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Numerical Methods for Structured Markov Chains$
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Dario A. Bini, Guy Latouche, and Beatrice Meini

Print publication date: 2005

Print ISBN-13: 9780198527688

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198527688.001.0001

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LOGARITHMIC REDUCTION AND CYCLIC REDUCTION

LOGARITHMIC REDUCTION AND CYCLIC REDUCTION

Chapter:
(p.179) 7 LOGARITHMIC REDUCTION AND CYCLIC REDUCTION
Source:
Numerical Methods for Structured Markov Chains
Author(s):

D. A. Bini

G. Latouche

B. Meini

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198527688.003.0007

This chapter is concerned with quadratically convergent algorithms for solving nonlinear matrix equations encountered in M/G/1, G/M/1, and QBD processes. The algorithm of logarithmic reduction for solving QBDs is described and analysed, and an algebraic proof of its convergence is provided. Then the cyclic reduction technique for QBDs is explained and related to logarithmic reduction. Cyclic reduction is extended to M/G/1 and G/M/1 processes, convergence and applicability results are proved. Computational issues concerning the implementation of cyclic reduction are treated by relying on the tools introduced in Chapters 2 and 3.

Keywords:   nonlinear matrix equations, M/G/1 Markov chains, G/M/1 Markov chains, QBD processes

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