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Rational Processes Related to Communicating Markov Processes

Part of: Stochastic processes Markov processes

Published online by Cambridge University Press:  04 February 2016

Peter Buchholz  and
Miklós Telek
Peter Buchholz*
Affiliation:
TU Dortmund
Miklós Telek*
Affiliation:
Technical University of Budapest
*
∗ Postal address: Informatik IV, TU Dortmund, D-44221 Dortmund, Germany. Email address: peter.buchholz@udo.edu
∗∗ Postal address: Department of Telecommunications, Technical University of Budapest, H-1521 Budapest, Hungary. Email address: telek@hit.hme.hu
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Abstract

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We define a class of stochastic processes, denoted as marked rational arrival processes (MRAPs), which is an extension of matrix exponential distributions and rational arrival processes. Continuous-time Markov processes with labeled transitions are a subclass of this more general model class. New equivalence relations between processes are defined, and it is shown that these equivalence relations are natural extensions of strong and weak lumpability and the corresponding bisimulation relations that have been defined for Markov processes. If a general rational process is equivalent to a Markov process, it can be used in numerical analysis techniques instead of the Markov process. This observation allows one to apply MRAPs like Markov processes and since the new equivalence relations are more general than lumpability and bisimulation, it is sometimes possible to find smaller representations of given processes. Finally, we show that the equivalence is preserved by the composition of MRAPs and can therefore be exploited in compositional modeling.

Keywords

Markov processrational processequivalencelumpability

MSC classification

Primary: 60G20: Generalized stochastic processes
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces
Type
Research Article
Information
Journal of Applied Probability , Volume 49 , Issue 1 , March 2012 , pp. 40 - 59
Copyright
© Applied Probability Trust 

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