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Spectral Theory for Skip-Free Markov Chains

Published online by Cambridge University Press:  27 July 2009

Joseph Abate  and
Ward Whitt
Joseph Abate
Affiliation:
AT&T Bell Laboratories Whippany, New Jersey07981
Ward Whitt
Affiliation:
AT&T Bell Laboratories Murray Hill, New Jersey07974
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Abstract

The distribution of upward first passage times in skip-free Markov chains can be expressed solely in terms of the eigenvalues in the spectral representation, without performing a separate calculation to determine the eigenvectors. We provide insight into this result and skip-free Markov chains more generally by showing that part of the spectral theory developed for birth-and-death processes extends to skip-free chains. We show that the eigenvalues and eigenvectors of skip-free chains can be characterized in terms of recursively defined polynomials. Moreover, the Laplace transform of the upward first passage time from 0 to n is the reciprocal of the nth polynomial. This simple relationship holds because the Laplace transforms of the first passage times satisfy the same recursion as the polynomials except for a normalization.

Type
Articles
Information
Probability in the Engineering and Informational Sciences , Volume 3 , Issue 1 , January 1989 , pp. 77 - 88
Copyright
Copyright © Cambridge University Press 1989

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