Abstract
Standard performance evaluation methods for discrete-state stochastic models such as Petri nets either generate the reachability graph followed by a numerical solution of equations, or use some variant of simulation. Both methods have characteristic advantages and disadvantages depending on the size of the reachability graph and type of performance measure. The paper proposes a hybrid performance evaluation algorithm for Stochastic Petri Nets that integrates elements of both methods. It automatically adapts its behavior depending on the available size of main memory and number of model states. As such, the algorithm unifies simulation and numerical analysis in a joint framework. It is proved to result in an unbiased estimator whose variance tends to zero with increasing simulation time; furthermore, its applicability is demonstrated through case studies.
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Notes
- 1.
We use the term particle here to denote one simulation state as part of a larger set, not in the sense of a multi-particle simulation in physics.
- 2.
Assuming that the initial transient phase of the simulation has passed at \(t = 0\).
- 3.
A filtration is a growing sequence of \(\sigma \)-algebras which may be interpreted as containing the information up to the corresponding time point.
- 4.
\({{\mathrm{\mathbf {E}}}}\) and \({{\mathrm{\mathbf {Var}}}}\) denote expected value and variance of the subsequent terms.
- 5.
This and the later second experiment model file as well as more numerical results are available at www.tu-ilmenau.de/sse/timenet/data-for-the-multi-trajectory- algorithm to support reproducibility. TimeNET can be obtained from timenet.tu-ilmenau.de.
References
Ajmone Marsan, M., Balbo, G., Conte, G., Donatelli, S., Franceschinis, G.: Modelling with Generalized Stochastic Petri Nets. Series in Parallel Computing. Wiley, Hoboken (1995)
Marsan, M.A.: Stochastic Petri nets: an elementary introduction. In: Rozenberg, G. (ed.) APN 1988. LNCS, vol. 424, pp. 1–29. Springer, Heidelberg (1990). doi:10.1007/3-540-52494-0_23
Buchholz, P.: A new approach combining simulation and randomization for the analysis of large continuous time Markov chains. ACM Trans. Model. Comput. Simul. 8, 194–222 (1998)
Canabal Lavista, A.: Multi-trajectory rare-event simulation of stochastic Petri nets. Master’s thesis, Technische Universität Ilmenau (2015)
Ciardo, G., German, R., Lindemann, C.: A characterization of the stochastic process underlying a stochastic Petri net. IEEE Trans. Softw. Eng. 20, 506–515 (1994)
German, R., Kelling, C., Zimmermann, A., Hommel, G.: TimeNET - a toolkit for evaluating non-Markovian stochastic Petri nets. Perform. Eval. 24, 69–87 (1995)
Gilmer Jr., J.B., Sullivan, F.J.: Combat simulation trajectory management. In: Chinni, M. (ed.) Proceedings of Military, Government, and Aerospace Simulation Conference, pp. 236–241. Society for Computer Simulation, San Diego (1996)
Kelling, C.: TimeNET\(_\text{sim}\) - a parallel simulator for stochastic Petri nets. In: Proceedings of 28th Annual Simulation Symposium, Phoenix, AZ, USA, pp. 250–258 (1995)
Kovalchuk, S.V., Boukhanovsky, A.V.: Towards ensemble simulation of complex systems. Procedia Comput. Sci. 51, 532–541 (2015)
Lazarova-Molnar, S., Horton, G.: Proxel-based simulation of stochastic Petri nets containing immediate transistion. In: Kemper, P. (Hrsg.) Workshop on Stachastic Petri Nets and Related Formalisms: Satellite Workshop of ICALP 2003, Eindhoven, Dortmund, The Netherland 28–29 June 2003 - on-site proceedings, pp. S.203–S.217 (2003)
McGeoch, C.: Analysing algorithms by simulation: variance reduction techniques and simulation speedups. ACM Comput. Surv. 24(2), 195–212 (1992)
Pawlikowski, K.: Steady-state simulation of queueing processes: survey of problems and solutions. ACM Comput. Surv. 22(2), 123–170 (1990)
Sanders, W.H., Meyer, J.F.: A unified approach for specifying measures of performance, dependability, and performability. In: Avizienis, A., Laprie, J. (eds.) Dependable Computing for Critical Applications, Dependable Computing and Fault-Tolerant Systems, vol. 4, pp. 215–237. Springer, Heidelberg (1991). doi:10.1007/978-3-7091-9123-1_10
Schwetman, H.D.: Hybrid simulation models of computer systems. Commun. ACM 21(9), 718–723 (1978)
Trivedi, K.S.: Probability and Statistics with Reliability, Queuing and Computer Science Applications, 2nd edn. Wiley, Hoboken (2002)
Tuffin, B., Trivedi, K.S.: Implementation of importance splitting techniques in stochastic Petri net package. In: Haverkort, B.R., Bohnenkamp, H.C., Smith, C.U. (eds.) TOOLS 2000. LNCS, vol. 1786, pp. 216–229. Springer, Heidelberg (2000). doi:10.1007/3-540-46429-8_16
Villén-Altamirano, M., Villén-Altamirano, J.: RESTART: a method for accelerating rare event simulations. In: Queueing, Performance and Control in ATM, pp. 71–76. Elsevier Science Publishers (1991)
Zimmermann, A.: Stochastic Discrete Event Systems. Springer, Berlin (2007)
Zimmermann, A.: Modeling and evaluation of stochastic Petri nets with TimeNET 4.1. In: Proceedings of 6th International Conference on Performance Evaluation Methodologies and Tools (VALUETOOLS), Corse, France, pp. 54–63 (2012)
Zimmermann, A., Reijsbergen, D., Wichmann, A.: Canabal Lavista, A.: Numerical results for the automated rare event simulation of stochastic Petri nets. In: 11th International Workshop on Rare Event Simulation (RESIM 2016), Eindhoven, Netherlands, pp. 1–10 (2016)
Acknowledgements
The authors would like to thank Florian Kelma and Thomas Böhme, both from the Institute for Mathematics, Technische Universität Ilmenau, for fruitful discussions on the mathematical treatment.
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Zimmermann, A., Hotz, T., Lavista, A.C. (2017). A Hybrid Multi-trajectory Simulation Algorithm for the Performance Evaluation of Stochastic Petri Nets. In: Bertrand, N., Bortolussi, L. (eds) Quantitative Evaluation of Systems. QEST 2017. Lecture Notes in Computer Science(), vol 10503. Springer, Cham. https://doi.org/10.1007/978-3-319-66335-7_7
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