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Advances in Model Representations

  • Markus Siegle
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2165)

Abstract

We review high-level specification formalisms for Markovian performability models, thereby emphasising the role of structuring concepts as realised par excellence by stochastic process algebras. Symbolic representations based on decision diagrams are presented, and it is shown that they quite ideally support compositional model construction and analysis.

Keywords

Transition System Symbolic Representation Parallel Composition Process Algebra Binary Decision Diagram 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    M. Ajmone Marsan, G. Balbo, and G. Conte. A Class of Generalized Stochastic Petri Nets for the Performance Evaluation of Multiprocessor Systems. ACM Transactions on Computer Systems, 2(2):93–122, May 1984.CrossRefGoogle Scholar
  2. 2.
    M. Ajmone Marsan, G. Balbo, G. Conte, S. Donatelli, and G. Franceschinis. Modelling with generalized stochastic Petri nets. Wiley, 1995.Google Scholar
  3. 3.
    F. Baccelli, W.A. Massey, and D. Towsley. Acyclic Fork-Join Queuing Networks. Journal of the ACM, 36(3):615–642, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    R.I. Bahar, E.A. Frohm, C.M. Gaona, G.D. Hachtel, E. Macii, A. Pardo, and F. Somenzi. Algebraic Decision Diagrams and their Applications. Formal Methods in System Design, 10(2/3):171–206, April/May 1997.CrossRefGoogle Scholar
  5. 5.
    F. Baskett, K.M. Chandy, R.R. Muntz, and F.G. Palacios. Open, Closed and Mixed Networks of Queues with Different Classes of Customers. Journal of the ACM, 22(2):248–260, 1975.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    M. Bernardo and R. Gorrieri. A Tutorial on EMPA: A Theory of Concurrent Processes with Nondeterminism, Priorities, Probabilities and Time. Theoretical Computer Science, 202:1–54, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    H. Bohnenkamp and B. Haverkort. Semi-Numerical Solution of Stochastic Process Algebra Models. In J.-P. Katoen, editor, ARTS’99, pages 228–243. Springer, LNCS 1601, 1999.Google Scholar
  8. 8.
    G. Bolch and S. Greiner. Modeling and Performance Evaluation of Production Lines Using the Modeling Language MOSEL. In Proc. of the 2nd IEEE/ECLA/IFIP Int. Conf. on Architectures and Design Methods for Balanced Automation Systems, pages 163–174, June 1996.Google Scholar
  9. 9.
    A. Bouali and R. de Simone. Symbolic Bisimulation Minimisation. In Computer Aided Verification, pages 96–108, 1992. LNCS 663.Google Scholar
  10. 10.
    M. Bozga and O. Maler. On the Representation of Probabilities over Structured Domains. In N. Halbwachs and D. Peled, editors, Int. Conf. on Computer-Aided Verification (CAV’99), pages 261–273. Springer, LNCS 1633, July 1999.CrossRefGoogle Scholar
  11. 11.
    K.S. Brace, R.L. Rudell, and R.E. Bryant. Efficient Implementation of a BDD Package. In 27th ACM/IEEE Design Automation Conf., pages 40–45, 1990.Google Scholar
  12. 12.
    R.E. Bryant. Graph-based Algorithms for Boolean Function Manipulation. IEEE Transactions on Computers, C-35(8):677–691, August 1986.CrossRefGoogle Scholar
  13. 13.
    P. Buchholz. Die strukturierte Analyse Markovscher Modelle. PhD thesis, Universität Dortmund, 1991 (in German).Google Scholar
  14. 14.
    P. Buchholz. A Hierarchical View of GCSPNs and Its Impact on Qualitative and Quantitative Analysis. Journal of Parallel and Distributed Computing, 15(3):207–224, July 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    P. Buchholz. Aggregation and Reduction Techniques for Hierarchical GCSPNs. In Proc. of PNPM’ 93, pages 216–225, Tolouse, October 1993.Google Scholar
  16. 16.
    P. Buchholz. Hierarchies in Colored GSPNs. In M. Ajmone Marsan, editor, 14th Int. Conf. on Application and Theory of Petri Nets, pages 106–125. Springer, LNCS 691, 1993.Google Scholar
  17. 17.
