning of the offered load would take packetization into account, perhaps using a fixed constant to decrease the target payload, or by studying the total size (including headers) of each connection, as replayed in the same testbed environment. In any case, and given the above good results, it seems reasonable to ignore these further refinements.
The results in Figure 7.12 provide examples of scaling down the load of a trace, since the original load of UNC 1 PM was 177.36 Mbps, and 9 of the 12 experiments had target offered loads below this value. Scaling down the load of a trace using byte-driven resampling simply requires to choose a target load below the original load , which in turns means that the of the resampling will be below the original . These results confirm the close approximation of the target loads in the testbed experiments, where offered load is measured from real TCP segments (rather than computed using Equation 7.1). The plot shows for example that the three resamplings with target load 177.36 Mbps achieved loads of 176.72, 178.23 and 182.45 respectively. The impact of the TCP headers, retransmission and the slightly underestimated duration mentioned in the previous section is therefore very small. The results in Figure 7.13 provide an example of scaling up the load of a trace, since they correspond to byte-driven Poisson resamplings of UNC 1 AM, which had an original load of 91.65 Mbps. The resulting loads also approximate the intended targets very closely.
Doctoral Dissertation: Generation and Validation of Empirically-Derived TCP Application Workloads
© 2006 Félix Hernández-Campos