# HypoexponentialDistribution

HypoexponentialDistribution[{λ1,,λm}]

represents an m-phase hypoexponential distribution with rates λ1, , λm.

# Details • HypoexponentialDistribution is also known as a sequential m-phase exponential distribution.
• An m-phase hypoexponential distribution can be interpreted as having m servers in series where the i server has service rate λi.
• The probability density for value and distinct rates is a linear combination of exponentials for and zero for .
• HypoexponentialDistribution[{λ1,,λm}] is equivalent to TransformedDistribution[x1++xm,] with each xi from .
• HypoexponentialDistribution allows λi to be any positive real numbers.
• HypoexponentialDistribution allows λi to be any quantities of the same unit dimensions. »
• HypoexponentialDistribution can be used with such functions as Mean, CDF, and RandomVariate.

# Examples

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## Basic Examples(4)

Probability density function:

Cumulative distribution function:

Mean and variance:

Median can be found exactly in some cases:

But the general closed form does not exist:

## Scope(8)

Generate a sample of pseudorandom numbers from a hypoexponential distribution:

Compare the histogram to the PDF:

Distribution parameters estimation:

Estimate the distribution parameters from sample data:

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

Skewness:

The limiting values:

When both parameters go to simultaneously:

Kurtosis:

The limiting values:

When both parameters go to simultaneously:

Different moments with closed forms as functions of parameters:

Closed form for symbolic order:

Hazard function:

Quantile function:

Consistent use of Quantity in parameters yields QuantityDistribution:

Convert to days:

Find the median service time:

## Applications(1)

A process is composed of three independent consecutive steps, each with mean time exponentially distributed with parameters 0.003 hr. , .002 hr. , and 0.01 hr. respectively. Find the probability that the process takes less than 500 hours:

Find the mean duration of the process:

Find the median duration of the process:

Simulate the duration of the process in 50 consecutive runs:

## Properties & Relations(11)

The variation coefficient of hypoexponential distribution is always less than the variation coefficient of ExponentialDistribution:

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

Theoretically there is no limit on the length of the vector of s:

Hypoexponential distribution is invariant under any permutation of the rates vector:

HypoexponentialDistribution is closed under scaling by a positive factor:

Relationships to other distributions:

Sum of independent variables following ExponentialDistribution has hypoexponential distribution:

Hypoexponential distribution with single rate reduces to ExponentialDistribution:

HypoexponentialDistribution is a special case of CoxianDistribution:

Hypoexponential distribution with all rates equal is ErlangDistribution:

Hypoexponential distribution for all rates equal is GammaDistribution:

## Neat Examples(1)

PDFs for different μ values with CDF contours:

Introduced in 2012
(9.0)
|
Updated in 2016
(10.4)