Journal of Applied Probability

Rational processes related to communicating Markov processes

Peter Buchholz and Miklós Telek

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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.

Article information

J. Appl. Probab., Volume 49, Number 1 (2012), 40-59.

First available in Project Euclid: 8 March 2012

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G20: Generalized stochastic processes
Secondary: 60J27: Continuous-time Markov processes on discrete state spaces

Markov process rational process equivalence lumpability


Buchholz, Peter; Telek, Miklós. Rational processes related to communicating Markov processes. J. Appl. Probab. 49 (2012), no. 1, 40--59. doi:10.1239/jap/1331216833.

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