Formal Aspects of Computing

, Volume 24, Issue 4–6, pp 497–518 | Cite as

Reconciling real and stochastic time: the need for probabilistic refinement

  • J. Markovski
  • P. R. D’Argenio
  • J. C. M. Baeten
  • E. P. de Vink
Open Access
Original Article


We conservatively extend an ACP-style discrete-time process theory with discrete stochastic delays. The semantics of the timed delays relies on time additivity and time determinism, which are properties that enable us to merge subsequent timed delays and to impose their synchronous expiration. Stochastic delays, however, interact with respect to a so-called race condition that determines the set of delays that expire first, which is guided by an (implicit) probabilistic choice. The race condition precludes the property of time additivity as the merger of stochastic delays alters this probabilistic behavior. To this end, we resolve the race condition using conditionally-distributed unit delays. We give a sound and ground-complete axiomatization of the process theory comprising the standard set of ACP-style operators. In this generalized setting, the alternative composition is no longer associative, so we have to resort to special normal forms that explicitly resolve the underlying race condition. Our treatment succeeds in the initial challenge to conservatively extend standard time with stochastic time. However, the ‘dissection’ of the stochastic delays to conditionally-distributed unit delays comes at a price, as we can no longer relate the resolved race condition to the original stochastic delays. We seek a solution in the field of probabilistic refinements that enable the interchange of probabilistic and nondeterministic choices.


Timed and stochastic process algebras Conservative extensions race condition Generally-distributed stochastic delays 



We thank Sonja Georgievska for insightful comments and discussions on early drafts of this paper.

Open Access

This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.


