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Poisson functionals of Markov processes and queueing networks

Published online by Cambridge University Press:  01 July 2016

Richard F. Serfozo
Richard F. Serfozo*
Affiliation:
Georgia Institute of Technology
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Abstract

We present conditions under which a point process of certain jump times of a Markov process is a Poisson process. The central idea is that if the Markov process is stationary and the compensator of the point process in reverse time has a constant intensity a, then the point process is Poisson with rate a. A known example is that the output flow from an M/M/1 queueing system is Poisson. We present similar Poisson characterizations of more general marked point process functionals of a Markov process. These results yield easy-to-use criteria for a collection of such processes to be multivariate Poisson, compound Poisson, or marked Poisson with a specified dependence or independence. We discuss several applications for queueing systems with batch arrivals and services and for networks of queues. We also indicate how our results extend to functionals of non-Markovian processes.

Keywords

POISSON PROCESSMULTIVARIATE COMPOUND POISSON PROCESSTIME REVERSAL
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 595 - 611
Copyright
Copyright © Applied Probability Trust 1989 

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Footnotes

This research was sponsored in part by Air Force Office of Scientific Research contracts 84–0367 and F49620 85 C 0144; it was partly carried out at the University of North Carolina at Chapel Hill.

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