Wikipedia / Hypoexponential distribution
Wikipedia / Hypoexponential distribution
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 hypoexponential 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. The hypoexponential is a series of k exponential distribution. If we have k, then the random variable,
.
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 until state k to the absorbing state k+1. This can be written in the form of a subgenerator matrix,
. If the probability of starting in each of the k states is
Two parameter case
) the explicit forms of the probability functions and the associated statistics are:
The coefficient of variation is always less than 1.
can be estimated as follows:
.
.
Characterization
has cumulative distribution function given by,
and density function,
is a column vector of ones of the size k is the matrix exponential of A, the density function can be written as
are the Lagrange basis polynomials.
The distribution has Laplace transform of
Which can be used to find moments,
General case
is given by
with
.
Uses
This distribution has been used in population genetics, cell biology, and queuing theory.
See also
References
Further reading
M. F. Neuts. (1981) Matrix-Geometric Solutions in Stochastic Models: an Algorithmic 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), 731757.
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