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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 hyper-exponential distribution which has coefficient of variation greater than one and the exponential distribution which has coefficient of variation of one.

Overview

The Erlang distribution is a series of k exponential distributions all with rate λ. The hypoexponential is a series of k exponential distributions each with their own rate λi, the rate of the ith exponential distribution. If we have k independently distributed exponential random variables Xi, then the random variable,

X=i=1kXi

is hypoexponentially distributed. The hypoexponential has a minimum coefficient of variation of 1/k.

Relation to the phase-type distribution

As a result of the definition it is easier to consider this distribution as a special case of the phase-type distribution. The phase-type 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 phase-type distributed. This becomes the hypoexponential if we start in the first 1 and move skip-free from state i to i+1 with rate λi until state k transitions with rate λk to the absorbing state k+1. This can be written in the form of a subgenerator matrix,

[λ1λ10000λ2λ20000λk2λk20000λk1λk10000λk].

For simplicity denote the above matrix ΘΘ(λ1,,λk). If the probability of starting in each of the k states is

α=(1,0,,0)

then Hypo(λ1,,λk)=PH(α,Θ).

Two parameter case

Where the distribution has two parameters (μ1μ2) the explicit forms of the probability functions and the associated statistics are[1]

CDF: F(x)=1μ2μ2μ1eμ1x+μ1μ2μ1eμ2x

PDF: f(x)=μ1μ2μ1μ2(exμ2exμ1)

Mean: 1μ1+1μ2

Variance: 1μ12+1μ22

Coefficient of variation: μ1+μ2μ1+μ2

The coefficient of variation is always < 1.

Given the sample mean (x¯) and sample coefficient of variation (c) the parameters μ1 and μ2 can be estimated:

μ1=2x¯[1+1+2(c21)]1

μ2=2x¯[11+2(c21)]1

Characterization

A random variable XHypo(λ1,,λk) has cumulative distribution function given by,

F(x)=1αexΘ1

and density function,

f(x)=αexΘΘ1,

where 1 is a column vector of ones of the size k and eA is the matrix exponential of A. When λiλj for all ij, the density function can be written as

f(x)=i=1kλiexλi(j=1,jikλjλjλi)=i=1ki(0)λiexλi

where 1(x),,k(x) are the Lagrange basis polynomials associated with the points λ1,,λk.

The distribution has Laplace transform of

{f(x)}=α(sIΘ)1Θ1

Which can be used to find moments,

E[Xn]=(1)nn!αΘn1.

General case

In the general case where there are a distinct sums of exponential distributions with rates λ1,λ2,,λa and a number of terms in each sum equals to r1,r2,,ra respectively. The cumulative distribution function for t0 is given by

F(t)=1(j=1aλjrj)k=1al=1rkΨk,l(λk)trklexp(λkt)(rkl)!(l1)!,

with

Ψk,l(x)=l1xl1(j=0,jka(λj+x)rj).

with the additional convention λ0=0,r0=1.

Uses

This distribution has been used in population genetics[2] and queuing theory[3][4]

See also

References

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Additional material

  • M. F. Neuts. (1981) Matrix-Geometric 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). Phase-type 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) Closed-form expressions for distribution of sum of exponential random variables,IEEE Trans. Reliab. 46, 519–522

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  1. Template:Cite doi
  2. Strimmer K, Pybus OG (2001) "Exploring the demographic history of DNA sequences using the generalized skyline plot", Mol Biol Evol 18(12):2298-305
  3. http://www.few.vu.nl/en/Images/stageverslag-calinescu_tcm39-105827.pdf
  4. Bekker R, Koeleman PM (2011) "Scheduling admissions and reducing variability in bed demand". Health Care Manag Sci, 14(3):237-249