    P. Buchholz. A class of hierarchical queueing networks and their analysis. Queueing Systems, 15: 59–80, 1994.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    P. Buchholz. Markovian Process Algebra: Composition and Equivalence. In U. Herzog and M. Rettelbach, editors, Proc. of the 2nd Workshop on Process Algebras and Performance Modelling, pages 11–30. Arbeitsberichte des IMMD No. 27/4, Universität Erlangen-Nürnberg, July 1994.Google Scholar
  19. 19.
    P. Buchholz. A Framework for the Hierarchical Analysis of Discrete Event DynamicSystems. Habilitation thesis, Universität Dortmund, 1996.Google Scholar
  20. 20.
    P. Buchholz, G. Ciardo, S. Donatelli, and P. Kemper. Complexity of memory-efficient Kronecker operations with applications to the solution of Markov models. INFORMS Journal of Computing, 12(3):203–222, Summer 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    P. Buchholz and P. Kemper. On Generating a Hierarchy for GSPN Analysis. Performance Evaluation Review (ACM Sigmetrics), 26(2):5–14, August 1998.CrossRefGoogle Scholar
  22. 22.
    P. Buchholz and P. Kemper. A Toolbox for the Analysis of Discrete Event Dynamic Systems. In N. Halbwachs and D. Peled, editors, Computer Aided Verification, pages 483–486. Springer, LNCS 1633, 1999.CrossRefGoogle Scholar
  23. 23.
    J.R. Burch, E.M. Clarke, K.L. McMillan, D.L. Dill, and L.J. Hwang. Symbolic Model Checking: 1020 States and Beyond. Information and Computation, (98):142–170, 1992.zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    J.P. Buzen. Computational Algorithms for Closed Queueing Networks with Exponential Servers. Communications of the ACM, 16:527–531, 1973.zbMATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    W.L. Cao and W.J. Stewart. Iterative Aggregation/Disaggregation Techniques for Nearly Uncoupled Markov Chains. Journal of the ACM, 32(3):702–719, July 1985.zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    G. Chiola, C. Dutheillet, G. Franceschinis, and S. Haddad. On Well-Formed Coloured Nets and their Symbolic Reachability Graph. In Proc. of the 11th Int. Conf. on Application and Theory of Petri Nets, pages 387–410, Paris, June 1990. Reprinted in High-level Petri Nets, K. Jensen, G. Rozenberg, eds., Springer 1991.Google Scholar
  27. 27.
    G. Ciardo. Analysis of Large Stochastic Petri Net Models. PhD thesis, Duke University, Durham, NC, USA, 1989.Google Scholar
  28. 28.
    G. Ciardo and A.S. Miner. A data structure for the e.cient Kronecker solution of GSPNs. In P. Buchholz and M. Silva, editors, PNPM’99, pages 22–31. IEEE computer society, 1999.Google Scholar
  29. 29.
    G. Ciardo and M. Tilgner. Parametric State Space Structuring. Technical Report ICASE Report No. 97-67, ICASE, 1997.Google Scholar
  30. 30.
    G. Ciardo and K.S. Trivedi. A decomposition approach for stochastic reward net models. Performance Evaluation, 18(1):37–59, July 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  31. 31.
    E.M. Clarke, K.L. McMillan, X. Zhao, M. Fujita, and J. Yang. Spectral Transforms for Large Boolean Functions with Applications to Technology Mapping. In 30th Design Automation Conf., pages 54–60. ACM/IEEE, 1993.Google Scholar
  32. 32.
    A.E. Conway and N.D. Georganas. Queueing Networks — Exact Computational Algorithms. MIT Press, 1989.Google Scholar
  33. 33.
    P.J. Courtois. Decomposability, queueing and computer system applications. ACM monograph series, 1977.Google Scholar
  34. 34.
    P.J. Courtois. On Time and Space Decomposition of Complex Structures. Communications of the ACM, 28(6):590–603, June 1985.CrossRefGoogle Scholar
  35. 35.
    J. Couvillion, R. Freire, R. Johnson, W.D. Obal II, M.A. Qureshi, M. Rai, W.H. Sanders, and J. Tvedt. Performability modeling with UltraSAN. IEEE Software, 8(5):69–80, September 1991.CrossRefGoogle Scholar
  36. 36.
