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##

Preliminaries

We first define the transition probability matrix, ,
such that its element,
, represents the
probability that (i) the sojourn time at state is AND
(ii) the first transition out of state is to state .
(Note that state denotes a state (), where
.
State is defined analogously.)
By the following Lemma, which follows from the memoryless property
of the exponential distribution,
is also the product of the
probabilities of events (i) and (ii).

**Lemma 7**
*Let and be independent exponential random variables.
Let
. Then
*
*
*
Matrix has the same structural shape as , although its
entries are different, and thus it can be represented in terms
of submatrices, analogous to those in , indicating backward transitions,
local transitions, and forward
transitions, as follows:

Specifically,
the element of
is the probability that the sojourn time in state is
AND the first transition out of state is to state .
Likewise,
the element of
(respectively,
)
is the probability that the sojourn time in state is
AND the first transition out of state is to state
(respectively, to state ).
Next, we define the -th moment of submatrices,
,
,
, as follows:

for , and , where an integral of a matrix is a
matrix of the integrals of the elements in .
For the repeating part, we define
,
, and
,
omitting the superscript.
We now define the limits as
of
,
, and
as follows:

for .
For the repeating part, we define
,
, and
,
omitting the superscript.

**Subsections**

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Takayuki Osogami
2005-07-19