Abstract
The paper presents a survey of out-of-core methods available for the analysis of large Markov chains on single workstations. First, we discuss the main sparse matrix storage schemes and review iterative methods for the solution of systems of linear equations typically used in disk-based methods. Next, various out-of-core approaches for the steady state solution of CTMCs are described. In this context, serial out-of-core algorithms are outlined and analysed with the help of their implementations. A comparison of time and memory requirements for typical benchmark models is given.
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Bahar, I., Frohm, E., Gaona, C., Hachtel, G., Macii, E., Pardo, A., Somenzi, F.: Algebraic Decision Diagrams and their Applications. In: ICCAD 1993, Santa Clara, pp. 188–191 (1993)
Barrett, R., Berry, M., Chan, T.F., Demmel, J., Donato, J.M., Dongarra, J., Eijkhout, V., Pozo, R., Romine, C., van der Vorst, H.: Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods. Philadalphia: Society for Industrial and Applied Mathematics (1994)
Baskett, F., Chandy, K.M., Muntz, R.R., Palacios, F.G.: Open, Closed, and Mixed Networks of Queues with Different Classes of Customers. Journal of ACM 22(2) (1975)
Bause, F.: Queueing Petri Nets: A Formalism for the Combined Qualitative and Quantitative Analysis of Systems. In: Proc. PNPM 1993. IEEE Computer Society Press, Los Alamitos (1993)
Bell, A.: Verteilte Bewertung Stochastischer Petrinetze, Diploma thesis, RWTH, Aachen, Department of Computer Science (March 1999)
Bell, A., Haverkort, B.R.: Serial and Parallel Out-of-Core Solution of Linear Systems arising from Generalised Stochastic Petri Nets. In: High Performance Computing 2001, Seattle, USA (April 2001)
Bernardo, M., Gorrieri, R.: Extended Markovian Process Algebra. In: Sassone, V., Montanari, U. (eds.) CONCUR 1996. LNCS, vol. 1119. Springer, Heidelberg (1996)
Buchholz, P., Kemper, P.: Kronecker based Matrix Representations for Large Markov Models. In: This Proceedings (2003)
Ciardo, G., Miner, A.: A Data Structure for the Efficient Kronecker Solution of GSPNs. In: Proc. PNPM 1999, Zaragoza (1999)
Ciardo, G., Tilgner, M.: On the use of Kronecker Operators for the Solution of Generalized Stochastic Petri Nets. ICASE Report 96-35, Institute for Computer Applications in Science and Engineering (1996)
Ciardo, G., Trivedi, K.S.: A Decomposition Approach for Stochastic Reward Net Models. Performance Evaluation 18(1), 37–59 (1993)
Clarke, E., Fujita, M., McGeer, P., Yang, J., Zhao, X.: Multi-Terminal Binary Decision Diagrams: An Effificient Data Structure for Matrix Representation. In: International Workshop on Logic Synthesis, IWLS 1993 (May 1993)
Conway, A.E., Georganas, N.D.: Queueing Networks - Exact Computational Algorithms: A Unified Theory based on Decomposition and Aggregation. MIT Press, Cambridge (1989)
Deavours, D.D., Sanders, W.H.: An Efficient Disk-based Tool for Solving Very Large Markov Models. In: Marie, R., Plateau, B., Calzarossa, M.C., Rubino, G.J. (eds.) TOOLS 1997. LNCS, vol. 1245, pp. 58–71. Springer, Heidelberg (1997)
Deavours, D.D., Sanders, W.H.: An Efficient Disk-based Tool for Solving Large Markov Models. Performance Evaluation 33(1), 67–84 (1998)
Deavours, D.D., Sanders, W.H.: “On-the-fly”Solution Techniques for Stochastic Petri Nets and Extensions. IEEE Transactions on Software Engineering 24(10), 889–902 (1998)
Hermanns, H., Meyer-Kayser, J., Siegle, M.: Multi Terminal Binary Decision Diagrams to Represent and Analyse Continuous Time Markov Chains. In: Proc. NSMC 1999, Zaragoza (1999)
Hermanns, H., Rettelbach, M.: Syntax, Semantics, Equivalences, and Axioms for MTIPP. In: Proc. PAPM 1994, Germany (1994)
Heroux, M.: A proposal for a sparse BLAS Toolkit, Technical Report TR/PA/92/90, Cray Research, Inc., USA (December 1992)
