Next: Properties of PH distributions Up: Brief tutorial on phase Previous: Definition of PH distribution   Contents

## Subclasses of PH distribution

By restricting the structure of the Markov chain, we can define a subclass of the PH distribution. Below, we define subclasses of the PH distribution that come up in this thesis.

Figure 2.8: Subclasses of the PH distribution.
 (a) acyclic PH
 (b) Coxian PH

 (c) generalized Erlang
 (d) mixed Erlang

First, if the Markov chain whose absorption time defines a PH distribution is acyclic (i.e., any state is never visited more than once in the Markov chain), the PH distribution is called an acyclic PH distribution. A three-phase acyclic PH distribution is illustrated in Figure 2.8(a) as the absorption time in a Markov chain.

An acyclic PH distribution is called a Coxian PH distribution if the Markov chain whose absorption time defines the acyclic PH distribution has the following two properties: (i) the initial non-absorbing state is unique (i.e., the initial state is either the unique non-absorbing state or the absorbing state), and (ii) for each state, the next non-absorbing state is unique (i.e., the next state is the unique non-absorbing state or the absorbing state). A three-phase Coxian PH distribution is illustrated in Figure 2.8(b). When a Coxian PH distribution does not have mass probability at zero (i.e., the initial state is not the absorbing state), we refer to the Coxian PH distribution as a Coxian PH distribution.

An acyclic PH distribution is called a hyperexponential distribution if the Markov chain whose absorption time defines the acyclic PH distribution has the following property: for any state, the next state is the absorbing state. That is, a mixture of exponential distributions is a hyperexponential distribution.

An acyclic PH distribution is called an Erlang distribution if the Markov chain whose absorption time defines the acyclic PH distribution has the following three properties: (i) the initial state is a unique non-absorbing state, (ii) for each state, the next state is unique, and (iii) the sojourn time distribution at each state is identical. That is, the sum of i.i.d. exponential random variables has an -phase Erlang distribution. An -phase Erlang distribution is also called an Erlang- distribution. An Erlang distribution is generalized to a generalized Erlang distribution [122] by allowing a transition from the initial state to the absorbing state. A three-phase generalized Erlang distribution is illustrated in Figure 2.8(c).

A mixture of Erlang distributions is called a mixed Erlang distribution [87,88]. A mixed Erlang distribution is illustrated in Figure 2.8(d), where an Erlang-2 distribution and an Erlang-3 distribution are mixed. A mixture of Erlang distributions with the same number of phases, is called a mixed Erlang distribution with common order [87,88].

Next: Properties of PH distributions Up: Brief tutorial on phase Previous: Definition of PH distribution   Contents
Takayuki Osogami 2005-07-19