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Random distribution dialog window

This window is aimed at defining the statistical distribution function to be associated with the given variable parameter, which value would be randomly generated by the Event Generation Engine each time when starting a new snapshot during simulation. This is the most oftenly used data type in SEAMCAT, as absolute majority of the parameters in workspace scenario may be defined via distributions, with those randomly generated parameters providing the statistical essence of Monte-Carlo simulations in SEAMCAT.

The distribution entry dialog window looks like this:

Generation of random parameter may be defined in this dialog window by setting one of the following distribution types:

• Constant: in this case a trial performed on this variable always returns the same constant value (may be integer or floating point);
• User-defined: continuous distribution defined by its cumulative distribution function, entered as pairs (x, y=F(x)):

Note: the definition range of y=(0, 1)

• Uniform: represents a continuous uniform distribution, with given min and max values, and all intermediate values having equal probability:

Note: the continuos nature of this function results in that the trial returns double floating point number within the range (min, max). This means that this function is not suitable e.g. to define frequency hopping pattern, since the latter would usually occur within a set of (pre-defined) discrete channels. Such cases can be modelled using Discrete uniform function described below.

• Gaussian: an ordinary Gaussian (Normal) distribution defined by mean m and standard deviation σ values:

• Rayleigh: an ordinary Rayleigh distribution defined via its min and standard deviation σ values:

• Uniform polar distance: is a distribution function designed to define a random positioning of transmitter along the radius of coverage cell, to achieve a random uniform distribution of transmitters within a circular area centred around a given zero-point. This function has one parameter - max distance - and probability of distribution along that distance is defined as:

Note: Uniform polar distance distribution is typically used for deriving distance factor used in calculation of the relative locations of transceivers  within a link and between victim and interfering links. The result of the trial on such a distribution, the distance factor, is then multiplied by a coverage radius or simulation radius. Hence the default maximum value of R is set to 1, meaning that after multiplication of this random factor with the radius value, the resulting distance will be distributed uniformly along the entire coverage/simulation radius.

• Uniform polar angle: to be used along with Uniform polar distance, this function is designed to describe a uniform distribution of transmitters within a circular area centred around a given zero-point. But whereas Uniform polar distance describes random distance to centre point, the Uniform polar angle function defines random angle (azimuth) of transmitter with regards to centre point. This function has one input parameter - maximum angle Amax - and generated random values will be placed with equal probability (uniform distrbution function) within the range -Amax...Amax;

• User-defined (stair): is the discrete alternative of continuous User-defined function described above. The Stair function is defined by a set of pairs (X_i, S(X_i)) where the set of X_i represents all possible values that might be assigne to the variable, whereas S(X_i) represents their cumulative probabilities:

Note: the definition range of S(X_i)=(0, 1)

• Discrete uniform: is the discrete alternative of the Uniform distribution described above. The Discrete uniform distribution is defined by the following parameters:
  • Lower bound X_min (OBS: not to be mixed with the smallest value of discrete variable X_i, see the illustration below)
  • Upper bound X_max (OBS: not to be mixed with the largest value of discrete variable X_i, see the illustration below)
  • Step S (e.g. channel spacing in the case of frequency distributions)

The relationship between these parameters and the discrete values of modeled random variable X_i is shown below:

As a result, the generated discrete random parameter will be taking the following values:

each value being assigned the same probability:

with

General notes:

1) Trials performed on all of the above distributions are based upon using internal Java pseudo-random number generator;
2) For entering user-defined distributions, enter the values in the table grid in form of pairs (x, y=P(X less than x)):

• To add a data pair click on Add button. Pairs are then automatically sorted by increasing x values,
• To suppress a selected data pair, click on Delete button,
• To symmetrize a distribution, click on Sym button, this results in generating for each pair (x, P(X)) a symetric pair (-x, (1-P(X)) if it doesn't already exist,
• Import/export: Click on the Load button to load the function values from an external text file. This file must contains one pair (x, P(X < x)) per line, TAB separated. Other way round, user can save the defined function by pressing on Save button to export the data to a text file.

Tip: user may test the result of generating random values using a particular distribution type via menu option Tools -> Test internal functions -> Test distribution. This could be e.g. used to analyse the statistical qualities of generated random variables.