Abstract
We consider parametric version of fixed-delay continuous-time Markov chains (or equivalently deterministic and stochastic Petri nets, DSPN) where fixed-delay transitions are specified by parameters, rather than concrete values. Our goal is to synthesize values of these parameters that, for a given cost function, minimise expected total cost incurred before reaching a given set of target states. We show that under mild assumptions, optimal values of parameters can be effectively approximated using translation to a Markov decision process (MDP) whose actions correspond to discretized values of these parameters. To this end we identify and overcome several interesting phenomena arising in systems with fixed delays.
The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA grant agreement \(\text {n}^\circ \) [291734]. This work is partly supported by the German Research Council (DFG) as part of the Transregional Collaborative Research Center AVACS (SFB/TR 14), by the EU 7th Framework Programme under grant agreement no. 295261 (MEALS) and 318490 (SENSATION), by the Czech Science Foundation, grant No. 15-17564S, and by the CAS/SAFEA International Partnership Program for Creative Research Teams.
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Notes
- 1.
More precisely, all but the largest probability in \(T(s,\mathbf {d})\) are rounded up, the largest probability is suitably rounded down so that the resulting vector adds up to 1.
References
Alur, R., Courcoubetis, C., Dill, D.: Model-checking for probabilistic real-time systems. In: Albert, J.L., Monien, B., Artalejo, M.R. (eds.) Automata, Languages and Programming. LNCS, vol. 510, pp. 115–136. Springer, Heidelberg (1991)
Alur, R., Henzinger, T.A., Vardi, M.Y.: Parametric real-time reasoning. In: STOC, pp. 592–601. ACM (1993)
Audsley, N.C., Grigg, A.: Timing analysis of the ARINC 629 databus for real-time applications. Microprocess. Microsyst. 21(1), 55–61 (1997)
Baier, C., Bertrand, N., Bouyer, P., Brihaye, T., Größer, M.: Probabilistic and topological semantics for timed automata. In: Arvind, V., Prasad, S. (eds.) FSTTCS 2007. LNCS, vol. 4855, pp. 179–191. Springer, Heidelberg (2007)
Baier, C., Bertrand, N., Bouyer, P., Brihaye, T., Größer, M.: Almost-sure model checking of infinite paths in one-clock timed automata. In: LICS, pp. 217–226. IEEE (2008)
Brázdil, T., Forejt, V., Krčál, J., Křetínský, J., Kučera, A.: Continuous-time stochastic games with time-bounded reachability. Inf. Comput. 224, 46–70 (2013)
Brázdil, T., Korenčiak, Ĺ., Krčál, J., Novotný, P., Řehák, V.: Optimizing performance of continuous-time stochastic systems using timeout synthesis. CoRR, abs/1407.4777 (2014)
Brázdil, T., Krčál, J., Křetínský, J., Kučera, A., Řehák, V.: Stochastic real-time games with qualitative timed automata objectives. In: Gastin, P., Laroussinie, F. (eds.) CONCUR 2010. LNCS, vol. 6269, pp. 207–221. Springer, Heidelberg (2010)
Brázdil, T., Krčál, J., Křetínský, J., Řehák, V.: Fixed-delay events in generalized semi-Markov processes revisited. In: Katoen, J.-P., König, B. (eds.) CONCUR 2011. LNCS, vol. 6901, pp. 140–155. Springer, Heidelberg (2011)
Buchholz, P., Hahn, E.M., Hermanns, H., Zhang, L.: Model checking algorithms for CTMDPs. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 225–242. Springer, Heidelberg (2011)
Carnevali, L., Ridi, L., Vicario, E.: A quantitative approach to input generation in real-time testing of stochastic systems. IEEE Trans. Softw. Eng. 39(3), 292–304 (2013)
Chen, T., Han, T., Katoen, J.-P., Mereacre, A.: Quantitative model checking of continuous-time Markov chains against timed automata specifications. In: LICS, pp. 309–318. IEEE (2009)
