Abstract
We study probabilistic safe Petri nets, a probabilistic extension of safe Petri nets interpreted under the true-concurrent semantics. In particular, the likelihood of processes is defined on partial orders, not on firing sequences.
We focus on memoryless probabilistic nets: we give a definition for such systems, that we call Markov nets, and we study their properties. We show that several tools from Markov chains theory can be adapted to this true-concurrent framework. In particular, we introduce stopping operators that generalize stopping times, in a more convenient fashion than other extensions previously proposed. A Strong Markov Property holds in the concurrency framework. We show that the Concurrent Strong Markov property is the key ingredient for studying the dynamics of Markov nets. In particular we introduce some elements of a recurrence theory for nets, through the study of renewal operators. Due to the concurrency properties of Petri nets, Markov nets have global and local renewal operators, whereas both coincide for sequential systems.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Proc. of the 16th Conference on Mathematical Theory of Networks and Systems (MTNS 2004), Leuven, Belgium (2004), ISBN 90-5682-517-8
Abbes, S.: Probabilistic model of distributed and concurrent systems. Limit theorems and Applications. PhD thesis, Université de Rennes 1, IRISA, Available from ftp://ftp.irisa.fr/techreports/theses/2004/abbes.pdf
Abbes, S., Benveniste, A.: Branching cells as local states for event structures and nets: probabilistic applications. In: Sassone, V. (ed.) FOSSACS 2005. LNCS, vol. 3441, pp. 95–109. Springer, Heidelberg (2005); Extended version available as Research Report INRIA RR-5347, http://www.inria.fr/rrrt/rr-5347.html
Benveniste, A., Haar, S., Fabre, E.: Markov nets: probabilistic models for distributed and concurrent systems. IEEE Trans. on Aut. Control 48(11), 1936–1950 (2003)
Benvensite, A., Fabre, E., Haar, S., Jard, C.: Distributed monitoring of concurrent and asynchronous systems. In: Amadio, R.M., Lugiez, D. (eds.) CONCUR 2003. LNCS, vol. 2761, pp. 1–26. Springer, Heidelberg (2003)
Breiman, L.: Probability. SIAM, Philadelphia (1992)
Engelfriet, J.: Branching processes of Petri nets. Acta Informatica 28, 575–591 (1991)
Girault, C., Valk, R.: Petri nets for systems engeneering. Springer, Heidelberg (2003)
Nielsen, M., Plotkin, G., Winskel, G.: Petri nets, event structures and domains, part 1. T.C.S. 13, 86–108 (1980)
Rabin, M.O.: Probabilistic automata. Inf. and Control 6(3), 230–245 (1963)
Revuz, D.: Markov chains. N.-H. Math. Library, vol. 11. North-Holland, Amsterdam (1975)
Varacca, D., Völzer, H., Winskel, G.: Probabilistic event structures and domains. In: Gardner, P., Yoshida, N. (eds.) CONCUR 2004. LNCS, vol. 3170, pp. 481–496. Springer, Heidelberg (2004)
Völzer, H.: Randomized non-sequential processes. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 184–201. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Abbes, S. (2005). The (True) Concurrent Markov Property and Some Applications to Markov Nets. In: Ciardo, G., Darondeau, P. (eds) Applications and Theory of Petri Nets 2005. ICATPN 2005. Lecture Notes in Computer Science, vol 3536. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11494744_6
Download citation
DOI: https://doi.org/10.1007/11494744_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26301-2
Online ISBN: 978-3-540-31559-9
eBook Packages: Computer ScienceComputer Science (R0)