Hostname: page-component-76fb5796d-vvkck Total loading time: 0 Render date: 2024-04-25T14:33:09.363Z Has data issue: false hasContentIssue false
  • English
  • Français

Exact and ordinary lumpability in finite Markov chains

Part of: Markov processes Computer system organization

Published online by Cambridge University Press:  14 July 2016

Peter Buchholz
Peter Buchholz*
Affiliation:
Universität Dortmund
*
∗ Postal address: Universität Dortmund, Informatik IV, D-44221 Dortmund, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

Exact and ordinary lumpability in finite Markov chains is considered. Both concepts naturally define an aggregation of the Markov chain yielding an aggregated chain that allows the exact determination of several stationary and transient results for the original chain. We show which quantities can be determined without an error from the aggregated process and describe methods to calculate bounds on the remaining results. Furthermore, the concept of lumpability is extended to near lumpability yielding approximative aggregation.

Keywords

AGGREGATIONSTATIONARY AND TRANSIENT ANALYSISPERFORMANCE AND RELIABILITY MODELLING

MSC classification

Primary: 60J10: Markov chains (discrete-time Markov processes on discrete state spaces)
Secondary: 68M20: Performance evaluation; queueing; scheduling
Type
Research Papers
Information
Journal of Applied Probability , Volume 31 , Issue 1 , March 1994 , pp. 59 - 75
Copyright
Copyright © Applied Probability Trust 1994 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aupperle, B. E. and Meyer, J. F. (1991) State space generation for degradable multiprocessor systems. Proc. FCTS 21, Montreal, Canada.Google Scholar
[2] Buchholz, P. (1990) The aggregation of Markovian submodels in isolation. Universität Dortmund, Fachbereich Informatik, Forschungsbericht No. 369.Google Scholar
[3] Buchholz, P. (1992) Hierarchical Markovian models – symmetries and aggregation. Proc. 6th Internat. Conf. on Modelling Tools and Techniques for Performance Evaluation. Edinburgh University Press.Google Scholar
[4] Chiola, G. and Franceschinis, G. (1989) Colored GSPN models and automatic symmetry detection. Proc. 3rd Internat. Workshop on Petri Nets and Performance Models. IEEE Press, New York.Google Scholar
[5] Chiola, G., Dutheillet, C., Franceschinis, G. and Haddad, S. (1991) Stochastic wellformed colored nets and multiprocessor modelling application. In High-Level Petri Nets, ed. Jensen, K. and Rosenberg, G. Springer-Verlag, Berlin.Google Scholar
[6] Courtois, P. J. (1982) Error minimization in decomposable stochastic models. In Applied Probability–Computer Science: The Interface, Vol. 1. Birkhäuser, Boston.Google Scholar
[7] Courtois, P. J. and Semal, P. (1984) Bounds for the positive eigenvectors of nonnegative matrices and their approximation by decomposition. J. Assoc. Comput. Mach. 31, 804–825.Google Scholar
[8] Kemeny, J. G. and Snell, J. L. (1976) Finite Markov Chains. Springer-Verlag, New York.Google Scholar
[9] Nicola, V. F. (1989) Lumping in Markov reward processes. IBM Research Report RC 14719.Google Scholar
[10] Plateau, B. D. and Tripathi, S. K. (1988) Performance analysis of synchronization for two communicating processes. Performance Evaluation 8, 305–320.CrossRefGoogle Scholar
[11] Rogers, L. C. G. and Pitman, J. W. (1981) Markov functions. Ann. Prob. 9, 573–582.Google Scholar
[12] Rubino, G. and Sericola, B. (1989) On weak lumpability in Markov chains. J. Appl. Prob. 26, 446–457.Google Scholar
[13] Rubino, G. and Sericola, B. (1989) Sojourn times in finite Markov processes. J. Appl. Prob. 26, 744–756.Google Scholar
[14] Sanders, W. H. and Meyer, J. F. (1991) Reduced base models construction methods for stochastic activity networks. IEEE J. Sel. Areas Commun. 9, 25–36.Google Scholar
[15] Schweitzer, P. (1984) Aggregation methods for large Markov chains. In Mathematical Computer Performance and Reliability, ed. Iazeolla, G. et al. Elsevier-North-Holland, Amsterdam.Google Scholar
[16] Sumita, U. and Rieders, M. (1989) Lumpability and time reversibility in the aggregation-disaggregation method for large Markov chains. Commun. Statist. – Stoch. Models 5, 63–81.Google Scholar