APPENDIX D Statistical Distributions

Most of the parameters for continuous distributions can be classified, on the basis of their physical or geometric interpretation, as being one of three basic types: location, scale, or shape parameters. The following discussion is taken from the book Simulation Modeling and Analysis (Third Edition) by Law and Kelton (2000).

A location parameter l specifies an abscissa (x axis) location point of a distribution's range of values; usually, l is the midpoint (e.g., the mean of a normal distribution) or lower endpoint (e.g., location for a Pearson type V distribution) of the distribution's range. As l changes, the associated distribution merely shifts left or right without otherwise changing. Also, if the distribution of the random variable X has a location parameter of 0, then the distribution of the random variable Y = X + l has a location parameter of l.

A scale parameter b determines the scale (or unit) of measurement of the values in the range of the distribution. (The standard deviation is a scale parameter for the normal distribution.) A change in b compresses or expands the associated distribution without altering its basic form. Also, if the random variable X has a scale parameter of 1, the distribution of the random variable Y = bX has a scale parameter of b.

A shape parameter a determines, distinct from location and scale, the basic form or shape of a distribution with the general family of distributions of interest. A change in a generally alters a distribution's properties (e.g., skewness) more fundamentally than a change in location or scale. Some distributions (e.g., exponential and normal) do not have a shape parameter, while others (e.g., beta and Pearson type VI) may have two.

The last parameter of each distribution is the stream number.This is the random-number stream that will be used to general random values from the distribution. Possible values for stream are 1, 2, ..., 215, with 1 being the default.

Uniform Distribution

The uniform distribution (continuous) is equally likely to take on any real number in the finite interval [minimum, maximum] (minimum >= 0.0 and maximum > minimum). The real numbers produced by a random-number generator (appear to) have a uniform distribution on the interval [0, 1].

Normal Distribution

Probability density function and cumulative density function for normal distribution function with mean = 10, standard deviation = 1. There are two normal distributions in SIMPROCESS. One only returns non-negative values (zero or higher), and the other will return negative values. For both it is required that standard deviation > 0.0.

This distribution is similar to the classical normal distribution, but if a negative value is generated, it is rejected and new values are generated until a non-negative value is generated. In general, this distribution will not be a good model for the time required to perform some task, since task-time distributions are almost always skewed to the right.

This is the classical normal distribution, which is found in most statistics books. It takes on real values between minus infinity and plus infinity. The density function is the familiar "bell-shaped" curve, which is symmetric about the mean. The probability that a value is between the mean minus 2 standard deviations and the mean plus two standard deviations is approximately 0.95. This distribution should not be used to model the time required to perform some task, since the normal distribution can take on negative values. Furthermore, as stated above, the distribution of the time to perform some task is almost always skewed to the right, rather than being symmetric.

Triangular Distribution

The triangular distribution (continuous) is typically used as a rough model for the time required to perform some task when no real-world data are available. A triangular distribution takes on values in the finite interval [minimum, maximum] (minimum >= 0.0, mode > minimum, and maximum > mode), with values near the mode being most likely to occur. Subjective estimates of the three parameters are obtained from subject-matter experts. The mean of a triangular distribution is only equal to the mode when the distribution is symmetric.

Exponential Distribution

The exponential distribution (continuous) is commonly used to model interarrival times of customers to some system when the arrival rate is approximately constant over the time period of interest. It is also sometimes used to model the time to failure of a piece of equipment. The mean of an exponential distribution is a scale parameter and must be greater than 0.0. An exponential distribution with mean = m is a gamma distribution with mean = m and shape = 1. An exponential distribution with mean = m is a Weibull distribution with shape = 1 and scale = m. If interarrival times of customers have an exponential distribution with mean = m, then the number of arrivals in any time interval of length t has a Poisson distribution (discrete) with mean = t/m.

Gamma Distribution

The gamma distribution (continuous) could be used to model the time required to perform some task. If a gamma distribution has parameters mean = m and shape = a, then b = m/a is a scale parameter. A gamma distribution with mean = m and shape = 1 is an exponential distribution with mean = m. When shape is a positive integer, the gamma distribution is an Erlang distribution. Parameter restrictions are mean >= 0.0 and 0.0 < shape < 100.0.

