Skip to main content
SpringerLink
Log in

On the relations between Markov chain lumpability and reversibility

  • Original Article
  • Published:
Acta Informatica Aims and scope Submit manuscript

Abstract

In the literature, the notions of lumpability and time reversibility for large Markov chains have been widely used to efficiently study the functional and non-functional properties of computer systems. In this paper we explore the relations among different definitions of lumpability (strong, exact and strict) and the notion of time-reversed Markov chain. Specifically, we prove that an exact lumping induces a strong lumping on the reversed Markov chain and a strict lumping holds both for the forward and the reversed processes. Based on these results we introduce the class of \(\lambda \rho \)-reversible Markov chains which combines the notions of lumping and time reversibility modulo state renaming. We show that the class of autoreversible processes, previously introduced in Marin and Rossi (Proceedings of the IEEE 21st international symposium on modeling, analysis and simulation of computer and telecommunication systems MASCOTS, pp. 151–160, 2013), is strictly contained in the class of \(\lambda \rho \)-reversible chains.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15

Similar content being viewed by others

References

  1. Baarir, S., Beccuti, M., Dutheillet, C., Franceschinis, G., Haddad, S.: Lumping partially symmetrical stochastic models. Perform. Eval. 68(1), 21–44 (2011)

    Article  Google Scholar 

  2. Baier, C., Haverkort, B., Hermanns, H., Katoen, J.P.: Model-checking algorithms for continuous-time Markov chains. IEEE Trans. Soft. Eng. 29(7), 524–541 (2003)

    Article  MATH  Google Scholar 

  3. Balsamo, S., Harrison, P.G., Marin, A.: A unifying approach to product-forms in networks with finite capacity constraints. In: Misra, V., Barford, P., Squillante, M.S. (eds.) Proceedings of the 2010 ACM SIGMETRICS International Conference on Measurement and Modeling of Computer Systems, pp. 25–36. New York, NY, USA (14–18 June 2010)

  4. Balsamo, S., Harrison, P.G., Marin, A.: Methodological construction of product-form stochastic Petri-nets for performance evaluation. J. Syst. Softw. 85(7), 1520–1539 (2012)

    Article  Google Scholar 

  5. Baskett, F., Chandy, K.M., Muntz, R.R., Palacios, F.G.: Open, closed, and mixed networks of queues with different classes of customers. J. ACM 22(2), 248–260 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  6. Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. J. Appl. Probab. 31, 59–75 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dovier, A., Piazza, C., Policriti, A.: An efficient algorithm for computing bisimulation equivalence. Theoret. Comput. Sci. 311(1–3), 221–256 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gates, D.J., Westcott, M.: Kinetics of polymer crystallization. Discrete and continuum models. Proc. R. Soc. Lond. 416, 443–461 (1988)

    Article  MathSciNet  Google Scholar 

  9. Gelenbe, E.: Random neural networks with negative and positive signals and product form solution. Neural Comput. 1(4), 502–510 (1989)

    Article  Google Scholar 

  10. Gelenbe, E.: Product form networks with negative and positive customers. J. Appl. Prob. 28(3), 656–663 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  11. Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems, 2nd edn. Imperial College Press, London (2010)

    Book  MATH  Google Scholar 

  12. Gelenbe, E., Schassberger, M.: Stability of product form G-networks. Prob. Eng. Inf. Sci. 6, 271–276 (1992)

    Article  MATH  Google Scholar 

  13. Gilmore, S., Hillston, J., Ribaudo, M.: An efficient algorithm for aggregating PEPA models. IEEE Trans. Softw. Eng. 27(5), 449–464 (2001)

    Article  Google Scholar 

  14. Harrison, P.G.: Turning back time in Markovian process algebra. Theoret. Comput. Sci. 290(3), 1947–1986 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  15. Harrison, P.G., Marin, A.: Product-forms in multi-way synchronizations. Comput. J. 57(11), 1693–1710 (2014)

