BUILT-IN WOLFRAM LANGUAGE SYMBOL

# HyperexponentialDistribution

HyperexponentialDistribution[{α_{1},…,α_{m}},{λ_{1},…,λ_{m}}]

represents an m-phase hyperexponential distribution with phase probabilities and rates .

## DetailsDetails

- HyperexponentialDistribution is also known as a mixed exponential or parallel m-phase exponential distribution.
- An m-phase hyperexponential distribution can be interpreted as having m servers in parallel where the i server has service rate and is picked with probability .
- The probability density for value in an exponential distribution is for , and is zero for .
- allows to be any non-negative numbers such that and can be any positive real numbers.
- HyperexponentialDistribution can be used with such functions as Mean, CDF, and RandomVariate.

## Background & ContextBackground & Context

- HyperexponentialDistribution[{α
_{1},…,α_{m}},{λ_{1},…,λ_{m}}] represents a continuous statistical distribution defined over the interval , parameterized by two vectors and , and known as an -phase hyperexponential distribution. The parameters are called "phase probabilities," have values in the interval , and satisfy , while the parameters are called "phase rates" and have positive real values. Together these parameters determine the overall shape of the probability distribution function (PDF), which in general is monotonic decreasing and has tails which are "thin" in the sense that the PDF decreases exponentially rather than decreasing algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) Random variables X satisfying are sometimes said to have a hyperexponential distribution of order . - Named because its coefficient of variation (the ratio of StandardDeviation to Mean) is always larger than 1 (which is the coefficient of variation for any exponential distribution), the hyperexponential distribution is an example of a mixture distribution (MixtureDistribution) and is often thought of as a generalization of ExponentialDistribution in the sense that its PDF is the sum of exponential density functions. Because it has thin tails, the hyperexponential distribution is an often-used model for studying queueing systems and sensor networks. One unique property of the hyperexponential distribution is that it can be used to approximate arbitrary probability distributions (even those with heavy tails), a fact which is used in the study of network performance models to capture performance data for various queueing models that cannot be quantitatively deduced using exact methods. The hyperexponential distribution has also been used in the study semiconductors, manufacturing systems, and computer hardware architecture.
- RandomVariate can be used to give one or more machine- or arbitrary-precision (the latter via the WorkingPrecision option) pseudorandom variates from a hyperexponential distribution. Distributed[x,HyperexponentialDistribution[{α
_{1},…,α_{m}},{λ_{1},…,λ_{m}}]], written more concisely as , can be used to assert that a random variable x is distributed according to a hyperexponential distribution. Such an assertion can then be used in functions such as Probability, NProbability, Expectation, and NExpectation. - The probability distribution and cumulative density functions may be given using PDF[HyperexponentialDistribution[{α
_{1},…,α_{m}},{λ_{1},…,λ_{m}}],x] and CDF[HyperexponentialDistribution[{α_{1},…,α_{m}},{λ_{1},…,λ_{m}}],x]. The mean, median, variance, raw moments, and central moments may be computed using Mean, Median, Variance, Moment, and CentralMoment, respectively. - DistributionFitTest can be used to test if a given dataset is consistent with a hyperexponential distribution, EstimatedDistribution to estimate a hyperexponential parametric distribution from given data, and FindDistributionParameters to fit data to a hyperexponential distribution. ProbabilityPlot can be used to generate a plot of the CDF of given data against the CDF of a symbolic hyperexponential distribution and QuantilePlot to generate a plot of the quantiles of given data against the quantiles of a symbolic hyperexponential distribution.
- TransformedDistribution can be used to represent a transformed hyperexponential distribution, CensoredDistribution to represent the distribution of values censored between upper and lower values, and TruncatedDistribution to represent the distribution of values truncated between upper and lower values. CopulaDistribution can be used to build higher-dimensional distributions that contain a hyperexponential distribution, and ProductDistribution can be used to compute a joint distribution with independent component distributions involving hyperexponential distributions.
- The hyperexponential distribution is related to a number of other distributions. HyperexponentialDistribution is an obvious generalization of ExponentialDistribution in that the PDF of HyperexponentialDistribution[{α
_{1},…,α_{m}},{λ_{1},…,λ_{m}}] is precisely that of MixtureDistribution[{α_{1},α_{2},…,α_{m}},{ExponentialDistribution[λ_{1}],ExponentialDistribution[λ_{2}],…,ExponentialDistribution[λ_{m}]}] and in particular, an exponential distribution ExponentialDistribution[λ_{1}] can be viewed both as a single-phase hyperexponential HyperexponentialDistribution[{1},{λ_{1}}] and as a hyperexponential HyperexponentialDistribution[{α_{1},α_{2},…,α_{m}},{λ_{1},λ_{1},…,λ_{1}}] in which the phase rates are all equal. In addition, HyperexponentialDistribution can be transformed into HypoexponentialDistribution (and vice versa); can be obtained from GammaDistribution, LaplaceDistribution, BenktanderWeibullDistribution, LogisticDistribution, ParetoDistribution, PearsonDistribution, PowerDistribution, and RayleighDistribution by composing transformations of ExponentialDistribution with TransformedDistribution and/or TruncatedDistribution; and is related to CoxianDistribution, ExtremeValueDistribution, GumbelDistribution, FrechetDistribution, and WeibullDistribution, among others.

Introduced in 2012

(9.0)

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