Beta Distribution

Probability density function and cumulative density function for a beta distribution with shape1 = 1.5, shape2 = 5.0, minimum = 0.0, and maximum = 1.0.

The beta distribution (continuous) could be used to model the time required to perform some task when the possible values are restricted to the finite interval [minimum, maximum] (minimum >= 0.0, maximum >= 1.0, and maximum > minimum). Parameter restrictions for shape1 and shape2 are shape1 >= 0.0 and shape2 > 0.0. The density function is skewed to the left, symmetric, or skewed to the right if shape1 > shape2, shape1 = shape2, or shape1 < shape2, respectively. A beta distribution with shape1 = shape2 = 1 is a uniform distribution with the interval [0, 1].

Erlang Distribution

The Erlang distribution (continuous) could be used to model the time required to perform some task. If an Erlang distribution has parameters mean = m and shape = a, then b = m/a is a scale parameter. An Erlang distribution is just a gamma distribution whose shape parameter is a positive integer. The sum of k exponential random variables with mean = m is an Erlang distribution with mean = km and shape = k. Parameter restrictions are mean >= 0.0 and 0.0 < shape < 100.0.

Weibull Distribution

The Weibull distribution (continuous) could be used to model the time required to perform some task. It is also sometimes used to model the time to failure of a piece of equipment. A Weibull distribution with parameters shape = 1 and scale = b is an exponential distribution with mean = b. The Weibull distribution is skewed to the left when shape > 3.6. Parameter restrictions are shape >= 0.0 and scale > 0.0.

Lognormal Distribution

The lognormal distribution could be used to model the time required to perform some task when "large" values sometimes occur. It is always skewed to the right and it has a longer right tail than the gamma or Weibull distributions. The lognormal distribution is closely related to the classical normal distribution - see the book Simulation Modeling and Analysis (Third Edition) by Law and Kelton (2000) for details. Furthermore, the parameters of the lognormal distribution, namely, mean and standard deviation, correspond to the lognormal distribution and are not the mean and standard deviation of the corresponding normal distribution. Parameter restrictions are mean > 0.0 and standard deviation > 0.0.

Poisson Distribution

The Poisson distribution (discrete) with mean = m is the distribution of the number of customers that arrive to some system in any time interval of length 1 when the interarrival times have an exponential distribution (continuous) with mean = 1/m. Mean must be greater than 0.0.

Hyper Exponential Distribution

The hyper exponential distribution (continuous) is a mixture of two exponential distributions. Specifically, a hyper exponential distribution with parameters mean1, mean2, and probability1 takes on values from an exponential distribution with parameter mean1 with a probability of probability1 and takes on values from an exponential distribution with parameter mean2 with a probability of 1 - probability1. A hyper exponential distribution with probability1 = 1 is an exponential distribution with parameter mean1. Parameter restrictions are mean1 >= 0.0, mean2 > 0.0, and 0.0 <= probability1 <= 1.0.

Uniform Integer Distribution

An uniform integer distribution (discrete) is equally likely to take on any integer in the finite interval [minimum, maximum], where minimum and maximum are integers with minimum >= 0 and minimum < maximum.

Geometric Distribution

The geometric distribution (discrete) with probability = p can be thought of as the distribution of the number of failures before the first success in a sequence of independent Bernoulli trials, where success occurs on each trial with a probability of p and failure occurs on each trial with a probability of 1 - p.

Pareto Distribution

The Pareto distribution (continuous) could be used to the model interarrival times of customers (e.g., messages) when the traffic is bursty. The mean and variance are finite only if shape > 2. Parameter restrictions are location >= 0.0 and shape > 0.0.

Binomial Distribution

The binomial distribution (discrete) with parameters trials = t (a positive integer) and probability = p can be thought of as the distribution of the number of successes in t independent Bernoulli trials, where success occurs on each trial with a probability of p and failure occurs on each trial with a probability of 1 - p. A binomial distribution with trials = 1 is called a Bernoulli distribution with probability = p.