Hillston, J.: A Compositional Approach to Performance Modelling. PhD thesis, University of Edinburgh (1994)
Ibe, O., Trivedi, K.: Stochastic Petri Net Models of Polling Systems. IEEE Journal on Selected Areas in Communications 8(9), 1649–1657 (1990)
Kahan, W.: Gauss-Seidel methods of solving large systems of linear equations. PhD thesis, University of Toronto (1958)
Knottenbelt, W.J., Harrison, P.G.: Distributed Disk-based Solution Techniques for Large Markov Models. In: Proc. NSMC 1999 (1999)
Kwiatkowska, M., Mehmood, R.: Out-of-core solution of large linear systems of equations arising from stochastic modelling. In: Hermanns, H., Segala, R. (eds.) PROBMIV 2002, PAPM-PROBMIV 2002, and PAPM 2002. LNCS, vol. 2399, p. 135. Springer, Heidelberg (2002)
Kwiatkowska, M., Mehmood, R., Norman, G., Parker, D.: A Symbolic Out-of- Core Solution Method for Markov Models. In: Proc. Parallel and Distributed Model Checking, PDMC 2002 (August 2002); Appeared in ENTCS 68(4), http://www.elsevier.nl/locate/entcs
Kwiatkowska, M., Norman, G., Parker, D.: PRISM: Probabilistic symbolic model checker. In: Field, T., Harrison, P.G., Bradley, J., Harder, U. (eds.) TOOLS 2002. LNCS, vol. 2324, p. 200. Springer, Heidelberg (2002)
Kwiatkowska, M., Norman, G., Parker, D.: Probabilistic symbolic model checking with PRISM: A hybrid approach. In: Katoen, J.-P., Stevens, P. (eds.) TACAS 2002. LNCS, vol. 2280, p. 52. Springer, Heidelberg (2002)
Marsan, M.A., Balbo, G., Conte, G.: A Class of Generalized Stochastic Petri Nets for the Performance Analysis of Multiprocessor Systems. ACM Transactions on Computer Systems 2(2) (1984)
Marsan, M.A., Balbo, G., Conte, G., Donatelli, S., Franceschinis, G., Kartson, D.: Modelling With Generalized Stochastic Petri Nets. John Wiley & Son Ltd, Chichester (1995)
Mehmood, R.: On the Development of Techniques for the Analysis of Large Markov Models. PhD thesis, University of Birmingham (2003) (to appear)
Miner, A., Parker, D.: Symbolic Representations and Analysis of Large State Spaces. In: This Proceedings (2003)
Molloy, M.K.: Performance Analysis using Stochastic Petri Nets. IEEE Trans. Comput. 31, 913–917 (1982)
Parker, D.: Implementation of Symbolic Model Checking for Probabilistic Systems. PhD thesis, University of Birmingham (August. 2002)
Plateau, B.: On the Stochastic Structure of Parallelism and Synchronisation Models for Distributed Algorithms. In: Proc. 1985 ACM SIGMETRICS Conference on Measurement and Modeling of Computer Systems (1985)
Plateau, B., Atif, K.: Stochastic Automata Network for Modeling Parallel Systems. IEEE Transactions on Software Engineering 17(10) (1991)
PRISM Web Page. http://www.cs.bham.ac.uk/~dxp/prism/
Saad, Y.: SPARSKIT: A basic tool kit for sparse matrix computations. Technical Report RIACS-90-20, NASA Ames Research Center, CA (1990)
Siegle, M.: Advances in model representations. In: de Luca, L., Gilmore, S. (eds.) PROBMIV 2001, PAPM-PROBMIV 2001, and PAPM 2001. LNCS, vol. 2165, p. 1. Springer, Heidelberg (2001)
Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton (1994)
Toledo, S.: A Survey of Out-of-Core Algorithms in Numerical Linear Algebra. In: External Memory Algorithms and Visualization, DIMACS Series in Discrete Mathematics and Theoretical Computer Science. American Mathematical Society Press, Providence (1999)
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Mehmood, R. (2004). Serial Disk-Based Analysis of Large Stochastic Models. In: Baier, C., Haverkort, B.R., Hermanns, H., Katoen, JP., Siegle, M. (eds) Validation of Stochastic Systems. Lecture Notes in Computer Science, vol 2925. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-24611-4_7
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DOI: https://doi.org/10.1007/978-3-540-24611-4_7
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