Abstract
Many man-made systems, especially those in the areas of computer and communication, are so complex that it is essential to study them with simplified mathematical models during their design, prototyping, and deployment.
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References
Ajmone Marsan, M., Balbo, G., Bobbio, A., Chiola, G., Conte, G., and Cumani, A. (1989). The effect of execution policies on the semantics and analyis of Stochastic Petri Nets. IEEE Trans. Softw. Eng., 15 (7): 832–846.
Ajmone Marsan, M., Balbo, G., Conte, G., Donatelli, S., and Franceschinis, G. (1995). Modelling with generalized stochastic Petri nets. John Wiley amp; Sons.
Amoia, V., De Micheli, G., and Santomauro, M. (1981). Computer-oriented formulation of transition-rate matrices via Kronecker algebra. IEEE Trans. Rel., 30: 123–132.
Baskett, F., Chandy, K. M., Muntz, R. R., and PalaciosGomez, F. (1975). Open, Closed, and Mixed networks of queues with different classes of customers. J. ACM, 22 (2): 335–381.
Bause, F., Buchholz, P., and Kemper, P. (1995). QPN-Tool for the specification and analysis of hierarchically combined queueing Petri nets. In Beilner, H. and Bause, F., editors, Quantitative Evaluation of Computing and Communication Systems, 8th Int. Conf. Modelling Techniques and Tools, LNCS 977, pages 224–238.
Berson, S., de Souza e Silva, E., and Muntz, R. R. (1991). A methodology for the specification and generation of Markov models. In Stewart, W. J., editor, Numerical Solution of Markov Chains, pages 11–36. Marcel Dekker, Inc., New York, NY.
Blum, A. M., Goyal, A., Heidelberger, P., and Lavenberg, S. S. (1994). Modeling and analysis of system dependability using the System Availability Estimator. In Proc. 24th Int. Symp. on Fault-Tolerant Computing, pages 137–141, Austin, TX.
Bobbio, A. and Cumani, A. (1984). Discrete state stochastic systems with phase-type distributed transition times. In Proc. 1984 AMSE Int. Conf. on Modelling and Simulation,Athens, Greece.
Buchholz, P. (1991). Numerical solution methods based on structured descriptions of Markovian models. In Balbo, G. and Serazzi, G., editors, Computer performance evaluation, pages 251–267. Elsevier Science Publishers B.V. ( North-Holland )
Buchholz, P. (1994). A class of hierarchical queueing networks and their analysis. Queueing Systems., 15: 59–80.
Buchholz, P. and Kemper, P. (1995). Numerical analysis of stochastic marked graphs. In Proc. 6th Int. Workshop on Petri Nets and Performance Models (PNPM’95), pages 32–41, Durham, NC. IEEE Comp. Soc. Press.
Caselli et al., 1995] Caselli, S., Conte, G., and Marenzoni, P. (1995). Parallel state space exploration for GSPN models. In De Michelis, G. and Diaz, M., editors, Application and Theory of Petri Nets 1995, Lecture Notes in Computer Science 935 (Proc. 16th Int. Conf. on Applications and Theory of Petri Nets, Turin, Italy),pages 181–200. Springer-Verlag.
Cassandras, C. G. (1993). Discrete Event Systems: Modeling and Performance Analysis. Aksen Associates.
Çinlar, E. (1975). Introduction to Stochastic Processes. Prentice-Hall.
Chen, P.-z., Bruell, S. C., and Balbo, G. (1989). Alternative methods for incorporating non-exponential distributions into stochastic timed Petri nets. In Proc. 3rd Int. Workshop on Petri Nets and Performance Models (PNPM’89),Kyoto, Japan. IEEE Comp. Soc. Press.
Chiola, G., Dutheillet, C., Franceschinis, G., and Haddad, S. (1993). Stochastic well-formed colored nets and symmetric modeling applications. IEEE Trans. Comp., 42 (11): 1343–1360.
Choi, H., Kulkarni, V. G., and Trivedi, K. S. (1994). Markov regenerative stochastic Petri nets. Perf. Eval., 20 (1–3): 337–357.
Ciardo, G. (1989). Analysis of large stochastic Petri net models. PhD thesis, Duke University, Durham, NC.