  1. ABC+94.
    Ajmone Marsan M, Bianco A, Ciminiera L, Sisto R, Valenzano A (1994) A LOTOS extension for the performance analysis of distributed systems. IEEE/ACM Trans Netw 2(2): 151–165CrossRefGoogle Scholar
  2. ABC+95.
    Ajmone Marsan M, Balbo G, Conte G, Donatelli S, Franceschinis G (1995) Modelling with generalized stochastic Petri nets. Wiley, New YorkMATHGoogle Scholar
  3. ABN11.
    Acciai L, Boreale M, De Nicola R (2011) Linear-time and may-testing in a probabilistic reactive setting. In: Proceedings of FMOODS 2011. Lecture Notes in Computer Science, vol 6722. Springer, Berlin, pp 29–43Google Scholar
  4. Bae05.
    Baeten JCM (2005) A brief history of process algebra. Theor Comput Sci 335: 131–146MathSciNetMATHCrossRefGoogle Scholar
  5. BBD03.
    Bryans J, Bowman H, Derrick J (2003) Model checking stochastic automata. ACM Trans Comput Logic 4(4): 452–492MathSciNetCrossRefGoogle Scholar
  6. BBG97.
    Bravetti M, Bernardo M, Gorrieri R (1997) From EMPA to GSMPA: allowing for general distributions. In: Proceedings of PAPM’97, Enschede, pp 17–33Google Scholar
  7. BBK87.
    Baeten JCM, Bergstra JA, Klop JW (1987) On the consistency of Koomen’s fair abstraction rule. Theor Comput Sci 51(1): 129–176MathSciNetMATHCrossRefGoogle Scholar
  8. BBR10.
    Baeten JCM, Basten T, Reniers MA (2010) Process algebra: equational theories of communicating processes. In: Cambridge tracts in theoretical computer science, vol 50. Cambridge University Press, CambridgeGoogle Scholar
  9. BD04.
    Bravetti M, D’Argenio PR (2004) Tutte le algebre insieme: concepts, discussions and relations of stochastic process algebras with general distributions. In: Validation of stochastic systems—a guide to current research. Lecture Notes in Computer Science, vol 2925. Springer, Berlin, pp 44–88Google Scholar
  10. BDHK06.
    Bohnenkamp HC, D’Argenio PR, Hermanns H, Katoen J-P (2006) MODEST: a compositional modeling formalism for hard and softly timed systems. IEEE Trans Softw Eng 32: 812–830CrossRefGoogle Scholar
  11. BG98.
    Bernardo M, Gorrieri R (1998) A tutorial on EMPA: a theory of concurrent processes with nondeterminism, priorities, probabilities and time. Theor Comput Sci 202(1–2): 1–54MathSciNetMATHCrossRefGoogle Scholar
  12. BKLL95.
    Brinksma E, Katoen J-P, Langerak R, Latella D (1995) A stochastic causality-based process algebra. Comput J 38(7): 552–565CrossRefGoogle Scholar
  13. BM02.
    Baeten JCM, Middelburg CA (2002) Process algebra with timing. In: Monographs in theoretical computer science. Springer, BerlinGoogle Scholar
  14. BR04.
    Baeten JCM, Reniers MA (2004) Timed process algebra (with a focus on explicit termination and relative timing). In: Proceedings of SFM 2004. Lecture Notes of Computer Science, vol 3185. Springer, Berlin, pp 59–97Google Scholar
  15. Bra02.
    Bravetti M (2002) Specification and analysis of stochastic real-time systems. PhD thesis, Università à di BolognaGoogle Scholar
  16. BW90.
    Baeten JCM, Weijland WP (1990) Process algebra. Cambridge tracts in theoretical computer science, vol 18. Cambridge University Press, CambridgeGoogle Scholar
  17. CSKN05.
    Cattani S, Segala R, Kwiatkowska M, Norman G (2005) Stochastic transition systems for continuous state spaces and non-determinism. In: Proceedings of FoSSaCS’05. Lecture Notes of Computer Science, vol 3441. Springer, Berlin, pp 125–139Google Scholar
  18. CSV07.
    Cheung L, Stoelinga M, Vaandrager F (2007) A testing scenario for probabilistic processes. J ACM 54Google Scholar
  19. D’A03.
    D’Argenio PR (2003) From stochastic automata to timed automata: abstracting probability in a compositional manner. In: Proceedings of WAIT 2003, Buenos AiresGoogle Scholar
  20. dAHJ01.
    de Alfaro L, Henzinger T, Jhala R (2001) Compositional methods for probabilistic systems. In: Proceedings of CONCUR 2001. Lecture Notes in Computer Science, vol 2154. Springer, Berlin, pp 351–365Google Scholar
  21. DK05a.
    D’Argenio PR, Katoen J-P (2005) A theory of stochastic systems, part I: stochastic automata. Inform Comput 203(1): 1–38MathSciNetMATHCrossRefGoogle Scholar
  22. DK05b.
    D’Argenio PR, Katoen J-P (2005) A theory of stochastic systems, part II: process algebra. Inform Comput 203(1): 39–74MathSciNetMATHCrossRefGoogle Scholar
  23. GA10.