    I. Davies. Symbolic techniques for the performance analysis of generalised stochastic Petri nets. Master’s thesis, University of Cape Town, Department of Computer Science, January 2001.Google Scholar
  37. 37.
    M. Davio. Kronecker Products and Shuffle Algebra. IEEE Transactions on Computers, C-30(2): 116–125, February 1981.MathSciNetGoogle Scholar
  38. 38.
    L. de Alfaro, M. Kwiatkowska, G. Norman, D. Parker, and R. Segala. Symbolic Model Checking for Probabilistic Processes using MTBDDs and the Kronecker Representation. In S. Graf and M. Schwartzbach, editors, TACAS’2000, pages 395–410, Berlin, 2000. Springer, LNCS 1785.Google Scholar
  39. 39.
    S. Donatelli. Superposed stochastic automata: a class of stochastic Petri nets with parallel solution and distributed state space. Performance Evaluation, 18(1):21–36, July 1993.zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    S. Donatelli. Superposed Generalized Stochastic Petri Nets: Definition and Efficient Solution. In M. Silva, editor, 15th Int. Conf. on Application and Theory of Petri Nets, Zaragoza, June 1994.Google Scholar
  41. 41.
    R. Enders, T. Filkorn, and D. Taubner. Generating BDDs for symbolicmo del checking in CCS. Distributed Computing, (6):155–164, 1993.Google Scholar
  42. 42.
    P. Fernandes, B. Plateau, and W.J. Stewart. Efficient Descriptor-Vector Multiplications in Stochastic Automata Networks. Journal of the ACM, 45(3):381–414, May 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    G. Franceschinis and M. Ribaudo. Efficient Performance Analysis Techniques for Stochastic Well-Formed Nets and Stochastic Process Algebras. In W. Reisig and G. Rozenberg, editors, Lectures on Petri Nets II: Applications, pages 386–437. Springer, LNCS 1492, 1998.Google Scholar
  44. 44.
    E. Frank. Codierung und numerische Analyse von Transitionssystemen unter Verwendung von MTBDDs. Student’s thesis, Universität Erlangen-Nürnberg, October 1999 (in German).Google Scholar
  45. 45.
    E. Frank. Erweiterung eines MTBDD-basierten Werkzeugs für die Analyse stochastischer Transitionssysteme. Technical Report Inf 7, 01/00, Universität Erlangen-Nürnberg, January 2000 (in German).Google Scholar
  46. 46.
    M. Fujita, P. McGeer, and J.C.-Y. Yang. Multi-terminal Binary Decision Diagrams: An efficient data structure for matrix representation. Formal Methods in System Design, 10(2/3):149–169, April/May 1997.CrossRefGoogle Scholar
  47. 47.
    R. German. Performance Analysis of Communication Systems — Modelling with Non-Markovian Stochastic Petri Nets. Wiley, 2000.Google Scholar
  48. 48.
    N. Götz, U. Herzog, and M. Rettelbach. Multiprocessor and Distributed System Design: The Integration of Functional Specification and Performance Analysis Using Stochastic Process Algebras. In Proc. of PERFORMANCE 1993, Tutorial, pages 121–146. Springer LNCS 729, 1993.Google Scholar
  49. 49.
    G.D. Hachtel, E. Macii, A. Pardo, and F. Somenzi. Probabilistic Analysis of Large Finite State Machines. In 31st Design Automation Conf., pages 270–275, San Diego, CA, June 1994. ACM/IEEE.Google Scholar
  50. 50.
    G.D. Hachtel, E. Macii, A. Pardo, and F. Somenzi. Symbolic Algorithms to Calculate Steady-State Probabilities of a Finite State Machine. In European Design Automation Conf., pages 214–218, Paris, February 1994. IEEE.Google Scholar
  51. 51.
    G.D. Hachtel, E. Macii, A. Pardo, and F. Somenzi. Markovian Analysis of Large Finite State Machines. IEEE Transactions on CAD, 15(12):1479–1493, Dec. 1996.Google Scholar
  52. 52.
