Hostname: page-component-7b7567c8c-7pwfn Total loading time: 0 Render date: 2026-11-15T04:04:33.744Z Has data issue: false hasContentIssue false
  • English
  • Français

Exponential ergodicity in Markov renewal processes

Published online by Cambridge University Press:  14 July 2016

Jozef L. Teugels
Jozef L. Teugels*
Affiliation:
University of Louvain, Belgium
Get accessRights & Permissions [Opens in a new window]

Extract

In [3], Kendall proved a solidarity theorem for irreducible denumerable discrete time Markov chains. Vere-Jones refined Kendall's theorem by obtaining uniform estimates [14], while Kingman proved analogous results for an irreducible continuous time Markov chain [4], [5].

We derive similar solidarity theorems for an irreducible Markov renewal process. The transient case is discussed in Section 3, and Section 4 deals with the positive recurrent case. Recently Cheong also proved solidarity theorems for Semi-Markov processes [1]. His theorems use the Markovian structure, while our emphasis is on the renewal aspects of Markov renewal processes.

An application to the M/G/1 queue is included in the last section.

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 5 , Issue 2 , August 1968, pp. 387 - 400
Copyright
Copyright © Applied Probability Trust 1968 

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

[1] Cheong, C. K. (1967) Geometric convergence of semi-Markov transition probabilities. Z. Wahrscheinlichkeitsth. 7, 122–330.CrossRefGoogle Scholar
[2] Karlin, S. and Mcgregor, J. L. (1958) Many server queueing processes with Poisson input and exponential service times. Pacific J. Math. 8, 87–118.CrossRefGoogle Scholar
[3] Kendall, D. G. (1959) Unitary dilations of Markov transition operators and the corresponding integral representation for transition-probability matrices. Probability and Statistics Editor: Grenander, U. Almqvist & Wiksell, Stockholm.Google Scholar
[4] Kingman, J. F. C. (1963) The exponential decay of Markov transition probabilities. Proc. Lond. Math. Soc. 13, 337–358.CrossRefGoogle Scholar
[5] Kingman, J. F. C. (1963) Ergodic properties of continuous-time Markov processes and their discrete skeletons. Proc. Lond. Math. Soc. 13, 593–604.CrossRefGoogle Scholar
[6] Neuts, M. F. (1966) The queue with Poisson input and general service times, treated as a branching process. Purdue Univ. Dept. of Stat. Mimeo Series 92.Google Scholar
[7] Neuts, M. F. and Teugels, J. L. (1967) Exponential ergodicity of the M/G/1 queue. Purdue Univ. Dept. of Stat. Mimeo Series 126.Google Scholar
[8] Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.CrossRefGoogle Scholar
[9] Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.CrossRefGoogle Scholar
[10] Pyke, R. and Schaufele, R. (1963) Limit theorems for Markov renewal processes. Ann. Math. Statist. 35, 1746–1764.CrossRefGoogle Scholar
[11] Tákacs, L. (1962) Introduction to the Theory of Queues. Oxford University Press, New York.Google Scholar
[12] Teugels, J. L. (1967) On the rate of convergence in renewal and Markov renewal processes. Purdue Univ. Dept. of Stat. Mimeo Series 107.Google Scholar
[13] Teugels, J. L. (1967) Exponential decay in renewal theorems. Bull. Soc. Math. Belg. 19, 259–276.Google Scholar
[14] Vere-Jones, D. (1962) Geometric ergodicity in denumerable Markov chains. Quart. J. Math. 13, 7–28.CrossRefGoogle Scholar
[15] Widder, D. (1947) The Laplace Transform. Princeton University Press, Princeton.Google ScholarPubMed