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#

Positive closed form solution

The `Positive` solution is built upon the `Complete`
solution, but does not have mass probability at zero. The key idea
in the design of the `Positive` solution is to match the input
distribution either by a mixture of an EC distribution (with no mass
probability at zero) and an exponential distribution,
or by the convolution of an EC distribution (with positive
mass probability at zero) and an exponential distribution. The use of these
types of distributions makes intuitive sense, since they can
approximate the EC distribution with mass probability at zero
arbitrarily closely by letting the rate of the exponential
distributions approach infinity.
Therefore, in this section, we extend the definition of the EC
distribution and use the extended EC distribution to well-represent
the input distribution.

**Definition 14**
*An extended EC distribution has
a distribution function either of the form
or of the form
,
where is an EC distribution with no mass probability at zero;
and are exponential distributions.*
See Figure 2.3
for the Markov chain whose absorption time defines
an extended EC distribution.
Note that the parameter in an extended EC distribution denotes the
number of phases in the EC portion of the extended EC
distribution.
Therefore, the total number of phases in the extended EC distribution is .

**Subsections**

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Takayuki Osogami
2005-07-19