    H. Hermanns. Interactive Markov Chains. PhD thesis, Universität Erlangen-Nürnberg, September 1998. Arbeitsberichte des IMMD No. 32/7.Google Scholar
  53. 53.
    H. Hermanns, U. Herzog, and J.-P. Katoen. Process algebra for performance evaluation. Theoretical Computer Science, 2001. to appear.Google Scholar
  54. 54.
    H. Hermanns, U. Herzog, U. Klehmet, V. Mertsiotakis, and M. Siegle. Compositional performance modelling with the TIPPtool. Performance Evaluation, 39(1–4):5–35, January 2000.zbMATHCrossRefGoogle Scholar
  55. 55.
    H. Hermanns, U. Herzog, and V. Mertsiotakis. Stochastic Process Algebras-Between LOTOS and Markov Chains. Computer Networks and ISDN systems (CNIS), 30(9–10):901–924, 1998.CrossRefGoogle Scholar
  56. 56.
    H. Hermanns, J.-P. Katoen, J. Meyer-Kayser, and M. Siegle. A Markov Chain Model Checker. In S. Graf and M. Schwartzbach, editors, TACAS’2000, pages 347–362, Berlin, 2000. Springer, LNCS 1785.Google Scholar
  57. 57.
    H. Hermanns, J.-P. Katoen, J. Meyer-Kayser, and M. Siegle. Towards model checking stochastic process algebra. In W. Grieskamp, T. Santen, and B. Stoddart, editors, 2nd Int. Conf. on Integrated Formal Methods, pages 420–439, Dagstuhl, November 2000. Springer, LNCS 1945.CrossRefGoogle Scholar
  58. 58.
    H. Hermanns, M. Kwiatkowska, G. Norman, D. Parker, and M. Siegle. On the use of MTBDDs for performability analysis and verification of stochastic systems. (in preparation).Google Scholar
  59. 59.
    H. Hermanns, J. Meyer-Kayser, and M. Siegle. Multi Terminal Binary Decision Diagrams to Represent and Analyse Continuous Time Markov Chains. In B. Plateau, W.J. Stewart, and M. Silva, editors, 3rd Int. Workshop on the Numerical Solution of Markov Chains, pages 188–207. Prensas Universitarias de Zaragoza, 1999.Google Scholar
  60. 60.
    H. Hermanns and M. Rettelbach. Syntax, Semantics, Equivalences, and Axioms for MTIPP. In U. Herzog and M. Rettelbach, editors, Proc. of the 2nd Workshop on Process Algebras and Performance Modelling, pages 71–88. Arbeitsberichte des IMMD No. 27/4, Universität Erlangen-Nürnberg, July 1994.Google Scholar
  61. 61.
    H. Hermanns and M. Siegle. Bisimulation Algorithms for Stochastic Process Algebras and their BDD-based Implementation. In J.-P. Katoen, editor, ARTS’99, 5th Int. AMAST Workshop on Real-Time and Probabilistic Systems, pages 144–264. Springer, LNCS 1601, 1999.Google Scholar
  62. 62.
    U. Herzog. Leistungsbewertung und Modellbildung für Parallelrechner. Informationstechnik (it), 31(1):31–38, 1989. (in German).Google Scholar
  63. 63.
    U. Herzog. Performance Evaluation and Formal Description. In V.A. Monaco and R. Negrini, editors, Advanced Computer Technology, Reliable Systems and Applications, Proceedings, pages 750–756. IEEE Comp. Soc. Press, 1991.Google Scholar
  64. 64.
    J. Hillston. A Compositional Approach to Performance Modelling. Cambridge University Press, 1996.Google Scholar
  65. 65.
    J. Hillston and V. Mertsiotakis. A Simple Time Scale Decomposition Technique for Stochastic Process Algebras. The Computer Journal, 38(7):566–577, December 1995. Special issue: Proc. of the 3rd Workshop on Process Algebras and Performance Modelling.CrossRefGoogle Scholar
  66. 66.
    O.C. Ibe and K.S. Trivedi. Stochastic Petri Net Models of Polling Systems. IEEE Journal on Selected Areas in Communications, 8(9):1649–1657, December 1990.CrossRefGoogle Scholar
  67. 67.
