Abstract
This tutorial presents an overview of model checking for both discrete and continuous-time Markov chains (DTMCs and CTMCs). Model checking algorithms are given for verifying DTMCs and CTMCs against specifications written in probabilistic extensions of temporal logic, including quantitative properties with rewards. Example properties include the probability that a fault occurs and the expected number of faults in a given time period. We also describe the practical application of stochastic model checking with the probabilistic model checker PRISM by outlining the main features supported by PRISM and three real-world case studies: a probabilistic security protocol, dynamic power management and a biological pathway.
Partly supported by EPSRC grants EP/D07956X and EP/D076625 and Microsoft Research Cambridge contract MRL 2005-44.
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
Alur, R., Henzinger, T.: Reactive modules. Formal Methods in System Design 15(1), 7–48 (1999)
Andova, S., Hermanns, H., Katoen, J.-P.: Discrete-time rewards model-checked. In: Larsen, K.G., Niebert, P. (eds.) FORMATS 2003. LNCS, vol. 2791, pp. 88–104. Springer, Heidelberg (2004)
Aspnes, J., Herlihy, M.: Fast randomized consensus using shared memory. Journal of Algorithms 15(1), 441–460 (1990)
Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Verifying continuous time Markov chains. In: Alur, R., Henzinger, T.A. (eds.) CAV 1996. LNCS, vol. 1102, pp. 269–276. Springer, Heidelberg (1996)
Aziz, A., Sanwal, K., Singhal, V., Brayton, R.: Model checking continuous time Markov chains. ACM Transactions on Computational Logic 1(1), 162–170 (2000)
Aziz, A., Singhal, V., Balarin, F., Brayton, R., Sangiovanni-Vincentelli, A.: It usually works: The temporal logic of stochastic systems. In: Wolper, P. (ed.) CAV 1995. LNCS, vol. 939, pp. 155–165. Springer, Heidelberg (1995)
Bahar, I., Frohm, E., Gaona, C., Hachtel, G., Macii, E., Pardo, A., Somenzi, F.: Algebraic decision diagrams and their applications. Formal Methods in System Design 10(2-3), 171–206 (1997)
Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: Model checking continuous-time Markov chains by transient analysis. In: Emerson, E.A., Sistla, A.P. (eds.) CAV 2000. LNCS, vol. 1855, pp. 358–372. Springer, Heidelberg (2000)
Baier, C., Haverkort, B.R., Hermanns, H., Katoen, J.-P.: On the Logical Characterisation of Performability Properties. In: Welzl, E., Montanari, U., Rolim, J.D.P. (eds.) ICALP 2000. LNCS, vol. 1853, pp. 780–792. Springer, Heidelberg (2000)
Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)
Baier, C., Katoen, J.-P., Hermanns, H., Haverkort, B.R.: Simulation for Continuous-Time Markov Chains. In: Brim, L., Jančar, P., Křetínský, M., Kucera, A. (eds.) CONCUR 2002. LNCS, vol. 2421, pp. 338–354. Springer, Heidelberg (2002)
Baier, C., Katoen, J.-P., Hermanns, H.: Approximate Symbolic Model Checking of Continuous-Time Markov Chains (Extended Abstract). In: Baeten, J.C.M., Mauw, S. (eds.) CONCUR 1999. LNCS, vol. 1664, pp. 146–161. Springer, Heidelberg (1999)
Baier, C., Kwiatkowska, M.: Model checking for a probabilistic branching time logic with fairness. Distributed Computing 11(3), 125–155 (1998)
Benini, L., Bogliolo, A., Paleologo, G., Micheli, G.D.: Policy optimization for dynamic power management. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 8(3), 299–316 (2000)
Bianco, A., de Alfaro, L.: Model checking of probabilistic and nondeterministic systems. In: Thiagarajan, P.S. (ed.) FSTTCS 1995. LNCS, vol. 1026, pp. 499–513. Springer, Heidelberg (1995)
Billingsley, P.: Probability and Measure. Wiley, Chichester (1995)
Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. Journal of Applied Probability 31, 59–75 (1994)
Cheung, L.: Randomized Wait-Free Consensus Using an Atomicity Assumption. In: Anderson, J.H., Prencipe, G., Wattenhofer, R. (eds.) OPODIS 2005. LNCS, vol. 3974, pp. 47–60. Springer, Heidelberg (2006)
Ciardo, G., Jones, R., Miner, A., Siminiceanu, R.: Logic and stochastic modeling with smart. Performance Evaluation 63(6), 578–608 (2006)
Clarke, E., Emerson, E., Sistla, A.: Automatic verification of finite-state concurrent systems using temporal logics. ACM Transactions on Programming Languages and Systems 8(2), 244–263 (1986)
Clarke, E., Fujita, M., McGeer, P., McMillan, K., Yang, J., Zhao, X.: Multi-terminal binary decision diagrams: An efficient data structure for matrix representation. Formal Methods in System Design 10(2-3), 149–169 (1997)
Courcoubetis, C., Yannakakis, M.: Verifying temporal properties of finite state probabilistic programs. In: Proc. 29th Annual Symposium on Foundations of Computer Science (FOCS’88), 1988, pp. 338–345. IEEE Computer Society Press, Los Alamitos (1988)
Courcoubetis, C., Yannakakis, M.: The complexity of probabilistic verification. Journal of the ACM 42(4), 857–907 (1995)
Daws, C., Kwiatkowska, M., Norman, G.: Automatic verification of the IEEE 1394 root contention protocol with KRONOS and PRISM. Int. Journal on Software Tools for Technology Transfer 5(2-3), 221–236 (2004)
Even, S., Goldreich, O., Lempel, A.: A randomized protocol for signing contracts. Communications of the ACM 28(6), 637–647 (1985)
Fokkink, W., Pang, J.: Variations on itai-rodeh leader election for anonymous rings and their analysis in prism. Journal of Universal Computer Science 12(8), 981–1006 (2006)
Fox, B., Glynn, P.: Computing Poisson probabilities. Communications of the ACM 31(4), 440–445 (1988)
German, R.: Performance Analysis of Communication Systems: Modeling with Non-Markovian Stochastic Petri Nets. John Wiley and Sons, Chichester (2000)
Hansson, H., Jonsson, B.: A logic for reasoning about time and reliability. Formal Aspects of Computing 6(5), 512–535 (1994)
Haverkort, B.: Performance of Computer Communication Systems: A Model-Based Approach. John Wiley & Sons, Chichester (1988)
Haverkort, B., Cloth, L., Hermanns, H., Katoen, J.-P., Baier, C.: Model checking performability properties. In: Proc. Int. Conf. Dependable Systems and Networks (DSN’02), 2002, IEEE Computer Society Press, Los Alamitos (2002)
Heath, J.K., Kwiatkowska, M., Norman, G., Parker, D., Tymchyshyn, O.: Probabilistic Model Checking of Complex Biological Pathways. In: Priami, C. (ed.) CMSB 2006. LNCS (LNBI), vol. 4210, pp. 32–47. Springer, Heidelberg (2006)
Hérault, T., Lassaigne, R., Magniette, F., Peyronnet, S.: Approximate probabilistic model checking. In: Steffen, B., Levi, G. (eds.) VMCAI 2004. LNCS, vol. 2937, pp. 73–84. Springer, Heidelberg (2004)
Hermanns, H., Katoen, J.-P., Meyer-Kayser, J., Siegle, M.: A Markov Chain Model Checker. In: Schwartzbach, M.I., Graf, S. (eds.) ETAPS 2000 and TACAS 2000. LNCS, vol. 1785, pp. 347–362. Springer, Heidelberg (2000)
Hermanns, H., Katoen, J.-P., Meyer-Kayser, J., Siegle, M.: Towards model checking stochastic process algebra. In: Grieskamp, W., Santen, T., Stoddart, B. (eds.) IFM 2000. LNCS, vol. 1945, pp. 420–439. Springer, Heidelberg (2000)
Hinton, A., Kwiatkowska, M., Norman, G., Parker, D.: PRISM: A Tool for Automatic Verification of Probabilistic Systems. In: Hermanns, H., Palsberg, J. (eds.) TACAS 2006 and ETAPS 2006. LNCS, vol. 3920, pp. 441–444. Springer, Heidelberg (2006)
IEEE standard for a high performance serial bus. IEEE Computer Society, IEEE Std. 1394-1995.
