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5.4 Modeling traffic to help identify the safeguard band

In the previous section, I used traces from real links in the network to predict the safeguard band that is required for a certain overflow probability. In most cases, it is not economical to have trace collecting equipment on every link, and so it may not be possible to obtain such detailed traces. For this reason, it is beneficial to have a simple model that requires less information to achieve the same goal. In addition, if the model is simple enough, one can also obtain formulae that predict the appropriate safeguard band based on a small number of network parameters.

I will now perform the same analysis as in the previous section on synthetic traffic traces that are generated using the distributions and statistics from the links under consideration. Notice that this stochastic information can be obtained with considerably less effort than a real trace because they can be estimated by sampling the traffic.

In a trace of active flows, one has three pieces of information per flow: the flow interarrival time, the flow duration and the flow average bandwidth.5.5 Flow interarrival times are essentially independent of each other and closely follow a Poisson process, as shown by the nearly exponential interarrival times in Figure 5.7. In the traces, the average arrival rates were between 124 and 594 flow/s. This hypothesis of Poisson-like arrivals is further supported by the wavelet estimator described by Abry and Veitch [1]: the Hurst parameter of the interarrival times is very close to 1/2, which suggests independence. Similar results have been reported by Fredj et al. [78] and by Cleveland et al. [33,45]. For this reason, for the synthetic trace, we can model flow arrivals as a Poisson process. Hence to parameterize the model we need only the average arrival rate of the flows.

Figure 5.7: Inverse cumulative histogram of the flow interarrivals for both TCP and non-TCP traffic in the Sprint traces. An exponential interarrival time would be represented as a straight line in this graph.

Figure 5.8:

Histograms of (a) the flow average bandwidth and (b) the flow duration for both TCP and non-TCP traffic in the Sprint traces. Single-packet flows have not been considered.

 

The flow average bandwidth (shown in Figure 5.8a) and the flow duration (shown in Figure 5.8b) have empirical distributions that are harder to model. Furthermore, the values are not independent of each other. The correlation coefficient between them in the Sprint traces was between -0.134 and -0.299,5.6 which is consistent with the work by Zhang et al. [189]. Figure 5.9 shows the joint histogram for the flow duration and average bandwidth, which makes their correlation clear. Jobs with more available bandwidth usually take less time to complete. In terms of successive arrivals, the autocorrelation function was almost zero, and so the arrivals can be considered independent. The flow average bandwidth and flow duration can then be modeled as a sequence of i.i.d. 2-dimensional random variables.

Figure 5.9: Joint histogram of flow durations and average bandwidths for both TCP and non-TCP traffic in the Sprint traces.

Even if the correlation between the flow average bandwidth and the flow duration is small, when the marginal distributions of the two magnitudes are used the results of the model and the traces diverge considerably for the low overflow probabilities. The reason is that short-duration and high-bandwidth flows occur more often in the synthetic traces created from the marginal distributions than in the real trace, and these flows can skew the results. Results are much closer to the trace-driven model when using Poisson arrivals and the empirical joint distribution for the flow duration and average rate. Figure 5.10 shows how the synthetic trace using the joint distribution produces results that are very close to those obtained with the real trace.

Figure 5.10: Safeguard band required for certain overflow probabilities and circuit-creation latencies for real traffic traces (solid line) and a simple traffic model (dashed line) with Poisson arrivals and flow characteristics that are drawn from an empirical distribution.

This model corresponds to an MB/G/$\infty$ system, where there are infinite parallel servers, arrivals are batched Poisson and service times are correlated with the batch size. As far as I know, there is no closed-form expression for the transition probabilities:


p

= 1 - P[N(t) - N(0) < STp . N(0),t [0, T)]  

= P[max{N(t) - N(0),t [0, T)} STp . N(0)]

 

where N(t) is the number of clients in the system at time t.

In summary, we can estimate the safeguard band that is required to avoid circuit overflows just by using the average flow rate and the joint distribution of the flow average bandwidth and the flow duration. This information can then be used to construct a set of curves like the one in Figure 5.10.


next up previous
Next: 5.5 Discussion Up: 5. Coarse circuit switching Previous: 5.3 Monitoring user flows
Copyright © Pablo Molinero-Fernández 2002-3