Abstract
In this paper, we show how to design a perfect sampling algorithm for stochastic Free-Choice Petri nets by backward coupling. For Markovian event graphs, the simulation time can be greatly reduced by using extremal initial states, namely blocking marking, although such nets do not exhibit any natural monotonicity property. Another approach for perfect simulation of non-Markovian event graphs is based on a (max,plus) representation of the system and the theory of (max,plus) stochastic systems. We also show how to extend this approach to one-bounded free choice nets to the expense of keeping all states. Finally, experimental runs show that the (max,plus) approach needs a larger simulation time than the Markovian approach.
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Notes
Throughout the paper, the self loops are not displayed in the figures in the sake of clarity. The only Petri net without implicit self-loops is displayed in Fig. 3.
by definition, \((A\otimes B)_{ij} = \bigvee_{k} (A_{ik} + B_{kj})\)
References
Baccelli F, Foss S, Mairesse J (1996) Stationary ergodic jackson networks: results and counter-examples. Stochastic networks, Oxford Univ. Press, pp 281–307
Baccelli F, Gohen G, Olsder G, Quadrat, J-P (1992) Synchronization and linearity. Wiley
Baccelli F, Jean-Marie A, Mitrani I (eds) (1995) Quantitative methods in parallel systems. Basic Research Series. Springer
Baccelli F, Mairesse J (1998) Idempotency. Chapter ergodic theory of stochastic operators and discrete event networks. Publications of the Isaac Newton Institute. Cambridge Univ. Press, pp 171–208
Bouillard A, Gaujal B (2001) Coupling time of a (max,plus) matrix. In: Workshop on Max-Plus algebras and their applications to discrete-event systems. Theoretical Computer Science, and Optimization, Prague. IFAC
Bouillard A, Gaujal B (2006) Backward coupling in petri nets. In: Valuetools, Pisa, Italy
Desel J, Esparza J (1995) Free choice Petri nets. Cambridge Tracts in Theorical Computer Science
Foss S, Tweedy R, Corcoran J (1998) Simulating the invariant measures of markov chains using backward coupling at regeneration time. Prob Eng Inf Sci 12:303–320
Gaubert S, Mairesse J (1999) Modeling and analysis of timed Petri nets using heaps of pieces. IEEE Trans Autom Control 44(4):683–697
Gaujal B, Haar S, Mairesse J (2003) Blocking a transition in a free choice net, and what it tells about its throughput. J Comput Syst Sci 66(3):515–548
Häggström O (2002) Finite Markov chains and Algorithmic Applications, vol 52 of Student texts. Cambridge Univ. Press
Mairesse J (1997) Products of irreducible random matrices in the (max,+) algebra. Adv Appl Prob 29(2):444–477
Propp D, Wilson J (1996) Exact sampling with coupled Markov chains and application to statistical mechanics. Random Struct Algorithms 9(1):223–252
Vincent J-M, Marchand C (2004) On the exact simulation of functionals of stationary Markov chains. Linear Algebra Appl 386:285–310
Walker A (1974) An efficient method for generating random variables with general distributions. ACM Trans Math Softw 253–256
Acknowledgement
The authors would like to thank Jean Mairesse for his precious advices.
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Bouillard, A., Gaujal, B. Backward Coupling in Bounded Free-Choice Nets Under Markovian and Non-Markovian Assumptions. Discrete Event Dyn Syst 18, 473–498 (2008). https://doi.org/10.1007/s10626-008-0041-8
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DOI: https://doi.org/10.1007/s10626-008-0041-8