Rational Generating Functions and Matrix-Geometric Invariant Measures

Abstract

The paper studies the connection between rational generating matrix-valued functions and not necessarily nonnegative matrix-geometric invariant measures for M/G/1 and G/M/1 type Markov chains. It characterizes the situation of the existence, and shows that in the transient QBD case not every positive invariant measure is matrix-geometric. A method for the determination of all invariant measures in the QBD case is presented. The questions of summability and generalized notions of matrix-geometricity in the spirit of system theory are studied. Main conclusion: invariant measures are not always matrix-geometric, but are very often generalized matrix-geometric.

Keywords:

• Block Toeplitz matrix
• Generalized matrix-geometric invariant measures
• McMillan degree
• M/G/1 and G/M/1 type Markov chains
• QBD process
• Rational generating functions
• Realization

2000 Mathematics Subject Classification:

• Primary 65C40
• Secondary 93A10

ACKNOWLEDGMENT

The authors are much indebted to the referees and editors who contributed by valuable remarks to the improvement of the presentation. The research was partially supported by the Hungarian National Foundation for Scientific Research, Grant No. T047276, by the TU Berlin and by the Budapest UTE.