Ciardo, G. (1994). Petri nets with marking-dependent arc multiplicity: properties and analysis. In Valette, R., editor, Application and Theory of Petri Nets 1994, Lecture Notes in Computer Science 815 (Proc. 15th Int. Conf. on Applications and Theory of Petri Nets, Zaragoza, Spain),pages 179–198. Springer-Verlag.
Ciardo, G. (1995). Discrete-time Markovian stochastic Petri nets. In Stewart, W. J., editor, Computations with Markov Chains, pages 339–358. Kluwer, Boston, MA.
Ciardo, G., Blakemore, A., Chimento, P. F. J., Muppala, J. K., and Trivedi, K. S. (1993a). Automated generation and analysis of Markov reward models using Stochastic Reward Nets. In Meyer, C. and Plemmons, R. J., editors, Linear Algebra, Markov Chains, and Queueing Models,volume 48 of IMA Volumes in Mathematics and its Applications,pages 145–191. Springer-Verlag.
Ciardo, G., German, R., and Lindemann, C. (1993b). A characterization of the stochastic process underlying a stochastic Petri net. In Proc. 5th Int. Workshop on Petri Nets and Performance Models (PNPM’93), pages 170–179, Toulouse, France. IEEE Comp. Soc. Press.
Ciardo, G., Gluckman, J., and Nicol, D. (1998). Distributed state-space generation of discrete-state stochastic models. INFORMS J. Comp., 10 (1): 82–93.
Ciardo, G. and Miner, A. S. (1997). Storage alternatives for large structured state spaces. In Marie, R., Plateau, B., Calzarossa, M., and Rubino, G., editors, Proc. 9th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, LNCS 1245, pages 44–57, St. Malo, France. Springer-Verlag.
Ciardo, G., Muppala, J. K., and Trivedi, K. S. (1991). On the solution of GSPN reward models. Perf. Eval., 12 (4): 237–253.
Ciardo, G. and Tilgner, M. (1996). 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, Hampton, VA.
Ciardo, G. and Zijal, R. (1996). Well-defined stochastic Petri nets. In Proc. 4th Int. Workshop on Modeling, Analysis and Simulation of Computer and Telecommunication Systems (MASCOTS’96), pages 278–284, San Jose, CA, USA. IEEE Comp. Soc. Press.
Clocksin, W. F. and Mellish, C. S. (1984). Programming in Prolog. Springer-Verlag.
Cox, D. (1955). A use of complex probabilities in the theory of stochastic processes. Proc. of the Cambridge Philosophical Society, 51: 313319.
Cumani, A. (1985). ESP - A package for the evaluation of stochastic Petri nets with phase-type distributed transitions times. In Proc. Int. Workshop on Timed Petri Nets,Torino, Italy.
Davio, M. (1981). Kronecker products and shuffle algebra. IEEE Trans. Comp., C-30: 116–125.
de Souza e Silva, E. and Gail, H. R. (1989). Calculating availability and performability measures of repairable computer systems using randomization. J. ACM., 36 (1): 171–193.
Deavours, D. D. and Sanders, W. H. (1997a). An efficient disk-based tool for solving very large Markov models. In Marie, R., Plateau, B., Calzarossa, M., and Rubino, G., editors, Proc. 9th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, LNCS 1245, pages 58–71, St. Malo, France. Springer-Verlag.
Deavours, D. D. and Sanders, W. H. (1997b). “On-the-fly” solution techniques for stochastic Petri nets and extensions. In Proc. 7th Int. Workshop on Petri Nets and Performance Models (PNPM’97), pages 132–141, St. Malo, France. IEEE Comp. Soc. Press.
Donatelli, S. (1991). Superposed Stochastic Automata: a class of stochastic Petri nets amenable to parallel solution. In Proc. 4th Int. Workshop on Petri Nets and Performance Models (PNPM’91), pages 54–63, Melbourne, Australia. IEEE Comp. Soc. Press.
Donatelli, S. (1994). Superposed generalized stochastic Petri nets: definition and efficient solution. In Valette, R., editor, Application and Theory of Petri Nets 199.4, Lecture Notes in Computer Science 815 (Proc. 15th Int. Conf. on Applications and Theory of Petri Nets), pages 258–277, Zaragoza, Spain. Springer-Verlag.