Negative Binomial Distribution

The negative binomial distribution (discrete) with parameters s (> 0.0) and probability = p can be thought of as the distribution of the number of failures before the sth success in a sequence of independent Bernoulli trials, where success occurs on each trial with a probability of p and failure occurs on each trial with a probability of 1 - p. A negative binomial distribution with parameters s = 1 and probability = p is a geometric distribution with probability = p.

Inverse Gaussian Distribution

The inverse Gaussian distribution (continuous) could be used to model the time required to perform some task. Parameter restrictions are scale > 0.0 and shape > 0.0.

Inverted Weibull

The inverted Weibull distribution (continuous) could be used to model the time required to perform some task. The mean and variance are finite only if shape > 2. If the random variable X has an inverted Weibull distribution with location = 0, scale = b, and shape = a, then Y = 1/X has a Weibull distribution with scale = 1/b and shape = a. (The location parameter is 0.) Parameter restrictions are scale > 0.0 and shape > 0.0.

Johnson SB Distribution

The Johnson SB distribution (continuous) could be used to model the time required to perform some task when the possible values are restricted to the finite interval [minimum, maximum]. The density function is skewed to the left, symmetric, or skewed to the right if shape1 > 0, shape1 = 0, or shape1 < 0, respectively. The Johnson SB distribution is closely related to the classical normal distribution - see the book Simulation Modeling and Analysis (Third Edition) by Law and Kelton (2000) for details. Parameter restrictions are shape2 > 0.0 and maximum > minimum.

Johnson SU Distribution

The Johnson SU distribution (continuous) could be used to model a random variable that can take on any value between minus infinity and plus infinity. The density function is skewed to the left, symmetric, or skewed to the right if shape1 > 0, shape1 = 0, or shape1 < 0, respectively. The Johnson SU distribution is closely related to the classical normal distribution - see the book Simulation Modeling and Analysis (Third Edition) by Law and Kelton (2000) for details. Parameter restrictions are scale > 0.0 and shape2 > 0.0.

Log-Logistic Distribution

The log-logistic distribution (continuous) could be used to model the time required to perform some task. The mean and variance are finite only if shape > 2. Parameter restrictions are scale > 0.0 and shape > 0.0.

Log-Laplace Distribution

The log-Laplace distribution (continuous) could be used to model the time required to perform some task. The mean and variance are finite only if shape > 2. Parameter restrictions are scale > 0.0 and shape > 0.0.

Pearson Type V Distribution

The Pearson type V distribution (continuous) could be used to model the time required to perform some task. The mean and variance are finite only if shape > 2. The Pearson type V distribution is closely related to the gamma distribution - see the book Simulation Modeling and Analysis (Third Edition) by Law and Kelton (2000) for details. Parameter restrictions are scale > 0.0 and shape > 0.0.

Pearson Type VI Distribution

Probability density function and cumulative density function for a Pearson type VI distribution with location = 0, scale = 1, shape1 = 3, and shape2 = 4.

The Pearson type VI distribution (continuous) could be used to model the time required to perform some task. The density function can take on a wide variety of shapes because it has two shape parameters shape1 and shape2. The mean and variance are finite only if shape2 > 2.The Pearson type VI distribution is closely related to the beta distribution - see the book Simulation Modeling and Analysis (Third Edition) by Law and Kelton (2000) for details. Parameter restrictions are scale > 0.0 and shape1 > 0.0.

Random Walk Distribution

The random walk distribution (continuous) could be used to model the time required to perform some task. Parameter restrictions are scale > 0.0 and shape > 0.0.

Empirical Distribution

Empirical distributions are not found on the SIMPROCESS distribution list. An empirical distribution can only be created by an Auto Fit distribution. (See "Auto Fits Distributions" on page 101.) Thus, the name of the Auto Fit distribution appears on the SIMPROCESS distribution list, not the empirical distribution itself.

Empirical distributions consist of pairs of cumulative probability and value and can be discrete or continuous. This is the same format as Tabular Distributions. (See "Tabular Distributions" on page 99.)