Traffic Modeling

Approaches to construct Traffic Models Trace-Driven: Collect traces from the network using a sniffing tool and utilize it directly in simulations. Empirical Distribution: Generate an empirical distribution from collected traces and accordingly generate random variables to drive the simulations. Distribution Fitting: Fitting collected traces to a well-known distribution. Use the fitted distribution for both simulations and analysis .

Trace-Driven Traffic Modeling time Packet Arrival Behavior over Time

Empirical Distribution Traffic Modeling time Packet Arrival Behavior over Time Cumulative Distribution Function (CDF)

Generate Samples of an Empirical Model Generate a uniform random variable between [0,1] A random sample of the empirical distribution is generated by selecting from the CDF c that corresponds to the generated uniform distribution output. Sample #2 Sample #1 Cumulative Distribution Function (CDF)

Distribution Fitting Traffic Modeling time Packet Arrival Behavior over Time

Advantages and Disadvantages Trace-Driven Empirical distribution Fitted standard Advantages Data is sure to be from the correct sample. Practical real-life results could be anticipated Sometimes there is not enough data to figure out the distribution accurately More flexibility in terms of data that can be generated Fairly simple to deduce from data Same advantages as the empirical distribution Irregularities can be smoothed out Disadvantages Simulation is limited to the results produced from the collected data Data may not be sufficient to do long-enough runs May have irregularities if the collected sample is not large enough (statistical abnormalities) The number of data points that can be generated may be limited depending on the original data samples Can be difficult to deduce if the available data is limited Always a chance of abnormalities that were not accounted for

Characterization of Distributions Mean and Central Moments of Distributions: Continuous Dist. Discrete Dist. Expected Value (Mean) 𝐸 𝑋 = 𝑥𝑓 𝑥 𝑑𝑥 𝐸 𝑋 = 𝑥𝑃 𝑋=𝑥 Variance 𝑉𝑎𝑟 𝑋 = 𝑥−𝐸 𝑋 2 𝑓 𝑥 𝑑𝑥 𝑉𝑎𝑟 𝑋 = 𝑥−𝐸 𝑋 2 𝑃 𝑋=𝑥 Skewness 𝑆𝑘𝑒𝑤 𝑋 = 𝑥−𝐸 𝑋 3 𝑓 𝑥 𝑑𝑥 𝑆𝑘𝑒𝑤 𝑋 = 𝑥−𝐸 𝑋 3 𝑃 𝑋=𝑥

Expected Value of Random Variables The average of the generated random samples If you would like to replace the whole distribution by ONE value, the mean will have the least mean square error

Variance of Random Variables Characterizes how far does the random variable deviate away from its mean value.

Skewness of Random Variables Skewness is a measure of the asymmetry of probability distributions Negative Skewness: The left tail is longer; the mass of the distribution is concentrated on the right of the distribution. Positive Skewness: The right tail is longer; the mass of the distribution is concentrated on the left of the distribution.

Other Parameters for Continuous Distributions Some additional parameters of continuous distribution can be helpful in guidance for distribution fitting Location Parameter (Shift Parameter): Specifies an abscissa (x coordinate) location point of a distribution’s range of values, Often some kind of midpoint of the distribution. Scale Parameter: Determine the scale of measurement or spread of a distribution. Shape Parameter: Determine the basic form or shape of a distribution within the family of distributions of interest.

Location Parameter Examples Normal Distribution Pareto Distribution 𝝁=𝟎 𝝁=𝟐 𝜽=𝟏𝟎 𝜽=𝟐𝟎 𝑓 𝑥 = 1 2𝜋 𝜎 2 𝑒 − 𝑥−𝝁 2 2 𝜎 2 𝑓 𝑥 = 1 𝜎 1+𝑘 𝑥−𝜽 𝜎 −1− 1 𝑘

Scale Parameter Example If X is a random variable with a scale parameter 1 then if there is a random variable Y = 𝛽X then its distribution will have scale parameter 𝛽 The standard deviation of a normal distribution is a scale parameter for it i.e., 𝛽=𝜎 𝑿 𝝈=𝟏 𝒀=𝝈𝑿 𝝈=𝟐

Shape Parameter Characteristics Normal and exponential distributions do not have a shape parameter other distributions such as beta distribution may have two shape parameters. A change in the shape parameter generally alter a distribution property more fundamentally than shift or scale parameters

Heavy-Tail Distributions A distribution with a tail heavier than the exponential Distributions where random variable values that are far from the “mean” of the distribution have a non-zero probability Pareto Principle: known as the 80-20 rule, i.e. 80% of the effects come from 20% of the causes

Heavy or Light Tailed? Pareto Distribution: 𝐶𝐷𝐹 𝐹 𝑋 =1− 𝑥 𝑚 𝑥 𝛼 ,𝑥≥ 𝑥 𝑚 Pareto Survivor Function: 1-𝐶𝐷𝐹 𝑥 𝑚 𝑥 𝛼 ,𝑥≥ 𝑥 𝑚 Plot on log-log scale: A heavy tailed survivor function would be linear in log-log domain Heavy Tail LightTail

Discrete Probability Distributions Binomial Distribution Uniform (Discrete) Distribution Geometric Distribution Negative Binomial Distribution Hypergeometric Distribution

Continuous Probability Distributions Uniform (Continuous) Distribution Triangular Distribution Normal Distribution Exponential Distribution Cauchy Distribution

Continuous Probability Distributions Lognormal Distribution Weibull Distribution Gamma Distribution

Estimation of Parameters Suppose that some distribution shape was deduced from the data set by any of the methods mentioned earlier. The data set X1,X2,…,Xn was used to deduce the distribution shape and can be used to estimate the parameters defining the distribution completely. There are many methods used to estimate the parameters. We will use the maximum likelihood estimator (MLE) method. The method can be explained as follows : suppose we have decided that a certain discrete distribution is the closest to the data set and that the distribution have one unknown parameter q The Likelihood L(q) function is defined as follows : This is basically the joint probability mass function since the data are assumed independent. This gives the probability of obtaining the data set as a whole if q is the value of the unknown parameter.

Maximum Likelihood Estimator The MLE of the unknown value q , which is denoted by q* is defined to be the value of q which maximizes L(q) In the continuous case the probability mass function is substituted with the chosen probability density function. Example : For an exponential distribution q = b and The likelihood function will be given by :

Maximum Likelihood Estimator (cont’d) Because most of the theoretical distributions include exponential functions it is often easier to maximize the logarithm of the likelihood function instead of L(q) itself. Define the log-likelihood function as: The problem reduces to maximizing the logarithmic function as the value of b which maximizes both functions has to be the same. The above equation equals zero if which means the sample mean of the sample set. This should be expected in the case of exponential random variables since they are fully characterized by their means

Maximum Likelihood Estimator (cont’d) Suppose that the distribution chosen is a geometric distribution which is a discrete distribution with pmf given by : The likelihood function will be given by : The log-likelihood function : By differentiating and equating to zero