Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T18:46:28.279Z Has data issue: false hasContentIssue false
  • English
  • Français

Spectral representations of the transition probability matrices for continuous time finite Markov chains

Part of: Markov processes

Published online by Cambridge University Press:  14 July 2016

Nan Fu Peng
Nan Fu Peng*
Affiliation:
National Chiao Tung University
*
∗Postal address: Institute of Statistics, National Chiao Tung University, 1001 TA Hsueh Road, Hsinchu, Taiwan.
Get access Rights & Permissions [Opens in a new window]

Abstract

Using an easy linear-algebraic method, we obtain spectral representations, without the need for eigenvector determination, of the transition probability matrices for completely general continuous time Markov chains with finite state space. Comparing the proof presented here with that of Brown (1991), who provided a similar result for a special class of finite Markov chains, we observe that ours is more concise.

Keywords

MARKOV CHAINSTRANSITION PROBABILITY MATRICESSPECTRAL REPRESENTATIONS

MSC classification

Primary: 60J35: Transition functions, generators and resolvents
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Papers
Information
Journal of Applied Probability , Volume 33 , Issue 1 , March 1996 , pp. 28 - 33
Copyright
Copyright © Applied Probability Trust 1996 

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.)

References

[1] Abate, J., Kijima, M. and Ward, W. (1991) Decompositions of the M/M/1 transition function. Queueing Systems 9, 323–336.Google Scholar
[2] Brown, M. (1991) Spectral analysis, without eigenvectors, for Markov chains. Prob. Eng. Inf. Sci. 5, 131–144.CrossRefGoogle Scholar
[3] Fill, J. A. (1992) Strong stationary duality for continuous-time Markov chains. Part I: Theory. J. Theor. Prob. 5, 45–70.Google Scholar
[4] Hoffman, K. and Kunze, R. (1971) Linear Algebra. Prentice Hall, New York.Google Scholar
[5] Keilson, J. (1979) Markov Chain Models — Rarity and Exponentiality. Springer, Berlin.Google Scholar
[6] Kijima, M. (1987) Spectral structure of the first-passage-time densities for classes of Markov chains. J. Appl. Prob. 24, 631–643.Google Scholar
[7] Kijima, M. (1992) A note on external uniformization for finite Markov chains in continuous time. Prob. Eng. Inf. Sci. 6, 127–131.Google Scholar
[8] Kijima, M. (1992) The transient solution to a class of Markovian queues. Comput. Math. Appl. 24, 17–24.Google Scholar
[9] Parthasarathy, P. R. and Sharafali, M. (1989) Transient solution to the many server Poisson queue. A simple approach. J. Appl. Prob. 26, 584–594.Google Scholar
[10] Ross, S. M. (1987) Approximating transition probabilities and mean occupation times in continuous-time Markov chains. Prob. Eng. Inf. Sci. 1, 251–264.Google Scholar
[11] Sharma, O. P. and Dass, S. (1988) Multiserver Markovian queues with finite waiting space. Sankhya B50, 428–431.Google Scholar
[12] Yoon, B. S. and Shanthikumar, J. G. (1989) Bounds and approximations for the transient behavior of continuous-time Markov chains. Prob. Eng. Inf. Sci 3, 175–198.Google Scholar