    P. Kemper. Reachability analysis based on structured representation. In J. Billington and W. Reisig, editors, Application and Theory of Petri Nets, pages 269–288. Springer, LNCS 1091, 1999.Google Scholar
  68. 68.
    C. Kim and A.K. Agrawala. Analysis of the Fork-Join Queue. IEEE Transactions on Computers, 38(2): 250–255, 1989.zbMATHCrossRefMathSciNetGoogle Scholar
  69. 69.
    W. Kleinöder. Stochastische Bewertung von Aufgabenstrukturen fur hierarchische Mehrrechnersysteme. PhD thesis, Universität Erlangen-Nürnberg, Arbeitsberichte des IMMD No. 15/10, August 1982 (in German).Google Scholar
  70. 70.
    W. Knottenbelt. Generalized Markovian Analysis of Timed Transition Systems. Master’s thesis, University of Cape Town, June 1996.Google Scholar
  71. 71.
    M. Kwiatkowska, G. Norman, and R. Segala. Automated Verification of a Randomised Distributed Consensus Protocol Using Cadence SMV and PRISM. Technical Report CSR-01-1, School of Computer Science, University of Birmingham, January 2001.Google Scholar
  72. 72.
    V. Mertsiotakis. Time Scale Decomposition of Stochastic Process Algebra Models. In E. Brinksma and A. Nymeyer, editors, Proc. of 5th Workshop on Process Algebras and Performance Modelling. CTIT Technical Report series, No. 97–14, University of Twente, June 1997.Google Scholar
  73. 73.
    V. Mertsiotakis. Approximate Analysis Methods for Stochastic Process Algebras. PhD thesis, Universität Erlangen-Nürnberg, 1998.Google Scholar
  74. 74.
    V. Mertsiotakis and M. Silva. Throughput Approximation of Decision Free Processes Using Decomposition. In Proc. of the 7th Int. Workshop on Petri Nets and Performance Models, pages 174–182, St. Malo, June 1997. IEEE CS-Press.Google Scholar
  75. 75.
    M. Meyer. Entwurf eines spezialisierten Coprozessors für die Manipulation von Binären Entscheidungsdiagrammen. Student’s thesis, Universität Erlangen-Nürnberg, January 2001 (in German).Google Scholar
  76. 76.
    A. Miner. Efficient solution of GSPNs using matrix diagrams. In Petri Nets and Performance models (PNPM). IEEE Computer Society Press, 2001. (to appear).Google Scholar
  77. 77.
    A. Miner and G. Ciardo. Efficient reachability set generation and storage using decision diagrams. In H. Kleijn and S. Donatelli, editors, Application and Theory of Petri Nets 1999, pages 6–25. Springer, LNCS 1639, 1999.CrossRefGoogle Scholar
  78. 78.
    A.S. Miner, G. Ciardo, and S. Donatelli. Using the exact state space of a Markov model to compute approximate stationary measures. Performance Evaluation Review, 28(1):207–216, June 2000. Proc. of ACM SIGMETRICS 2000.CrossRefGoogle Scholar
  79. 79.
    M.K. Molloy. Performance Analysis Using Stochastic Petri Nets. IEEE Transactions on Computers, C-31: 913–917, September 1982.CrossRefGoogle Scholar
  80. 80.
    R. Nelson and A.N. Tantawi. Approximate Analysis of Fork/Join Synchronization in Parallel Queues. IEEE Transactions on Computers, 37(6):739–743, 1988.CrossRefGoogle Scholar
  81. 81.
    D. Parker. Implementation of symbolic model checking for probabilistic systems. PhD thesis, School of Computer Science, University of Birmingham, 2001. (to appear).Google Scholar
  82. 82.
    B. Plateau. On the Synchronization Structure of Parallelism and Synchronization Models for Distributed Algorithms. In Proc. of ACM SIGMETRICS, pages 147–154, Austin, TX, August 1985.Google Scholar
  83. 83.
    B. Plateau and K. Atif. Stochastic Automata Network for Modeling Parallel Systems. IEEE Transactions on Software Engineering, 17(10):1093–1108, 1991.CrossRefMathSciNetGoogle Scholar
  84. 84.