Choi, H., Kulkarni, V.G., Trivedi, K.S.: Transient analysis of deterministic and stochastic Petri nets. In: Marsan, M.A. (ed.) Application and Theory of Petri Nets, vol. 691, pp. 166–185. Springer, Heidelberg (1993)
Etessami, K., Wojtczak, D., Yannakakis, M.: Recursive stochastic games with positive rewards. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 711–723. Springer, Heidelberg (2008)
Feller, W.: An Introduction to Probability Theory and Its Applications, vol. 1. Wiley, New York (1968)
Guet, C.C., Gupta, A., Henzinger, T.A., Mateescu, M., Sezgin, A.: Delayed continuous-time Markov chains for genetic regulatory circuits. In: Madhusudan, P., Seshia, S.A. (eds.) CAV 2012. LNCS, vol. 7358, pp. 294–309. Springer, Heidelberg (2012)
Haase, C., Kreutzer, S., Ouaknine, J., Worrell, J.: Reachability in succinct and parametric one-counter automata. In: Bravetti, M., Zavattaro, G. (eds.) CONCUR 2009. LNCS, vol. 5710, pp. 369–383. Springer, Heidelberg (2009)
Hahn, E.M., Hermanns, H., Zhang, L.: Probabilistic reachability for parametric Markov models. STTT 13(1), 3–19 (2011)
Han, T., Katoen, J.P., Mereacre, A.: Approximate parameter synthesis for probabilistic time-bounded reachability. In: Real-Time Systems Symposium, pp. 173–182. IEEE (2008)
Jensen, P.G., Taankvist, J.H.: Learning optimal scheduling for time uncertain settings. Aalborg University, Student project (2014)
Jha, S.K., Langmead, C.J.: Synthesis and infeasibility analysis for stochastic models of biochemical systems using statistical model checking and abstraction refinement. TCS 412(21), 2162–2187 (2011)
Khaksari, M., Fischione, C.: Performance analysis and optimization of the joining protocol for a platoon of vehicles. In: ISCCSP, pp. 1–6. IEEE (2012)
Korenčiak, Ĺ., Krčál, J., Řehák, V.: Dealing with zero density using piecewise phase-type approximation. In: Horváth, A., Wolter, K. (eds.) EPEW 2014. LNCS, vol. 8721, pp. 119–134. Springer, Heidelberg (2014)
Kwiatkowska, M., Norman, G., Segala, R., Sproston, J.: Verifying quantitative properties of continuous probabilistic timed automata. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 123–137. Springer, Heidelberg (2000)
Lindemann, C.: An improved numerical algorithm for calculating steady-state solutions of deterministic and stochastic Petri net models. Perform. Eval. 18(1), 79–95 (1993)
Marsan, M.A., Chiola, G.: On Petri nets with deterministic and exponentially distributed firing times. In: Rozenberg, G. (ed.) Advances in Petri Nets, pp. 132–145. Springer, Heidelberg (1987)
Neuhäusser, M.R., Zhang, L.: Time-bounded reachability probabilities in continuous-time Markov decision processes. In: QEST, pp. 209–218. IEEE (2010)
Neuts, M.F.: Matrix-geometric Solutions in Stochastic Models: An Algorithmic Approach. Courier Dover Publications, Mineola (1981)
Norris, J.R.: Markov Chains. Cambridge University Press, Cambridge (1998)
Obermaisser, R.: Time-Triggered Communication. CRC Press, Boca Raton (2011)
Puterman, M.L.: Markov Decision Processes. Wiley, Hoboken (1994)
Ramamritham, K., Stankovic, J.A.: Scheduling algorithms and operating systems support for real-time systems. Proc. IEEE 82(1), 55–67 (1994)
Tiassou, K.B.: Aircraft operational reliability - a model-based approach and case studies. Ph.D. thesis, Universié de Toulouse (2013)
Wolovick, N., D’Argenio, P.R., Qu, V: Optimizing probabilities of real-time test case execution. In: ICST, pp. 446–455. IEEE (2009)
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Brázdil, T., Korenčiak, Ľ., Krčál, J., Novotný, P., Řehák, V. (2015). Optimizing Performance of Continuous-Time Stochastic Systems Using Timeout Synthesis. In: Campos, J., Haverkort, B. (eds) Quantitative Evaluation of Systems. QEST 2015. Lecture Notes in Computer Science(), vol 9259. Springer, Cham. https://doi.org/10.1007/978-3-319-22264-6_10
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