Structured analysis approaches for large Markov chains

Applied Numerical Mathematics

Volumn 31, Issue 4, 1999, Pages 375-404

  Buchholz, Peter ^a  

^a TU DORTMUND UNIVERSITY   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

ITERATIVE METHODS; MATRIX ALGEBRA; STATE SPACE METHODS; TENSORS;

CONTINUOUS TIME MARKOV CHAINS (CTMC);

MARKOV PROCESSES;

EID: 0033338997     PISSN: 01689274     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0168-9274(99)00005-7     Document Type: Article

References (48)

1. M. Ajmone-Marsan, G. Balbo, G. Conte, S. Donatelli, G. Franceschinis, Modelling with Generalized Stochastic Petri Nets, Wiley, New York, 1995.
2. M. Ajmone-Marsan, S. Donatelli, F. Neri, GSPN models of Markovian multiserver multiqueue systems, Performance Evaluation 11 (1990) 227-240.
3. V. Amoia, G.De Micheli, M. Santomauro, Computer-oriented formulation of transition-rate matrices via Kronecker algebra, IEEE Trans. Reliability 30 (2) (1981) 123-132.
4. F. Bause, P. Buchholz, P Kemper, QPN-tool for the specification and analysis of hierarchically combined queueing Petri nets, in: H. Beilner, F. Bause (Eds.), Quantitative Evaluation of Computing and Communication Systems, Lecture Notes in Computer Science, Vol. 977, Springer, Berlin, 1995, pp. 224-238.
5. C. Beounes, Stochastic Petri net modeling for dependability evaluation of complex computer systems, in: Proc. 1st Internat. Workshop on Petri Nets and Performance Models, IEEE CS Press, 1985, pp. 191-198.
6. G. Birkhoff, R.E. Lynch, Numerical Solution of Elliptic Problems, SIAM, Philadelphia, PA, 1984.
7. P. Bjorstad, O. Widlund, Iterative methods for the solution of elliptic problems on regions partioned into substructures, SIAM J. Numer. Anal. 23 (1986) 1097-1120.
8. A. Brandt, Multi-level adaptive solutions to boundary value problems, Math. Comp. 31 (1977) 333-390.
9. P. Buchholz, Die Strukturierte Analyse Markovscher Modelle, IFB 282, Springer, Berlin, 1991.
10. P. Buchholz, Numerical solution methods based on structured descriptions of Markovian models, in: G. Balbo, G. Serazzi (Eds.), Computer Performance Evaluation - Modelling Techniques and Tools, Elsevier, Amsterdam, 1992, pp. 251-267.
11. P. Buchholz, Markovian process algebra: Composition and equivalence, in: U. Herzog, M. Rettelbach (Eds.), Proc. of The 2nd Workshop on Process Algebras and Performance Modelling, Arbeitsberichte des IMMD, University of Erlangen, No. 27, 1994, pp. 11-30.
12. P. Buchholz, An aggregation/disaggregation algorithm for stochastic automata networks, Probab. Engrg. Inform. Sci., 11 (2) (1997) 229-253.
13. P. Buchholz, Hierarchical structuring of superposed GSPNs, in: Proc. 7th Internat. Workshop on Petri Nets and Performance Models, IEEE CS Press, 1997, pp. 81-90.
14. P. Buchholz, An adaptive aggregation/disaggregation algorithm for hierarchical Markovian models, European J. Oper. Res. 116 (3) (1999) 85-104.
15. P. Buchholz, A class of hierarchical queueing networks and their analysis, Queueing Systems 15 (1) (1994) 59-80.
16. P. Buchholz, G. Ciardo, S. Donatelli, P. Kemper, Complexity of Kronecker operations and sparse matrices with applications to the solution of Markov models, ICASE Report 97-66, NASA ICASE (submitted for publication), 1997.
17. R. Chan, Iterative methods for overflow queueing networks I, Numer. Math. 51 (1987) 143-180.
18. R. Chan, Iterative methods for overflow queueing networks II, Numer. Math. 54 (1988) 57-78.
19. G. Chiola, G. Franceschinis, R. Gaeta, M. Ribaudo, GreatSPN 1.7: Graphical editor and analyzer for timed and stochastic Petri nets, Performance Evaluation 24 (1/2) (1995) 47-68.
20. G. Ciardo, J. Muppala, K.S. Trivedi, SPNP - Stochastic Petri net package, in: Proc. 3rd Internat. Workshop on Petri-nets and Performance Models, IEEE CS Press, 1989, pp. 142-151.
21. M. Davio, Kronecker products and shuffle algebra, IEEE Trans. Comput. 30 (1981) 116-125.