    B. Plateau and J.-M. Fourneau. A Methodology for Solving Markov Models of Parallel Systems. Journal of Parallel and Distributed Computing, 12:370–387, 1991.CrossRefGoogle Scholar
  85. 85.
    B. Plateau, J.-M. Fourneau, and K.-H. Lee. PEPS: A Package for Solving Complex Markov Models of Parallel Systems. In Proc. of the 4th Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, pages 341–360, Palma (Mallorca), September 1988.Google Scholar
  86. 86.
    M. Reiser and S. Lavenberg. Mean Value Analysis of Closed Multichain Queueing Networks. Journal of the ACM, 27(2):313–322, 1980.zbMATHCrossRefMathSciNetGoogle Scholar
  87. 87.
    J.A. Rolia and K.C. Sevcik. The Method of Layers. IEEE Transactions on Software Engineering, 21(8):689–700, August 1995.CrossRefGoogle Scholar
  88. 88.
    W.H. Sanders and J.F. Meyer. Reduced Base Model Construction Methods for Stochastic Activity Networks. IEEE Journal on Selected Areas in Communications, 9(1):25–36, January 1991.CrossRefGoogle Scholar
  89. 89.
    C. Sauer and E. McNair. The Evolution of the Research Queueing Package RESQ. In Proc. of the First Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, Paris, May 1984.Google Scholar
  90. 90.
    M. Sczittnick. Techniken zur funktionalen und quantitativen Analyse von Markoffschen Rechensystemmodellen. Master’s thesis, Universitat Dortmund, August 1987 (in German).Google Scholar
  91. 91.
    M. Siegle. Using Structured Modelling for Efficient Performance Prediction of Parallel Systems. In G.R. Joubert, D. Trystram, F.J. Peters, and D.J. Evans, editors, Parallel Computing: Trends and Applications, Proc. of the Int. Conf. ParCo93, pages 453–460. North-Holland, 1994.Google Scholar
  92. 92.
    M. Siegle. Beschreibung und Analyse von Markovmodellen mit großem Zustandsraum. PhD thesis, Universität Erlangen-Nürnberg, 1995 (in German).Google Scholar
  93. 93.
    M. Siegle. Compositional Representation and Reduction of Stochasitic Labelled Transition Systems based on Decision Node BDDs. In D. Baum, N. Müller, and R. Rödler, editors, MMB’99, pages 173–185, Trier, September 1999. VDE Verlag.Google Scholar
  94. 94.
    M. Siegle. Behaviour analysis of communication systems: Stochastic modelling and analysis. Habilitation thesis, University of Erlangen-Nürnberg, 2001 (to appear).Google Scholar
  95. 95.
    H.A. Simon and A. Ando. Aggregation of Variables in Dynamic Systems. Econometrica, 29:111–138, 1961.zbMATHCrossRefGoogle Scholar
  96. 96.
    F. Somenzi. CUDD: Colorado University Decision Diagram Package, Release 2.3.0. User’s Manual and Programmer’s Manual, September 1998.Google Scholar
  97. 97.
    W. Stewart, K. Atif, and B. Plateau. The Numerical Solution of Stochastic Automata Networks. Rapport Apache 6, Institut IMAG, LGI, LMC, Grenoble, November 1993.Google Scholar
  98. 98.
    W.J. Stewart. MARCA: Markov Chain Analyzer, A Software Package for Markov Modeling. In W.J. Stewart, editor, Numerical Solution of Markov Chains. Marcel Dekker, 1991.Google Scholar
  99. 99.
    W.J. Stewart. Introduction to the numerical solution of Markov chains. Princeton University Press, 1994.Google Scholar
  100. 100.
    M. Veran and D. Potier. QNAP2: A Portable Environment for Queueing Systems Modelling. In Proc. of the First Int. Conf. on Modelling Techniques and Tools for Computer Performance Evaluation, Paris, May 1984.Google Scholar
  101. 101.
    M. Woodside, J.E. Neilson, D.C. Petriu, and S. Majumdar. The Stochastic Rendezvous Network Model for Performance of Synchronous Client-Server-like Distributed Software. IEEE Transactions on Computers, 44(1): 20–34, January 1995.zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • Markus Siegle
    • 1
  1. 1.Lehrstuhl für Informatik 7University of Erlangen-NürnbergGermany

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