Modified Traffic Model

In order to improve the model, we suggest an empirical approach in which packet size distribution and packet inter-arrival distribution are estimated per slot size. Hence, we divide the slots into N groups, and we estimate the empirical packet size distribution for each group of slots. Besides, we calculate the mean and variance of packet inter-arrivals for each group of slots.

Chapter 4. IP Traffic Modelling 61

Figure 4.7: CDF of Packet Inter-arrivals (during one slot)

Chapter 4. IP Traffic Modelling 62

Figure 4.9: Suggested Packet Generation Process Inside Slots

Let us denote P S the packet size distribution over the whole trace, and {P Si, i = 1, ..., N}

the packet size distribution per slot size group. We have:

E(P S) = N X i=1 E(P Si) N (4.12)

Similarly, let IA be the packet inter-arrival distribution over the whole trace, and {IAi, i = 1, ..., N} the packet inter-arrival distribution per slot size group, then:

E(IA) = N X i=1 E(IAi) N (4.13)

P S and P Si are estimated by discrete distributions directly from the trace. On the

other hand, we evaluate the average and variance of packet inter-arrivals for each group of slots and we generate packet inter-arrivals with distributions having the same average and variance. Two heuristics are proposed: first, we approximate the packet inter-arrivals inside slots by a Pareto distribution of the same average and variance, and second we generate packets according to an ON-OFF process (see Figure 4.9). In the later, we consider that packet inter-arrivals are constant (t) during ON period while there are no packet arrivals during OFF period. The duration of ON and OFF periods are calculated with respect to measured average and variance of packet inter-arrivals per slot size group.

According to the proposed model, we have:

E(λIAi) = 1 t ∗ TON TON + TOF F (4.14)

Chapter 4. IP Traffic Modelling 63

V ar(λIAi) = E(λ2_IAi) − (E(λIAi))

2 =1_t2_∗ TON TON+TOF F  −1t ∗ TON TON+TOF F 2 (4.15)

Note that TON + TOF F = T is constant by construction (slot duration), and after some

manipulations, we obtain: TON = (E(λIAi)) 2 V ar(λIAi) + (E(λIAi)) 2 ! ∗ T (4.16) λIAi = 1 t = V ar(λIAi) + (E(λIAi)) 2 E(λIAi) ! (4.17)

Once the values of TON and t are estimated, the generation process is completely specified.

We summarize in the following the complete estimation process as well as the generation process:

1. Divide the trace into slots of equal durations T .

2. Estimate the correlation structure of the time series corresponding to the size of slots.

3. Estimate the probability distribution of slot sizes. 4. Group slots into N sets by size.

5. For each set of grouped slots:

• Estimate the empirical packet size distribution.

• Calculate the mean and variance of packet inter-arrivals.

The generation process in the modified model is identical to the previous case. In fact, the M/G/∞ model will generate slot sizes respecting both probability distribution and correlation structure in the estimated traffic trace. The difference concerns only the way slot sizes are transformed into packet sizes with specified inter-arrivals. The following steps will take place after the generation of a size slot:

1. Determine the set to which belongs the generated slot.

2. Generate packets according to the packet size distribution of this set of slots. 3. Generate packet inter-arrivals respecting the estimated values per slot size, using

Chapter 4. IP Traffic Modelling 64

Table 4.5: Load in Packets and Loss Rate for ρ = 0.9 (Modified Model) Stats CAIDA ρ = 0.9 MAWI ρ = 0.9

Trace Model Trace Model

Loss rate % 5.2 4.9 5.31 5.1

Queue Load (Packets) 24.4 23.9 25.6 24.2 Mean Rate (Kbps) 90865.5 91292 25983.8 25311

4.3.5.1 Validation

The modified traffic model has the same statistical behaviour as the standard model. In fact, the modifications concerns only packet generation inside slots not the slot generation process. In Table 4.5 we show the simulation results using the modified model with the ON-OFF packet generation inside slots.

The modified model performs better in network simulation. The average queue length is almost the same in both cases. Similar results were obtained when using the Pareto distribution for packet inter-arrivals (not presented here). It seems that the model per- formance is influenced by the average and variance of packet inter-arrivals more than the distribution choice.

In fact, the accuracy of the model is achieved at the cost of estimating more param- eters (N packet size and packet inter-arrival distributions, N equals to 10 in our case). This may slow down the execution of the corresponding traffic generator which must be considered as another factor in designing efficient traffic models.

However, the packet size distribution and packet inter-arrival distribution during slots is a problematic issue in similar aggregate traffic models (FGN, FARIMA . . . ). In fact, many models succeed to capture correctly slot size correlations and probability distributions. Unfortunately, the model performance in a network environment is always optimistic as it does not take into consideration the variation of packet size distribution and packet inter-arrival distribution in function of the slot size.