    Article  Google Scholar 

  16. Hermanns, H.: Interactive Markov Chains and the Quest for Quantified Quality, LNCS, vol. 2428. Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  17. Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge Press, Cambridge (1996)

    Book  MATH  Google Scholar 

  18. Hillston, J., Thomas, N.: A syntactical analysis of reversible PEPA models. In: Proceedings of of 6th International Workshop on Process Algebra and Performance Modelling, pp. 37–49 (1998)

  19. Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979)

    MATH  Google Scholar 

  20. Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, New York (1976)

    MATH  Google Scholar 

  21. King, W.F.: Analysis of paging algorithms. In: Proceedings of IFIP Congress (1971)

  22. Lazowska, E.D., Zahorjan, J.L., Graham, G.S., Sevcick, K.C.: Quantitative System Performance: Computer System Analysis Using Queueing Network Models. Prentice Hall, Englewood Cliffs (1984)

    Google Scholar 

  23. Mairesse, J., Nguyen, H.T.: Deficiency zero Petri nets and product form. In: Proceedings of the 30th International Conference on Application and Theory of Petri Nets, PETRI NETS ’09, pp. 103–122. Springer-Verlag, Paris, France (2009)

  24. Marin, A., Rossi, S.: Autoreversibility: exploiting symmetries in Markov chains. In: Proceedings of the IEEE 21st International Symposium on Modeling, Analysis & Simulation of Computer and Telecommunication Systems MASCOTS, pp. 151–160. IEEE Computer Society (2013)

  25. Marin, A., Rossi, S.: On discrete time reversibility modulo state renaming and its applications. In: Proceedings of the 8th International Conference on Performance Evaluation Methodologies and Tools, VALUETOOLS, pp. 1–8 (2014)

  26. Marin, A., Rossi, S.: On the relations between lumpability and reversibility. In: Proc. of the IEEE 22nd International Symposium on Modelling, Analysis & Simulation of Computer and Telecommunication Systems, MASCOTS, pp. 427–432. IEEE Computer Society (2014)

  27. Marin, A., Rossi, S.: Lumping-based equivalences in Markovian automata and applications to product-form analyses. In: Proceedings of the 12th International Conference on Quantitative Evaluation of Systems, QEST, LNCS, vol. 9259, pp. 160–175. Springer (2015)

  28. Marsan, M.A., Conte, G., Balbo, G.: A class of generalized stochastic Petri nets for the performance evaluation of multiprocessor systems. ACM Trans. Comput. Syst. 2(2), 93–122 (1984)

    Article  Google Scholar 

  29. Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)

    MATH  Google Scholar 

  30. Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Application. Marcel Dekker, New York (1989)

    MATH  Google Scholar 

  31. Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  32. Schweitzer, P.: Aggregation methods for large Markov chains. In: Proceedings of the International Workshop on Computer Performance and Reliability, pp. 275–286. North Holland (1984)

  33. Sereno, M.: Towards a product form solution for stochastic process algebras. Comput. J. 38(7), 622–632 (1995)

    Article  Google Scholar 

  34. Sumita, U., Rieders, M.: Lumpability and time reversibility in the aggregation–disaggregation method for large Markov chains. Stoch. Models 5(1), 63–81 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  35. Takahashi, Y.: A lumping method for numerical calculations of statioanry distributions of Markov chains. In: Technical Report B-18, Department of Information Sciences, Tokyo Institute of Technology (1975)

  36. Whittle, P.: Systems in stochastic equilibrium. Wiley, New York (1986)

    MATH  Google Scholar 

  37. Yap, V.: Similar states in continuous-time Markov chains. J. Appl. Probab. 46, 497–506 (2009)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Università Ca’ Foscari Venezia, Venezia, Italy

    A. Marin & S. Rossi

Authors
  1. A. Marin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. S. Rossi
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. Marin.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marin, A., Rossi, S. On the relations between Markov chain lumpability and reversibility. Acta Informatica 54, 447–485 (2017). https://doi.org/10.1007/s00236-016-0266-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00236-016-0266-1

Search

Navigation