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Moment matching algorithm by Bobbio, Horváth, and Telek
In this section, we summarize the recent results by Bobbio et al.
[22] on characterization of PH distributions and moment
matching algorithms, which build upon our results in
Chapter 2. Recall that
denotes the set
of distributions that are well represented by an phase acyclic PH
distribution (Definition 5). Bobbio et al. provide
exact conditions for a distribution to be in set
(see Theorem 21)
as well as exact conditions for
(see Theorem 20), where
is defined as follows:
Definition 20
Let
denote the set of distributions that are
wellrepresented by an phase acyclic PH distribution
with no mass probability at zero for positive
integer .
Further, Bobbio et al. provide a closed form solution for mapping
any
to a minimalphase acyclic PH distribution without
mass probability at zero^{B.1} (see Theorem 22).
As we define the EC distribution and map an input distribution
to an EC distribution in Chapter 2, Bobbio et al. also define a subset
of PH distributions, and map an input distribution to
a PH distribution in the subset.
Specifically, Bobbio et al. map an input distribution
to an ErlangExp distribution (see Figure B.1) or an ExpErlang distribution
(see Figure B.2).
Figure B.1:
The Markov chain whose absorption time defines an ErlangExp distribution.

Figure B.2:
The Markov chain whose absorption time defines an ExpErlang distribution.

Theorem 20
[22]
A distribution is in set
iff its normalized moments
and satisfy the following conditions:
and
where and are defined as follows:
where
Theorem 21
[22]
A distribution is in set
iff its normalized moments
and satisfy the following conditions:
and
where is the same as in Theorem 20.
Next: Properties of Markovian arrival
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Previous: Proof of Theorem 11
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Takayuki Osogami
20050719