Abstract: We shall consider the problem of identifying the b-invariant measures and hence the quasistationary distributions of an absorbing level-dependent quasi-birth-and-death process (QBD), that is, an absorbing Markov chain with a block-tridiagonal transition matrix P. We examine successive lower truncations of P, obtained by removing rows and columns corresponding to levels.
The crucial factors in our technique are the Perron-Frobenius
eigenvalue of a fundamental matrix and the
sequence
of convergence norms of the successive
lower truncations:
is the convergence norm of the transient
class. We construct a
-invariant measure for all
. When
, we show that a QBD admits one of
two types of
-invariant measure: which type depends on
whether
or
. Together with a knowledge of
whether
or
, this is
sufficient to give the
-classification of the
process.
Keywords:
Quasi-birth-and-death process;
-invariant measures.
Quasi-birth-and-death process.
Acknowledgement: This worked was funded by the Australian Research Council.
The authors:
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Last modified: 28th January 1997