Traffic Modeling

The tool can calculate statistics from a pre-recorded stream (trace). The module accepts as input, for instance, the number of bytes per a given interval and produces first and second order statistics as output. The first-order statistics are: average traffic rate (bits/time unit), variance, and burstiness. The second order statistics are: autocovariance, autocorrelation, and index of dispersion per count. The $\textrm{\index*{IDC}}(t)$ for instance, is obtained, for a finite data set, by dividing the set into non-overlapping intervals and taking these intervals as different sample paths for the random process $\mathcal{N}$. In the interface we call ``window'' the non-overlapping intervals. We can also obtain other measures, such as the fraction of time above a given rate.

Let $X(n)$ be the $n$th sample of your trace file.

• The autocorrelation is given by
  

  $$\textrm{autocor}(\tau) = \frac{E[X(n)X(n+\tau)]-E[X(n)]^2} {E[X(n)^{2}]-E[X(n)]^{2}}.$$ (6.1)

  
  

• The autocovariance is given by
  

  $$\textrm{autocov}(\tau) = E[X(n)X(n+\tau )] - E[X(n)]^2.$$ (6.2)

  
  

• The IDC is given by
  

  $$\textrm{IDC}(\tau) = \frac{E[N(k\tau )^2]-E[N(k\tau )]^2}{E[N(k\tau )]},$$ (6.3)

  
  
  where $N=\{N(k\tau ),\, k\tau \geq 0\}$ is a stationary process that counts the amount of data in the interval $[0,k\tau ]$, with $k\in [0,\texttt{Number of points}]$. As your trace is a sample path, not a stationary process, it will be split into several windows (sample paths).

Figure shows the interface to compute some statistics from a pre-recorded stream.

Figure: Interface to obtain traffic statistics from a trace

The parameters to be given are:

Trace Name

The name of the file that contains the pre-recorded stream.

Number of samples

Number of samples in the trace file.

Time Scale

Time between two consecutive samples in the trace ($\Delta t$).

Maximum Time Lag

The measure will be computed from $0$ to this value.

Number of points

The total number of observation points, from zero to maximum time lag. Note that $\tau =\textrm{Maximum time lag}/\textrm{Number of points}$.

Input Values

• Sample: $X(n)=\texttt{sample}$, where sample are the sample values given in your trace.
• Sample/Time scale: $X(n)=\texttt{sample}/\Delta t$.

Interval between windows

The interval between two consecutive windows: it can be an exponential random variable or zero. It is used for the computation of IDC.

Notice that $\Delta t$ and $\tau$ are related. It does not make sense, for example, to set $\Delta t=2$ and $\tau =1$, as there is no sample at odd times. For these mistakes, Tangram II prints an error on your terminal window.
NOTES:

1. Your trace file should contain just the sample values. The tool assumes that they are evenly-spaced by $\Delta t$.
2. When a time scale does not make sense, as in calculating the autocorrelation of a delay trace, set $\Delta t$ to 1.

If the traffic model is Markovian, first and second order statistics can be obtained using the recursions given in [29]. The overall model containing the traffic model plus the network resources model can be built using TANGRAM-II. TANGRAM-II also calculates several measures of interest, such as loss probabilities.

Figure shows the interface to compute some statistics from a Markovian model. The parameters Initial Probability, Time Intervals, and Precision have to be provided for second-order statistics only. The parameter Reward Name is the name of the reward in the model that will be used to calculate the traffic model statistics (in general, the reward represents the traffic source rate).

Figure: Interface to obtain traffic statistics from a markovian model

The parameters to be given are:

Initial Probability

This specifies the probability vector at time zero:

• Equiprobable All states in the model have the same initial probability.
• Initial State The initial state specified in the model has initial probability 1, and all other states have initial probability 0.
• Equiprobable Set Each state in the set specified has the same initial probability.

Time Intervals

The intervals at which to compute the traffic model statistics:

• Initial Time The first observation point; this time must be greater than zero.
• Final Time The last observation point.
• Number of points The total number of observation points in a time interval, including the initial and the final time.

Precision

Error bound.

The user is not limited to Markovian traffic models. The TANGRAM-II simulator allows the specification of inter-event times obtained from samples from FARIMA or FBM processes. In this case the user must specify the mean rate, variance, time scale, and Hurst parameter.

From either the Markovian models or the FARIMA Distributions, FARIMA and FBM Distributions, FBM, second order statistics can be obtained, by direct recursions (if the model is Markovian) or from a trace generated by the simulator.