Abstract
We consider continuous-time Markov chains (CTMC) with very large or infinite state spaces which are, for instance, used to model biological processes or to evaluate the performance of computer and communication networks. We propose a numerical integration algorithm to approximate the probability that a process conforms to a specification that belongs to a subclass of deterministic timed automata (DTAs). We combat the state space explosion problem by using a dynamic state space that contains only the most relevant states. In this way we avoid an explicit construction of the state-transition graph of the composition of the DTA and the CTMC. We also show how to maximize the probability of acceptance of the DTA for parametric CTMCs and substantiate the usefulness of our approach with experimental results from biological case studies.
Notes
For simplicity, we choose a fixed step size here. In our implementation h is adaptive.
Note that the chain rule is applicable, since Q as well as π(t) depend on λ.
References
Alur R, Dill D (1994) A theory of timed automata. Theor Comput Sci 126(2):183–235
Andreychenko A, Mikeev L, Spieler D, Wolf V (2011) Parameter identification for Markov models of biochemical reactions. In: Proc of 23rd international conference on computer aided verification (CAV’11). LNCS, vol 6806. Springer, Berlin, pp 83–98
Anderson W (1991) Continuous-time Markov chains: an applications-oriented approach. Springer, Berlin
Barbot B, Chen T, Han T, Katoen J-P, Mereacre A (2011) Efficient CTMC model checking of linear real-time objectives. In: Proc of 17th international conference on tools and algorithms for the construction and analysis of systems (TACAS’11). LNCS, vol 6605. Springer, Berlin, pp 128–142
Baier C, Haverkort B, Hermanns H, Katoen J-P (2003) Model-checking algorithms for continuous-time Markov chains. IEEE Trans Softw Eng 29:524–541
Barkai N, Leibler S (2000) Biological rhythms: Circadian clocks limited by noise. Nature 403:267–268
Ballarini P, Mardare R, Mura I (2009) Analysing biochemical oscillation through probabilistic model checking. In: Proc 2nd workshop from biology to concurrency and back (FBTC’08). ENTCS, vol 229. Elsevier, Amsterdam, pp 3–19
Chen T, Han T, Katoen J-P, Mereacre A (2011) Model checking of continuous-time Markov chains against timed automata specifications. Log Methods Comput Sci 7:1–34
Donatelli S, Haddad S, Sproston J (2009) Model checking timed and stochastic properties with CSLTA. IEEE Trans Softw Eng 35:224–240
Dayar T, Hermanns H, Spieler D, Wolf V (2011) Bounding the equilibrium distribution of Markov population models. In: Numerical linear algebra with applications
Elowitz M, Leibler S (2000) A synthetic oscillatory network of transcriptional regulators. Nature 403(6767):335–338
German R (2000) Performance analysis of communication systems with non-Markovian stochastic Petri nets. Wiley, New York
Hairer E, Norsett S, Wanner G (2008) Solving ordinary differential equations I: nonstiff problems. Springer, Berlin
Kwiatkowska M, Norman G, Parker D (2011) PRISM 4.0: verification of probabilistic real-time systems. In: Gopalakrishnan G, Qadeer S (eds) Proc 23rd international conference on computer aided verification (CAV’11). LNCS, vol 6806. Springer, Berlin, pp 585–591
Loinger A, Lipshtat A, Balaban NQ, Biham O (2007) Stochastic simulations of genetic switch systems. Phys Rev E 75(2):021904
Mateescu M, Wolf V, Didier F, Henzinger T (2010) Fast adaptive uniformisation of the chemical master equation. IET Syst Biol J 4(6):441–452
Acknowledgements
This research has been partially funded by the German Research Council (DFG) as part of the Cluster of Excellence on Multimodal Computing and Interaction at Saarland University and the Transregional Collaborative Research Center “Automatic Verification and Analysis of Complex Systems” (SFB/TR 14 AVACS).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Mikeev, L., Neuhäußer, M.R., Spieler, D. et al. On-the-fly verification and optimization of DTA-properties for large Markov chains. Form Methods Syst Des 43, 313–337 (2013). https://doi.org/10.1007/s10703-012-0165-1
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10703-012-0165-1