Itai, A., Rodeh, M.: Symmetry breaking in distributed networks. Information and Computation 88(1), 60–87 (1990)
Katoen, J.-P., Khattri, M., Zapreev, I.: A Markov reward model checker. In: Proc. Second Int. Conf. Quantitative Evaluation of Systems (QEST 05), 2005, pp. 243–244. IEEE Computer Society Press, Los Alamitos (2005)
Katoen, J.-P., Kwiatkowska, M., Norman, G., Parker, D.: Faster and Symbolic CTMC Model Checking. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, pp. 23–38. Springer, Heidelberg (2001)
Kemeny, J., Snell, J., Knapp, A.: Denumerable Markov Chains, 2nd edn. Springer, Heidelberg (1976)
Kwiatkowska, M., Norman, G., Pacheco, A.: Model Checking CSL until Formulae with Random Time Bounds. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, pp. 152–168. Springer, Heidelberg (2002)
Kwiatkowska, M., Norman, G., Pacheco, A.: Model checking expected time and expected reward formulae with random time bounds. In: Proc. 2nd Euro-Japanese Workshop on Stochastic Risk Modelling for Finance, Insurance, Production and Reliability (September 2002)
Kwiatkowska, M., Norman, G., Pacheco, A.: Model checking expected time and expected reward formulae with random time bounds. Computers & Mathematics with Applications 51(2), 305–316 (2006)
Kwiatkowska, M., Norman, G., Parker, D.: PRISM users’ guide. Available from http://www.cs.bham.ac.uk/~dxp/prism
Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. Int. Journal on Software Tools for Technology Transfer 6(2), 128–142 (2004)
Kwiatkowska, M., Norman, G., Parker, D., Sproston, J.: Performance analysis of probabilistic timed automata using digital clocks. Formal Methods in System Design 29, 33–78 (2006)
Kwiatkowska, M., Parker, D., Zhang, Y., Mehmood, R.: Dual-processor parallelisation of symbolic probabilistic model checking. In: DeGroot, D., Harrison, P. (eds.) Proc. 12th Int. Symposium on Modeling, Analysis, and Simulation of Computer and Telecommunication Systems (MASCOTS’04), pp. 123–130. IEEE Computer Society Press, Los Alamitos (2004)
Larsen, K., Skou, A.: Bisimulation through probabilistic testing. Information and Computation 94, 1–28 (1991)
Norman, G., Parker, D., Kwiatkowska, M., Shukla, S., Gupta, R.: Using probabilistic model checking for dynamic power management. Formal Aspects of Computing 17(2), 160–176 (2005)
Norman, G., Shmatikov, V.: Analysis of probabilistic contract signing. Journal of Computer Security 14(6), 561–589 (2006)
Parker, D.: Implementation of Symbolic Model Checking for Probabilistic Systems. PhD thesis, University of Birmingham (2002)
PRISM web site: http://www.cs.bham.ac.uk/~dxp/prism
Qiu, Q., Wu, Q., Pedram, M.: Stochastic modeling of a power-managed system: Construction and optimization. In: Proc. Int. Symposium on Low Power Electronics and Design (1999)
Qiu, Q., Wu, Q., Pedram, M.: Stochastic modeling of a power-managed system: construction and optimization. IEEE Transactions on Computer Aided Design 20(10), 1200–1217 (2001)
Rutten, J., Kwiatkowska, M., Norman, G., Parker, D.: Mathematical Techniques for Analyzing Concurrent and Probabilistic Systems. CRM Monograph Series, vol. 23. American Mathematical Society, New York (2004)
Segala, R., Lynch, N.: Probabilistic simulations for probabilistic processes. In: Jonsson, B., Parrow, J. (eds.) CONCUR 1994. LNCS, vol. 836, pp. 481–496. Springer, Heidelberg (1994)
Somenzi, F.: CUDD: Colorado University decision diagram package. Public software, Colorado Univeristy, Boulder (1997), http://vlsi.colorado.edu/~fabio/
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton (1994)
Tarjan, R.: Depth-first search and linear graph algorithms. SIAM Journal on Computing 1, 146–160 (1972)
Trivedi, K.: Probability and Statistics with Reliability, Queuing, and Computer Science Applications. John Wiley & Sons, Chichester (2001)
Vardi, M.: Automatic verification of probabilistic concurrent finite state programs. In: Proc. 26th Annual Symposium on Foundations of Computer Science (FOCS’85), 1985, pp. 327–338. IEEE Computer Society Press, Los Alamitos (1985)
Younes, H., Kwiatkowska, M., Norman, G., Parker, D.: Numerical vs. statistical probabilistic model checking. Int. Journal on Software Tools for Technology Transfer 8(3), 216–228 (2006)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this chapter
Cite this chapter
Kwiatkowska, M., Norman, G., Parker, D. (2007). Stochastic Model Checking. In: Bernardo, M., Hillston, J. (eds) Formal Methods for Performance Evaluation. SFM 2007. Lecture Notes in Computer Science, vol 4486. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-72522-0_6
Download citation
DOI: https://doi.org/10.1007/978-3-540-72522-0_6
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-72482-7
Online ISBN: 978-3-540-72522-0
eBook Packages: Computer ScienceComputer Science (R0)