Parameters  rates (real)  

Support  
Expressed as a phasetype distribution Has no other simple form; see article for details  
CDF 
Expressed as a phasetype distribution  
Mean  
Median  General closed form does not exist^{[1]}  
Mode  if , for all k  
Variance  
Skewness  
Ex. kurtosis  no simple closed form  
MGF  
CF 
In probability theory the hypoexponential distribution or the generalized Erlang distribution is a continuous distribution, that has found use in the same fields as the Erlang distribution, such as queueing theory, teletraffic engineering and more generally in stochastic processes. It is called the hypoexponetial distribution as it has a coefficient of variation less than one, compared to the hyperexponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.
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Transcription
Contents
Overview
The Erlang distribution is a series of k exponential distributions all with rate . The hypoexponential is a series of k exponential distributions each with their own rate , the rate of the exponential distribution. If we have k independently distributed exponential random variables , then the random variable,
is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of .
Relation to the phasetype distribution
As a result of the definition it is easier to consider this distribution as a special case of the phasetype distribution. The phasetype distribution is the time to absorption of a finite state Markov process. If we have a k+1 state process, where the first k states are transient and the state k+1 is an absorbing state, then the distribution of time from the start of the process until the absorbing state is reached is phasetype distributed. This becomes the hypoexponential if we start in the first 1 and move skipfree from state i to i+1 with rate until state k transitions with rate to the absorbing state k+1. This can be written in the form of a subgenerator matrix,
For simplicity denote the above matrix . If the probability of starting in each of the k states is
then
Two parameter case
Where the distribution has two parameters () the explicit forms of the probability functions and the associated statistics are^{[2]}
CDF:
PDF:
Mean:
Variance:
Coefficient of variation:
The coefficient of variation is always < 1.
Given the sample mean () and sample coefficient of variation (), the parameters and can be estimated as follows:
The resulting parameters and are real values if .
Characterization
A random variable has cumulative distribution function given by,
and density function,
where is a column vector of ones of the size k and is the matrix exponential of A. When for all , the density function can be written as
where are the Lagrange basis polynomials associated with the points .
The distribution has Laplace transform of
Which can be used to find moments,
General case
In the general case where there are distinct sums of exponential distributions with rates and a number of terms in each sum equals to respectively. The cumulative distribution function for is given by
with
with the additional convention .
Uses
This distribution has been used in population genetics^{[3]} cell biology ^{[4]}^{[5]} and queuing theory^{[6]}^{[7]}
See also
References
 ^ https://reference.wolfram.com/language/ref/HypoexponentialDistribution.html. Missing or empty
title=
(help)  ^ Bolch, Gunter; Greiner, Stefan; de Meer, Hermann; Trivedi, Kishor Shridharbhai (2006). "Chapter 1. Introduction". Queueing Networks and Markov Chains: Modeling and Performance Evaluation with Computer Science Applications (2nd ed.). WileyBlackwell. doi:10.1002/0471200581.ch1. ISBN 9780471565253.
 ^ Strimmer K, Pybus OG (2001) "Exploring the demographic history of DNA sequences using the generalized skyline plot", Mol Biol Evol 18(12):2298305
 ^ Yates, Christian A. (21 April 2017). "A Multistage Representation of Cell Proliferation as a Markov Process". Bulletin of Mathematical Biology. 79 (1). doi:10.1007/s1153801703564.
 ^ Gavagnin, Enrico (14 October 2018). "The invasion speed of cell migration models with realistic cell cycle time distributions". Journal of Theoretical Biology. 79 (1). arXiv:1806.03140. doi:10.1016/j.jtbi.2018.09.010.
 ^ http://www.few.vu.nl/en/Images/stageverslagcalinescu_tcm39105827.pdf
 ^ Bekker R, Koeleman PM (2011) "Scheduling admissions and reducing variability in bed demand". Health Care Manag Sci, 14(3):237249
Further reading
 M. F. Neuts. (1981) MatrixGeometric Solutions in Stochastic Models: an Algorthmic Approach, Chapter 2: Probability Distributions of Phase Type; Dover Publications Inc.
 G. Latouche, V. Ramaswami. (1999) Introduction to Matrix Analytic Methods in Stochastic Modelling, 1st edition. Chapter 2: PH Distributions; ASA SIAM,
 Colm A. O'Cinneide (1999). Phasetype distribution: open problems and a few properties, Communication in Statistic  Stochastic Models, 15(4), 731–757.
 L. Leemis and J. McQueston (2008). Univariate distribution relationships, The American Statistician, 62(1), 45—53.
 S. Ross. (2007) Introduction to Probability Models, 9th edition, New York: Academic Press
 S.V. Amari and R.B. Misra (1997) Closedform expressions for distribution of sum of exponential random variables,IEEE Trans. Reliab. 46, 519–522
 B. Legros and O. Jouini (2015) A linear algebraic approach for the computation of sums of Erlang random variables, Applied Mathematical Modelling, 39(16), 4971–4977