22. S. Donatelli, Superposed generalized stochastic Petri nets: Definition and efficient solution, in: R. Valette (Ed.), Application and Theory of Petri Nets 1994, Lecture Notes in Computer Science, Vol. 815, Springer, Berlin, 1994, pp. 258-277.
23. S. Donatelli, Superposed stochastic automata: A class of stochastic Petri nets amenable to parallel solution, Performance Evaluation 18 (1994) 21-36.
24. P. Fernandes, B. Plateau, W.J. Stewart, Efficient descriptor-vector multiplication in stochastic automata networks, J. ACM 45 (3) (1998) 381-414.
25. A. Graham, Kronecker Products and Matrix Calculus with Applications, Ellis Howard, 1981.
26. H. Hermanns, U. Herzog, V. Mertsiotakis, Stochastic process algebras - between lotos and Markov chains, Computer Networks and ISDN Systems 30 (9/10) (1998) 901-924.
27. J. Hillston, Compositional Markovian modelling using a process algebra, in: W.J. Stewart (Ed.), Computations with Markov Chains, Kluwer Academic, Dordrecht, 1995, pp. 177-196.
28. G. Horton, S. Leutenegger, A multi-level solution algorithm for steady state Markov-chains, ACM Performance Evaluation Rev. 22 (1994) 191-200.
29. K. Kant, Introduction to Computer System Performance Evaluation, Mc Graw-Hill, 1992.
30. L. Kaufman, Matrix methods for queueing problems, SIAM J. Sci. Comput. 4 (1983) 525-552.
31. P. Kemper, Numerical analysis of superposed GSPNs, IEEE Trans. Software Engrg. 22 (9) (1996) 615-628.
32. R.L. Klevans, W.J. Stewart, From queueing networks to Markov chains: The XMARCA interface, Performance Evaluation 24 (1/2) (1996).
33. U. Krieger, B. Müller-Clostermann, M. Sczittnick, Modeling and analysis of communication systems based on computational methods for Markov chains, IEEE J. Selected Areas Commun. 8 (9) (1990) 1630-1648.
34. R.E. Lynch, J.R. Rice, D.H. Thomas, Tensor product analysis of partial difference equations, Bull Amer. Math. Soc. 70 (1964) 378-384.
35. D. Mitra, I. Mitrani, Analysis of a kanban discipline for cell coordination in production lines II: Stochastic demands, Oper. Res. 39 (1991) 807-823.
36. M.F. Neuts, Matrix-geometric Solutions in Stochastic Models, Johns Hopkins University Press, Baltimore, MD, 1981.
37. B. Philippe, Y. Saad, W.J. Stewart, Numerical methods in Markov chain modeling, Oper. Res. 40 (1996) 1156-1179.
38. B. Plateau, On the stochastic structure of parallelism and synchronisation models for distributed algorithms, Performance Evaluation Rev. 13 (1985) 142-154.
39. B. Plateau, J.M. Fourneau, K.H. Lee, Peps: A package for solving complex Markov models of parallel systems, in: R. Puijanger (Ed.), Proc. 4th Internat. Conf. on Modelling Tools and Techniques for Computer Performance Evaluation, Plenum Publishers, 1988.
40. B. Plateau, J.M. Fourneau, A methodology for solving Markov models of parallel systems, J. Parallel Distributed Comput. 12 (1991) 370-387.
41. P.A. Regalia, S.K. Mitra, Kronecker products, unitary matrices and signal processing applications, SIAM Rev. 31 (4) (1989) 586-613.
42. Y. Saad, Projection methods for the numerical solution of Markov models, in: W.J. Stewart (Ed.), Numerical Solution of Markov Chains, Marcel Dekker, New York, 1991, pp. 455-472.
43. Y Saad, M. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Comput. 7 (1986) 856-869.
44. P. Schweitzer, A survey of aggregation-disaggregation algorithms, in: W.J. Stewart (Ed.), Numerical Solution of Markov Chains, Marcel Dekker, New York, 1991, pp. 63-88.
45. W.J. Stewart, Introduction to The Numerical Solution of Markov Chains, Princeton University Press, Princeton, 1994.
46. W.J. Stewart, K. Atif, B. Plateau, The numerical solution of stochastic automata networks, European J. Oper. Res. 86 (1995) 503-525.
47. E. Uysal, T. Dayar, Iterative methods based on splittings for stochastic automata networks, European J. Oper. Res. 110 (1) (1998) 166-186.
48. U.M. Yang, Preconditioned Conjugate Gradient Like Methods for Nonsymmetric Linear Systems, Technical Report 1210, Center of Supercomp. Res. and Develop., University of Illinois at Urbana-Champaign, 1994.