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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 well-represented 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 minimal-phase acyclic PH distribution without mass probability at zeroB.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 Erlang-Exp distribution (see Figure B.1) or an Exp-Erlang distribution (see Figure B.2).

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.

Theorem 22   [22] Let be a distribution in . If or , then is well-represented by the Erlang-Exp distribution with the following parameters:

If and , then is well-represented by the Exp-Erlang distribution with the following parameters:

and is given by the following stepsB.2:

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Takayuki Osogami 2005-07-19