# Caos y orden by Antonio Escohotado

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As a second example, we consider the hyperexponential distribution for all ), which is the convex mixture of n exponential distributions. 2. 2: Phase-type representation of the hyperexponential distribution. 3. PH Random Variables 39 We are now ready to consider the general framework of PH distributions along the lines of Neuts [74, 79]. 3 PH Random Variables Since we only wish to rely on the connection to a Markov process, there is no need to restrict the transitions as in the examples of the previous section, and we may as well consider the general case.

The process {J(£i), J f a ] , . • } is a discrete time Markov chain with one absorbing state, and N is the number of transitions before absorption. Thus, N has a discrete PH distribution. To determine the matrix S of transition probabilities between transient states, we condition on the length U of the interval between two Poisson epochs and find that Since J(0) is chosen with the distribution r, the distribution at the first Poisson epoch is (1 — rSl, rS). 6 Closure Properties We have stated earlier that it is advantageous to use PH distributions because there is a connection with Markov processes.

Erlang and generalized by M. F. Neuts. The key idea is to model random time intervals as being made up of a (possibly random) number of exponentially distributed segments and to exploit the resulting Markovian structure to simplify the analysis. We start with a chapter on PH distributions for several reasons. First and foremost, they provide us with the simplest introduction to matrix analytic methods: the distribution of a random variable is defined through a matrix; its density function, moments, etc.