March 2012 Rational processes related to communicating Markov processes
Peter Buchholz, Miklós Telek
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J. Appl. Probab. 49(1): 40-59 (March 2012). DOI: 10.1239/jap/1331216833


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.


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Peter Buchholz. Miklós Telek. "Rational processes related to communicating Markov processes." J. Appl. Probab. 49 (1) 40 - 59, March 2012.


Published: March 2012
First available in Project Euclid: 8 March 2012

zbMATH: 1237.60028
MathSciNet: MR2952881
Digital Object Identifier: 10.1239/jap/1331216833

Primary: 60G20
Secondary: 60J27

Keywords: equivalence , lumpability , Markov process , rational process

Rights: Copyright © 2012 Applied Probability Trust


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Vol.49 • No. 1 • March 2012
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