Hostname: page-component-8448b6f56d-sxzjt Total loading time: 0 Render date: 2024-04-19T01:50:29.760Z Has data issue: false hasContentIssue false
  • English
  • Français

Some duality results for a class of multivariate semi-markov processes

Published online by Cambridge University Press:  14 July 2016

J. Janssen  and
J. M. Reinhard
J. Janssen*
Affiliation:
Université Libre de Bruxelles
J. M. Reinhard*
Affiliation:
Université Libre de Bruxelles
*
∗Postal address: Departement de Mathématique, Université Libre de Bruxelles, C.P. 210, Campus Plaine, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
∗Postal address: Departement de Mathématique, Université Libre de Bruxelles, C.P. 210, Campus Plaine, Boulevard du Triomphe, B-1050 Bruxelles, Belgium.
Get access Rights & Permissions [Opens in a new window]

Abstract

The duality results well known for classical random walk and generalized by Janssen (1976) for (J-X) processes (or sequences of random variables defined on a finite Markov chain) are extended to a class of multivariate semi-Markov processes. Just as in the classical case, these duality results lead to connections between some models of risk theory and queueing theory.

Keywords

MULTIVARIATE SEMI-MARKOV PROCESSESDUALITYWAITING TIMES FOR QUEUES WITH MANY SERVERSRISK THEORY FOR SEVERAL TYPES OF CLAIMS
Type
Research Paper
Information
Journal of Applied Probability , Volume 19 , Issue 1 , March 1982 , pp. 90 - 98
Copyright
Copyright © Applied Probability Trust 1982 

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

Cheong, C. K. and Teugels, J. L. (1973) On a semi-Markov generalization of the random walk. Stoch. Proc. Appl. 1, 53–66.CrossRefGoogle Scholar
Feller, W. (1971) An Introduction to Probability Theory and its Applications, Vol. II. Wiley, New York.Google Scholar
Janssen, J. (1969) Les processus (J-X) . Cahiers du Centre d'Etudes de Recherche Opérationnelle 11, 181–214.Google Scholar
Janssen, J. (1976) Some duality results in semi-Markov chain theory. Rev. Roumaine Math. Pures Appl. 21, 429–441.Google Scholar
Kolmogorov, A. N. and Fomin, S. V. (1957) Functional Analysis, Vol. 1. Graylock Press, Rochester.Google Scholar
Newbould, M. (1973) A classification of a random walk defined on a finite Markov chain. Z. Wahrscheinlichkeitsth. 26, 95–104.CrossRefGoogle Scholar
Pyke, R. (1961) Markov renewal processes: definitions and preliminary properties. Ann. Math. Statist. 32, 1231–1242.CrossRefGoogle Scholar
Pyke, R. (1961) Markov renewal processes with finitely many states. Ann. Math. Statist. 32, 1243–1259.CrossRefGoogle Scholar
Stone, L. D. (1968) On the distribution of the maximum of a semi-Markov process. Ann. Math. Statist. 39, 947–956.CrossRefGoogle Scholar