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

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 threephase 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 nonabsorbing state is unique (i.e., the initial state is either the unique nonabsorbing
state or the absorbing state), and
(ii) for each state, the next nonabsorbing state is unique (i.e., the next state is the unique nonabsorbing state or the absorbing state).
A threephase 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 nonabsorbing 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 threephase 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 Erlang2 distribution and an Erlang3 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
20050719