 
Summary: NUMERICAL SOLUTION of MARKOV CHAINS, p. 1{15
The rst Laurent series coeÆcients for singularly perturbed
stochastic matrices
Konstantin E. Avrachenkov 1 , Moshe Haviv 2
1 INRIA Sophia Antipolis, 2004 Route des Lucioles, B.P.93, 06902, France, email:
k.avrachenkov@sophia.inria.fr
2 Department of Statistics, The Hebrew University, 91905, Jerusalem, Israel, email: haviv@mscc.huji.ac.il
key words: Markov chains, mean rst passage times, deviation matrix, singular perturbations,
aggregation/disaggregation, Laurent series
ABSTRACT
There are a few procedures for computing the Laurent series expansions for the mean passage time
matrix and for the deviation matrix of a singularly perturbed Markov chain. We suggest here a method
for computing the rst terms in these expansions in a way which highlights the system dynamics in
various time scales.
1. Introduction
There are a few procedures for computing the Laurent series expansions for the mean passage
time matrix and for the deviation matrix of a singularly perturbed Markov chain. We suggest
here a method for computing the rst terms in these expansions in a way that highlights the
system dynamics in various time scales.
One usually refers to singular perturbation in Markov chains as the case in which a slight
