** Next:** Computing various performance measures
** Up:** Moments of inter-level passage
** Previous:** Extension to nonrepeating part
** Contents**

##

Generalization

So far, we have derived the first passage time from level to level
. In this section, we extend this to the first passage time
from level to level for
.
Let
be the time to go from state
to state , and let
be the event
that state is the first state reached in level
when starting in state , for
. Observe
that
and
.
Then, our goal is to
derive the
matrix,
, where
is
the -th moment of
given event
,
for each , , and
.
Notice that
.

Observe that

Hence, it suffices to derive two quantities:

for
.
Matrices
and
can be derived recursively from

via

Finally, we mention some other generalizations that Neuts' algorithm
allows. (i) We restricted ourselves to the first three moments, but
this approach can be generalized to any higher moments. (ii) We
restricted to QBD processes, but this can be generalized to M/G/1 type
semi-Markov processes. (iii) We restricted ourselves to the moments of
the distribution of the duration of busy periods, but this can be
generalized to the moments of the joint distribution of the duration
of a busy period and the number of transitions during the busy period.

** Next:** Computing various performance measures
** Up:** Moments of inter-level passage
** Previous:** Extension to nonrepeating part
** Contents**
Takayuki Osogami
2005-07-19