    Georgievska S, Andova S (2010) Retaining the probabilities in probabilistic testing theory. In: Proceedings of FOSSACS 2010. Lecture Notes in Computer Science, vol 6014. Springer, Berlin, pp 79–93Google Scholar
  24. GA12.
    Georgievska S, Andova S (2012) Probabilistic CSP: preserving the laws via restricted schedulers. In: Proceedings of MMB 2012. VDE Verlag (to appear). Available from:
  25. GD09.
    Giro S, D’Argenio PR (2009) On the expressive power of schedulers in distributed probabilistic systems. EPTCS 253: 45–71Google Scholar
  26. Geo11.
    Georgievska S (2011) Probability and hiding in concurrent processes. PhD thesis, Eindhoven University of TechnologyGoogle Scholar
  27. Gly89.
    Glynn PW (1989) A GSMP formalism for discrete event systems. Proc IEEE 77(1): 14–23CrossRefGoogle Scholar
  28. Her02.
    Hermanns H (2002) Interactive Markov chains: the quest for quantified quality. Lecture Notes in Computer Science, vol 2428. Springer, BerlinGoogle Scholar
  29. Hil96.
    Hillston J (1996) A Compositional approach to performance modelling. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  30. HMR94.
    Hermanns H, Mertsiotakis V, Rettelbach M (1994) Performance analysis of distributed systems using TIPP. In: Proceedings of UKPEW’94. University of Edinburgh, pp 131–144Google Scholar
  31. Hoa85.
    Hoare CAR (1985) Communicating sequential processes. Prentice Hall, Englewood CliffsMATHGoogle Scholar
  32. How71.
    Howard RA (1971) Dynamic probabilistic systems. Wiley, New YorkGoogle Scholar
  33. KD01.
    Katoen JP, D’Argenio PR (2001) General distributions in process algebra. In: Lectures on formal methods and performance analysis. Lecture Notes in Computer Science, vol 2090. Springer, Berlin, pp 375–429Google Scholar
  34. KNP02.
    Kwiatkowska M, Norman G, Parker D (2002) PRISM: probabilistic symbolic model checker. In: Proceedings of TOOLS 2002. Lecture Notes in Computer Science, vol 2324. Springer, Berlin, pp 200–204Google Scholar
  35. LN00.
    López N, Núñez M (2000) NMSPA: a non-Markovian model for stochastic processes. In: Proceedings of ICDS 2000. IEEE, pp 33–40Google Scholar
  36. Low93.
    Lowe G (1993) Representing nondeterministic and probabilistic behaviour in reactive processes. Technical report PRG-TR-11-93, Oxford University Computing LabsGoogle Scholar
  37. Mar08.
    Markovski J (2008) Real and stochastic time in processs algebras for performance evaluation. PhD thesis, Eindhoven University of TechnologyGoogle Scholar
  38. MMSS96.
    Morgan C, McIver A, Seidel K, Sanders JW (1996) Refinement-oriented probability for CSP. Formal Aspects Comput 8: 617–647MATHCrossRefGoogle Scholar
  39. MV06.
    Markovski J, de Vink EP (2006) Embedding real-time in stochastic process algebras. In: Proceedings of EPEW 2006. Lecture Notes of Computer Science, vol 4054. Springer, Berlin, pp 47–62Google Scholar
  40. MV07.
    Markovski J, de Vink EP (2007) Real-time process algebra with stochastic delays. In: Proceedings of ACSD 2007. IEEE, pp 177–186Google Scholar
  41. MV08.
    Markovski J, de Vink EP (2008) Extending timed process algebra with discrete stochastic time. In: Proceedings of AMAST 2008. Lecture Notes of Computer Science, vol 5140. Springer, Berlin, pp 268–283Google Scholar
  42. MV09.
    Markovski J, de Vink EP (2009) Performance evaluation of distributed systems based on a discrete real- and stochastic-time process algebra. Fundam Inform 95(1): 157–186MATHGoogle Scholar
  43. NS92.
    Nicollin X, Sifakis J (1992) An overview and synthesis of timed process algebras. In: Real-time: theory in practice. Lecture Notes of Computer Science, vol 600. Springer, Berlin, pp 526–548Google Scholar
  44. Seg95.
    Segala R (1995) Modeling and Verification of randomized distributed real-time systems. Phd thesis, MITGoogle Scholar
  45. Yi91.
    Yi W (1991) CCS + time z = an interleaving model for real-time systems. In: Proceedings of ICALP’91. Lecture Notes of Computer Science, vol 510. Springer, Berlin, pp 217–228Google Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • J. Markovski
    • 1
  • P. R. D’Argenio
    • 2
  • J. C. M. Baeten
    • 1
    • 3
  • E. P. de Vink
    • 1
    • 3
  1. 1.Eindhoven University of TechnologyEindhovenThe Netherlands
  2. 2.FaMAF, Universidad Nacional de CórdobaCórdobaArgentina
  3. 3.Centrum Wiskunde & InformaticaAmsterdamThe Netherlands

Personalised recommendations