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Marked point processes as limits of Markovian arrival streams

Part of:Special processesStochastic processes

Published online by Cambridge University Press:  14 July 2016

Søren Asmussen and
Ger Koole
Søren Asmussen*
Affiliation:
Aalborg University
Ger Koole*
Affiliation:
University of Leiden
*
∗ Postal address: Institute of Electronic Systems, Aalborg University, Fr. Bajersv. 7, DK-9220 Aalborg, Denmark.
∗∗ Postal address: Department of Mathematics and Computer Science, University of Leiden, P.O. Box 9512, 2300 RA Leiden, The Netherlands.
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Abstract

A Markovian arrival stream is a marked point process generated by the state transitions of a given Markovian environmental process and Poisson arrival rates depending on the environment. It is shown that to a given marked point process there is a sequence of such Markovian arrival streams with the property that as m →∞. Various related corollaries (involving stationarity, convergence of moments and ergodicity) and counterexamples are discussed as well.

Keywords

MARKOVIAN POINT PROCESSMARKOV-MODULATED POISSON PROCESSWEAK CONVERGENCE OF POINT PROCESSESCONTINUITY OF QUEUES

MSC classification

Primary:60G55: Point processes
Secondary:60K25: Queueing theory

Information

Type
Research Papers
Information
Journal of Applied Probability , Volume 30 , Issue 2 , June 1993, pp. 365 - 372
Copyright
Copyright © Applied Probability Trust 1993 

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