Feller, W. (1962). An Introduction to Probability Theory and Its Applications. John Wiley, New York. Second Edition.
Fernandes, P., Plateau, B., and Stewart, W. J. (1996). Numerical issue for stochastic automata networks. In Proc. of the Ord Workshop on Process Algebra and Performance Modelling (PAPM), Torino, Italy. CLUT.
Grassmann, W. K. (1991). Finding transient solutions in Markovian event systems through randomization. In Stewart, W. J., editor, Numerical Solution of Markov Chains, pages 357–371. Marcel Dekker, Inc., New York, NY.
Grassmann, W. K. and Wang, Y. (1995) Immediate events in Markov chains. In Stewart, W. J., editor, Computations with Markov Chains, pages 163–176. Kluwer, Boston, MA.
Gross, D. and Miller, D. (1984). The randomization technique as a modeling tool and solution procedure for transient Markov processes. Oper. Res., 32: 343–361.
Jensen, A. (1953). Markoff Chains as an Aid in the Study of Markoff Processes. Skand. Aktuarietidskr., 36: 87–91.
Jensen, K. (1987). Coloured Petri nets. In Petri Nets: Central Models and Their Properties, Lecture Notes in Computer Science 254, pages 248–299. Springer-Verlag.
Kemper, P. (1996). Numerical analysis of superposed GSPNs. IEEE Trans. Softw. Eng., 22 (4): 615–628.
Molloy, M. K. (1985). Discrete time stochastic Petri nets. IEEE Trans. Softw. Eng., 11 (4): 417–423.
Murata, T. (1989). Petri Nets: properties, analysis and applications. Proc. of the IEEE, 77 (4): 541–579.
Page, T. W., Berson, S. E., Cheng, W. C., and Muntz, R. R. (1989). An object-oriented modeling environment. In OOPSLA ‘89 Proc.
Plateau, B. (1985). On the stochastic structure of parallelism and synchronisation models for distributed algorithms. In Proc. 1985 ACM SIGMETRICS Conf. on Measurement and Modeling of Computer Systems, pages 147–153, Austin, TX, USA.
Plateau, B. and Atif, K. (1991). Stochastic Automata Network for modeling parallel systems. IEEE Trans. Softw. Eng., 17 (10): 1093–1108.
Plateau, B., Fourneau, J.-M., and Lee, K. H. (1988). PEPS: a package for solving complex Markov models of parallel systems. In Puigjaner, R., editor, Proc. 4th Int. Conf. Modelling Techniques and Tools, pages 341–360.
Qureshi, M. A., Sanders, W. H., van Morsel, A. P. A., and German, R. (1995). Algorithms for the generation of state-level representations of stochastic activity networks with general reward structures. In Proc. 6th Int. Workshop on Petri Nets and Performance Models (PNPM’95), pages 180–190, Durham, NC. IEEE Comp. Soc. Press.
Sanders, W. H. (1988). Construction and solution of performability models based on Stochastic Activity Networks. PhD thesis, Department of Computer Science and Engineering, University of Michigan, Ann Arbor, MI.
Sanders, W. H. and Meyer, J. F. (1991). Reduced base model construction methods for stochastic activity networks. IEEE J. Sel. Areas in Comm., 9 (1): 25–36.
Stewart, W. J. (1991). MARCA: Markov Chain Analyser, a software package for Markov modelling In Stewart, W. J., editor, Numerical Solution of Markov Chains, pages 37–61. Marcel Dekker, Inc., New York, NY.
Stewart, W. J., Atif, K., and Plateau, B. (1995). The numerical solution of stochastic automata networks. Europ. J. of Oper. Res., 86: 503–525.
Van Dijk, N. M. (1991). Truncation of Markov chains with applications to queueing. Operations Research, 39 (6): 1018–1026.
Wallace, V. L. and Rosenberg, R. (1994). RQA1, The recursive queue analyser. Technical Report 2, Systems Engineering Laboratory, University of Michigan, Ann Arbor.
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Ciardo, G. (2000). Tools for Formulating Markov Models. In: Grassmann, W.K. (eds) Computational Probability. International Series in Operations Research & Management Science, vol 24. Springer, Boston, MA. https://doi.org/10.1007/978-1-4757-4828-4_2
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