 
Summary: Applied Probability Trust (August 7, 1997)
THE FUNDAMENTAL MATRIX OF SINGULARLY PERTURBED MARKOV CHAINS
KONSTANTIN E. AVRACHENKOV, \Lambda University of South Australia
JEAN B. LASSERRE, \Lambda\Lambda LAASCNRS
Abstract
We consider a singularly perturbed (finite state) Markov chain and provide a
complete characterization of the fundamental matrix. In particular, we obtain a
formula for the regular part simpler than a previous Schweitzer's formula, and
the singular part is obtained via a reduction process similar to Delebecque's
reduction for the stationary distribution. In contrast to previous approaches,
one works with aggregate Markov chains of much smaller dimension than the
original chain, an essential feature for practical computation. An application to
mean firstpassage times is also presented.
MARKOV CHAINS; FUNDAMENTAL MATRIX; SINGULAR PERTURBATION
AMS 1991 SUBJECT CLASSIFICATION: PRIMARY 60J10;47A55
SECONDARY 60J35;15A51
1. Introduction
Singularly perturbed Markov chains have been studied in the pioneering works of Simon and
Ando [31], Schweitzer [27], Courtois [6], Pervozvanskii et al [10, 22, 23], Korolyuk and Turbin [17],
Delebecque and Quadrat [7, 8], Phillips and Kokotovic [24] and later by Coderch et al [5], Rohlicek
