In this paper we present a first approach to the definition of conformance testing relations for systems presenting stochastic timed behavior. By stochastic time we mean that the probability of performing an event may vary according to the elapsed time. In particular, we will consider delays specified by means of random variables.

In order to define our formal model, we will provide a stochastic extension of the notion of finite state machine. We will give a first implementation relation and we will discuss its practical drawbacks. That is, we will show that this relation cannot be appropriately checked under a black/grey-box testing methodology. We will also present other alternative implementation relations that can be checked up to a certain degree of confidence. We will define test cases and how they are applied to implementations. Finally, we will give a test generation algorithm providing complete, up to a degree of confidence, test suites.


Conformance testing test theory performance testing 


  1. 1.
    Ajmone Marsan, M., Bianco, A., Ciminiera, L., Sisto, R., Valenzano, A.: A LOTOS extension for the performance analysis of distributed systems. IEEE/ACM Transactions on Networking 2(2), 151–165 (1994)CrossRefGoogle Scholar
  2. 2.
    Ajmone Marsan, M., Conte, G., Balbo, G.: A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems. ACM Transactions on Computer Systems 5(2), 93–122 (1984)CrossRefGoogle Scholar
  3. 3.
    Alur, R., Dill, D.: A theory of timed automata. Theoretical Computer Science 126, 183–235 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bernardo, M., Cleaveland, W.R.: A theory of testing for markovian processes. In: Palamidessi, C. (ed.) CONCUR 2000. LNCS, vol. 1877, pp. 305–319. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  5. 5.
    Bernardo, M., Gorrieri, R.: A tutorial on EMPA: A theory of concurrent processes with nondeterminism, priorities, probabilities and time. Theoretical Computer Science 202, 1–54 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bosik, B.S., Uyar, M.U.: Finite state machine based formal methods in protocol conformance testing. Computer Networks & ISDN Systems 22, 7–33 (1991)CrossRefGoogle Scholar
  7. 7.
    Bravetti, M., Gorrieri, R.: The theory of interactive generalized semi-Markov processes. Theoretical Computer Science 282(1), 5–32 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Clarke, D., Lee, I.: Automatic generation of tests for timing constraints from requirements. In: 3rd Workshop on Object-Oriented Real-Time Dependable Systems (1997)Google Scholar
  9. 9.
    de Nicola, R., Hennessy, M.C.B.: Testing equivalences for processes. Theoretical Computer Science 34, 83–133 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    En-Nouaary, A., Dssouli, R., Khendek, F.: Timed Wp-method: Testing real time systems. IEEE Transactions on Software Engineering 28(11), 1024–1039 (2002)CrossRefGoogle Scholar
  11. 11.
    Götz, N., Herzog, U., Rettelbach, M.: Multiprocessor and distributed system design: The integration of functional specification and performance analysis using stochastic process algebras. In: Donatiello, L., Nelson, R. (eds.) SIGMETRICS 1993 and Performance 1993. LNCS, vol. 729, pp. 121–146. Springer, Heidelberg (1993)CrossRefGoogle Scholar
  12. 12.
    Harrison, P.G., Strulo, B.: SPADES – a process algebra for discrete event simulation. Journal of Logic Computation 10(1), 3–42 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hennessy, M.: Algebraic Theory of Processes. MIT Press, Cambridge (1988)zbMATHGoogle Scholar
  14. 14.
    Hermanns, H., Herzog, U., Katoen, J.-P.: Process algebra for performance evaluation. Theoretical Computer Science 274(1-2), 43–87 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Higashino, T., Nakata, A., Taniguchi, K., Cavalli, A.: Generating test cases for a timed I/O automaton model. In: 12th Workshop on Testing of Communicating Systems, pp. 197–214. Kluwer Academic Publishers, Dordrecht (1999)CrossRefGoogle Scholar
  16. 16.
    Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge University Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  17. 17.
    Lai, R.: A survey of communication protocol testing. Journal of Systems and Software 62, 21–46 (2002)CrossRefGoogle Scholar
  18. 18.
    Lee, D., Yannakakis, M.: Principles and methods of testing finite state machines: A survey. Proceedings of the IEEE 84(8), 1090–1123 (1996)CrossRefGoogle Scholar
  19. 19.
    López, N., Núñez, M.: A testing theory for generally distributed stochastic processes. In: Larsen, K.G., Nielsen, M. (eds.) CONCUR 2001. LNCS, vol. 2154, pp. 321–335. Springer, Heidelberg (2001)CrossRefGoogle Scholar
  20. 20.
    López, N., Núñez, M., Rubio, F.: Stochastic process algebras meet Eden. In: Butler, M., Petre, L., Sere, K. (eds.) IFM 2002. LNCS, vol. 2335, pp. 29–48. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  21. 21.
    Mandrioli, D., Morasca, S., Morzenti, A.: Generating test cases for real time systems from logic specifications. ACM Transactions on Computer Systems 13(4), 356–398 (1995)CrossRefGoogle Scholar
  22. 22.
    M. Núñez and I. Rodríguez. Encoding PAMR into (timed) EFSMs. In FORTE 2002, LNCS 2529, pages 1–16. Springer, 2002. Google Scholar
  23. 23.
    Springintveld, J., Vaandrager, F., D’Argenio, P.R.: Testing timed automata. Theoretical Computer Science 254(1-2), 225–257 (2001)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© IFIP International Federation for Information Processing 2003

Authors and Affiliations

  • Manuel Núñez
    • 1
  • Ismael Rodríguez
    • 1
  1. 1.Dept. Sistemas Informáticos y Programación, Facultad de InformáticaUniversidad Complutense de MadridMadridSpain

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