Acta Informatica

, Volume 54, Issue 5, pp 447–485 | Cite as

On the relations between Markov chain lumpability and reversibility

  • A. MarinEmail author
  • S. Rossi
Original Article


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.


  1. 1.
    Baarir, S., Beccuti, M., Dutheillet, C., Franceschinis, G., Haddad, S.: Lumping partially symmetrical stochastic models. Perform. Eval. 68(1), 21–44 (2011)CrossRefGoogle Scholar
  2. 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)CrossRefzbMATHGoogle Scholar
  3. 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)Google Scholar
  4. 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)CrossRefGoogle Scholar
  5. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buchholz, P.: Exact and ordinary lumpability in finite Markov chains. J. Appl. Probab. 31, 59–75 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Dovier, A., Piazza, C., Policriti, A.: An efficient algorithm for computing bisimulation equivalence. Theoret. Comput. Sci. 311(1–3), 221–256 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gates, D.J., Westcott, M.: Kinetics of polymer crystallization. Discrete and continuum models. Proc. R. Soc. Lond. 416, 443–461 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gelenbe, E.: Random neural networks with negative and positive signals and product form solution. Neural Comput. 1(4), 502–510 (1989)CrossRefGoogle Scholar
  10. 10.
    Gelenbe, E.: Product form networks with negative and positive customers. J. Appl. Prob. 28(3), 656–663 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gelenbe, E., Mitrani, I.: Analysis and Synthesis of Computer Systems, 2nd edn. Imperial College Press, London (2010)CrossRefzbMATHGoogle Scholar
  12. 12.
    Gelenbe, E., Schassberger, M.: Stability of product form G-networks. Prob. Eng. Inf. Sci. 6, 271–276 (1992)CrossRefzbMATHGoogle Scholar
  13. 13.
    Gilmore, S., Hillston, J., Ribaudo, M.: An efficient algorithm for aggregating PEPA models. IEEE Trans. Softw. Eng. 27(5), 449–464 (2001)CrossRefGoogle Scholar
  14. 14.
    Harrison, P.G.: Turning back time in Markovian process algebra. Theoret. Comput. Sci. 290(3), 1947–1986 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Harrison, P.G., Marin, A.: Product-forms in multi-way synchronizations. Comput. J. 57(11), 1693–1710 (2014)CrossRefGoogle Scholar
  16. 16.
    Hermanns, H.: Interactive Markov Chains and the Quest for Quantified Quality, LNCS, vol. 2428. Springer, Berlin (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Hillston, J.: A Compositional Approach to Performance Modelling. Cambridge Press, Cambridge (1996)CrossRefzbMATHGoogle Scholar
  18. 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)Google Scholar
  19. 19.
    Kelly, F.: Reversibility and Stochastic Networks. Wiley, New York (1979)zbMATHGoogle Scholar
  20. 20.
    Kemeny, J.G., Snell, J.L.: Finite Markov Chains. Springer, New York (1976)zbMATHGoogle Scholar
  21. 21.
    King, W.F.: Analysis of paging algorithms. In: Proceedings of IFIP Congress (1971)Google Scholar
  22. 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. 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)Google Scholar
  24. 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)Google Scholar
  25. 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)Google Scholar
  26. 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)Google Scholar
  27. 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)Google Scholar
  28. 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)CrossRefGoogle Scholar
  29. 29.
    Milner, R.: Communication and Concurrency. Prentice-Hall, Englewood Cliffs (1989)zbMATHGoogle Scholar
  30. 30.
    Neuts, M.F.: Structured Stochastic Matrices of M/G/1 Type and Their Application. Marcel Dekker, New York (1989)zbMATHGoogle Scholar
  31. 31.
    Paige, R., Tarjan, R.E.: Three partition refinement algorithms. SIAM J. Comput. 16(6), 973–989 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 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)Google Scholar
  33. 33.
    Sereno, M.: Towards a product form solution for stochastic process algebras. Comput. J. 38(7), 622–632 (1995)CrossRefGoogle Scholar
  34. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 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)Google Scholar
  36. 36.
    Whittle, P.: Systems in stochastic equilibrium. Wiley, New York (1986)zbMATHGoogle Scholar
  37. 37.
    Yap, V.: Similar states in continuous-time Markov chains. J. Appl. Probab. 46, 497–506 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Università Ca’ Foscari VeneziaVeneziaItaly

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