HyperexponentialDistribution

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`HyperexponentialDistribution`

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HyperexponentialDistribution[{α₁,…,α_m},{λ₁,…,λ_m}]

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

Details

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 λ_i and is picked with probability α_i.
The probability density for value in an exponential distribution is for , and is zero for .
HyperexponentialDistribution allows α_i to be any non-negative numbers such that α₁+⋯+α_m1 and λ_i can be any positive real numbers.
HyperexponentialDistribution allows λ_i to be any quantities of the same unit dimensions and α_i to be dimensionless quantities. »
HyperexponentialDistribution can be used with such functions as Mean, CDF, and RandomVariate.

Background & Context

HyperexponentialDistribution[{α₁,…,α_m},{λ₁,…,λ_m}] represents a continuous statistical distribution defined over the interval , parameterized by two vectors (α₁,…,α_m) and (λ₁,…,λ_m), and known as an -phase hyperexponential distribution. The parameters α_i are called "phase probabilities", have values in the interval , and satisfy , while the parameters λ_i are called "phase rates" and have positive real values. Together these parameters determine the overall shape of the probability density function (PDF), which in general is monotonic decreasing and has tails that are "thin" in the sense that the PDF decreases exponentially rather than algebraically for large values of . (This behavior can be made quantitatively precise by analyzing the SurvivalFunction of the distribution.) Random variables satisfying XHyperexponentialDistribution[{α₁,…,α_m},{λ₁,…,λ_m}] 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 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 that 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 of 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[{α₁,…,α_m},{λ₁,…,λ_m}]], written more concisely as xHyperexponentialDistribution[{α₁,…,α_m},{λ₁,…,λ_m}], 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 density and cumulative distribution functions may be given using PDF[HyperexponentialDistribution[{α₁,…,α_m},{λ₁,…,λ_m}],x] and CDF[HyperexponentialDistribution[{α₁,…,α_m},{λ₁,…,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 an exponential distribution ExponentialDistribution[λ₁] can be viewed both as a single-phase hyperexponential HyperexponentialDistribution[{1},{λ₁}] and as a hyperexponential HyperexponentialDistribution[{α₁,α₂,…,α_m},{λ₁,λ₁,…,λ₁}] in which the phase rates λ₁ are all equal. HyperexponentialDistribution can also be realized as a mixture distribution whose weights are each ExponentialDistribution. 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.

Examples

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Basic Examples (4)Summary of the most common use cases

Probability density function:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-pjhn1b

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-0pwyru

Out[2]=2

Cumulative distribution function:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-5uc3m0

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-9f1y07

Out[2]=2

Mean:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-3zjfmj

Out[1]=1

Variance:

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-m5wu23

Out[2]=2

Median can be found numerically:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-mhezfd

Out[1]=1

Scope (8)Survey of the scope of standard use cases

Generate a sample of pseudorandom numbers from a hyperexponential distribution:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-tscvhm

Compare the histogram to the PDF:

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-fw0ala

Out[2]=2

Distribution parameters estimation:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-45b7g2

Estimate the distribution parameters from sample data:

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-epi747

Out[2]=2

Compare a density histogram of the sample with the PDF of the estimated distribution:

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-f8ui5o

Out[3]=3

Skewness:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-eiql9h

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-fmqn15

Out[2]=2

The limiting values:

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-et38nc

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-svsi0u

Out[4]=4

When both rates go to simultaneously:

In[5]:=5

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-1q97xl

Out[5]=5

When both rates go to 0 simultaneously:

In[6]:=6

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-dz8vtx

Out[6]=6

Kurtosis:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-kvrrjx

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-cwxfx5

Out[2]=2

The limiting values:

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-tj81vf

Out[3]=3

In[4]:=4

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-q38y6b

Out[4]=4

When both rates go to simultaneously:

In[5]:=5

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-t1rs8n

Out[5]=5

When both rates go to 0 simultaneously:

In[6]:=6

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-gu6max

Out[6]=6

Different moments with closed forms as functions of parameters:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-js043h

Moment:

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-rx074o

Out[2]=2

Closed form for symbolic order:

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-ga3jkf

Out[3]=3

CentralMoment:

In[4]:=4

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-pknsqa

Out[4]=4

Closed form for symbolic order:

In[5]:=5

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-xz2sdr

Out[5]=5

FactorialMoment:

In[6]:=6

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-zg9ct4

Out[6]=6

Cumulant:

In[7]:=7

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-9gzmth

Out[7]=7

Hazard function:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-0gsst0

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-1ri4li

Out[2]=2

Quantile function:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-txywq

Out[1]=1

Consistent use of Quantity in parameters yields QuantityDistribution:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-din0xr

Out[1]=1

Find median time:

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-ci2f4

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-i2dlw8

Out[3]=3

Applications (1)Sample problems that can be solved with this function

Suppose a customer is buying an appliance and is choosing at random between an appliance with an average lifetime of 10 years and an appliance with an average lifetime of 12 years. Assuming the lifetime of this appliance follows an exponential distribution, find the lifetime distribution of the purchased appliance:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-uiofjv

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-7ogeyf

Out[2]=2

Find the average lifetime:

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-bfjzi

Out[3]=3

Find the probability that the appliance will work for more than 15 years:

In[4]:=4

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-44qdg7

Out[4]=4

Properties & Relations (6)Properties of the function, and connections to other functions

The variation coefficient of the hyperexponential distribution is always greater than the variation coefficient of ExponentialDistribution:

In[1]:=1

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-bwqtwt

In[2]:=2

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-grqt1w

Out[2]=2

In[3]:=3

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https://wolfram.com/xid/0bhgxm2pkjs633era8uy-nj2b1h

Out[3]=3

There are no valid parameters such that the variation coefficient of the hyperexponential distribution is less than or equal to the variation coefficient of an exponential distribution:

In[4]:=4