Hostname: page-component-89b8bd64d-72crv Total loading time: 0 Render date: 2026-05-07T00:12:10.206Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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.

Information

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Abate, J. & Whitt, W. (1988). Simple spectral representations for the M/M/I queue. Queueing Systems 3: 321–346.CrossRefGoogle Scholar
Brown, M. & Shao, Y. (1987). Identifying coefficients in the spectral representation for first passage time distributions. Probability in the Engineering and Information Sciences 1: 69–74.CrossRefGoogle Scholar
Feller, W. (1968). An introduction to probability theory and its applications, Third Edition. New York: Wiley.Google Scholar
Karlin, S. & McGregor, J.L. (1957). The differential equations of birth-and-death processes, and the Stieltjes moment problem. Transactions of the American Mathematical Society 85: 489–546.CrossRefGoogle Scholar
Karlin, S. & McGregor, J.L. (1959). A characterization of birth-and-death processes. Proceedings of the National Academy of Sciences (U.S.A.) 45: 375–379.CrossRefGoogle ScholarPubMed
Keilson, J. (1979). Markov chain models–rarity and exponentiality. New York: Springer-Verlag.CrossRefGoogle Scholar
Kendall, D.G. (1959). Unitary dilations of one-parameter semigroups of Markov transition operators, and the corresponding integral representations for Markov processes with a countable infinity of states. Proceedings of the London Mathematical Society 3: 417–431.CrossRefGoogle Scholar
Ledermann, W. & Reuter, G.E.H. (1954). Spectral theory for the differential equations of simple birth-and-death processes. Philosophical Transactions of the Royal Society of London, Series A 246: 321–369.Google Scholar
van Doom, E. (1980). Stochastic monotonicity and queueing applications of birth-death processes. New York: Springer-Verlag.Google Scholar